Good morning, everyone. Before we start on today's content, two minutes on what is happening with these sessions, because some of you are wondering.

I am taking over the sessions from here. I have gone through everything that has been covered with you so far — all of it — so I know where you have reached. But I was not in the room for those sessions. So if I repeat something you have already done, tell me and I will move on. And if I assume something you have not covered, tell me faster, because that one is worse.

Second thing. There are two batches sitting in this room today. Some of you finished Pandas already. Some of you started NumPy last week and have not touched Pandas at all. I know that, and it is fine.

Here is how we handle it. Today I am going to teach eight commands. That is all. Eight. If you already know them, you will find today easy, and I am going to use you — you will be coaching the person next to you in the last fifteen minutes. If you have never seen them, by the end of today you will have used every one of them at least once.

And for the next six days, the July batch has one extra session each day to close the gap on NumPy and Pandas. From day seven, everybody is on exactly the same page and there are no extra sessions. Six days. There is an end date.

Now, today's actual subject. We are starting statistics. And I want to start by telling you what statistics is for, because if I open with formulas you will write them down and never use them again.

Suppose you join a company on Monday as a data analyst. Your manager walks over and says: what is the average salary in this company?

What do you do? You open the employee file, you find the salary column, you add it all up and divide by how many people there are. Done. Fifteen seconds.

Now — is that answer certain? Could anybody argue with it?

No. It is exact. You had every single employee in that file, you did the arithmetic, and there is no argument to have. That is called descriptive statistics. You are describing data that you actually have in front of you.

Now the manager comes back and asks a second question: what is the average salary of every data analyst in India?

Stop and think about that one. Where is that file?

There is no file. Nobody has it. There is no database anywhere containing the salary of every analyst in this country, and there never will be. That number exists in reality — every analyst has a salary, so an average exists — but nobody can compute it.

And yet companies decide what to pay you based on that number. Salary surveys publish it every year. Job sites quote it. So how do they get it?

They ask a few thousand people. And from those few thousand, they make a careful claim about the whole country. That is inferential statistics — using the data you have to say something about the data you do not have.

And notice what has to change when you do that. The first answer was exact. The second answer can never be exact. It comes with a range, and a level of confidence, and a chance of being wrong. Learning to state that honestly is most of this course.

Descriptive: summarise the data you have. Inferential: make a claim about data you do not have.

These ten sessions walk from the left of that board to the right. Roughly the first half is descriptive — we get very good at describing what is in front of us. Then we cross over, and everything after that is about making claims under uncertainty.

One more thing before we touch a keyboard. Notice that the difference is not the amount of data. If I count all six hundred million votes, that is still descriptive — enormous, but descriptive. If I survey four hundred people and make a claim about ten million, that is inferential with a tiny dataset.

The difference is the claim you are making, not the size of the file. Hold onto that, because students get this wrong every year.

Everything we do for the next three months happens inside one library, called Pandas. So let us start there.

For those of you who have not used it — Pandas gives you one thing, and that thing is a table. Rows and columns, exactly like a spreadsheet, except it lives in memory and you talk to it with code instead of a mouse. That table is called a DataFrame.

And the good news for the July batch is that you already know how to build one, you just do not know it yet. You have done dictionaries. Watch.

Look at what happened. I gave it a dictionary — keys on the left, lists on the right — and it gave me back a table. The keys became the column headings. The lists became the columns.

That is the whole idea. A dictionary of lists is almost a table already. Pandas just makes it official and gives you a few hundred useful functions on top.

One thing to notice, because somebody always asks. See that column on the far left with 0, 1, 2? I did not create that. Pandas added it.

That is called the index. It is the label for each row. If you do not give Pandas labels it just numbers them from zero. For now, that is all you need to know about it — it is a name tag on each row.

Different types. The dictionary is still a dictionary. Pandas made a new object out of it.

Now — June batch. You have seen all of this. Do not switch off, because I want you watching for something specific. In about twenty minutes I am going to run one command, and that single command is going to contain the entire syllabus for the next ten sessions. See if you spot it before I tell you.

This is a file of two hundred employees from a company. I did not clean it. It is the kind of thing that actually lands in your inbox.

And I want to start with a rule that sounds obvious and that almost nobody follows. When a new dataset arrives, the first thing you do is not analysis. It is looking.

You would not sign a contract without reading it. Do not analyse a file you have not looked at. There are four commands, you run them in the same order every single time, and it takes about twenty seconds.

First command. read_csv, and the name of the file in quotes. That is it — the whole file is now in memory as a table.

Second, head. That gives you the first five rows. Why five? Because somebody who wrote Pandas decided five was a sensible default. You can ask for ten by putting ten in the brackets.

And look at what you learn in one second. What the columns are called. What kind of thing is in them. Whether the file loaded correctly at all. If you had the wrong separator, or the header row was wrong, you would see it here immediately.

shape gives you two hundred and one rows, twelve columns. Rows first, then columns — always that order, and you will use this constantly to check that an operation did what you expected.

Now notice something. head had brackets. shape does not. That is not a typo and it is not random.

head() has brackets because it goes and does a job for you — it fetches rows. shape has no brackets because it is just a fact the table already knows about itself, sitting there waiting to be read.

You will get this wrong for about a week and then never again. If you type df.shape() you will get an error saying a tuple is not callable, and now you know what that means.

info gives you the types and something more valuable — how many values are actually present in each column.

Look at the age column. Two hundred non-null. And shape just told us there are two hundred and one rows. So one age is missing. Look at salary — a hundred and ninety-nine. Two salaries missing.

That subtraction is how you find missing data. You do not need a special command. Compare the count in info against the row count in shape, and the gap is your missing values.

Right. Come back to dtypes for a second. I want you to look at one column.

There is a column called dept_id. Department ID. One for Sales, two for Tech, three for HR, four for Marketing. And Python has looked at it and decided it is int64 — a number.

So let me treat it like a number.

Two point three two.

The average department in this company is two point three two.

Now — what is two point three two? Sales is one. Tech is two. HR is three. What is two point three two? Is it a department? Can I go and sit in it?

Python did nothing wrong. It cannot tell the difference between a number that measures something and a number that is just a label. To Python, one two three four is one two three four.

The mistake was mine. I asked a meaningless question and got a meaningless answer, and nothing warned me. There was no error. No red text. Just a confident, useless number that I could have put straight into a report.

Knowing which columns you are allowed to average is your job. Not the computer's. And that sentence is the whole of tomorrow's session.

June batch — this is the one. This single command is the syllabus for the next ten sessions, already computed, sitting on the screen.

Let me read it to you, top to bottom, and tell you which session each row belongs to.

Count. How many values are actually there. Two hundred ages out of two hundred and one rows. A hundred and ninety-nine salaries. That is missing data, and handling it is a session on its own.

Mean. The average. Day three is built on this row — and on when it lies to you.

Std. Standard deviation. How spread out the values are. Same day — day three covers centre and spread together, because you should never report one without the other.

Min and max. Smallest and largest.

Twenty-five per cent, fifty per cent, seventy-five per cent. Those are the quartiles. The fifty per cent row is the median. Those three plus min and max are called the five-number summary, and that is day four.

So you have not learned statistics yet. But you are already reading it. Every number in this output is a topic we are going to spend ninety minutes on.

But before all that, this output does something more immediately useful. Look at the age column. Just look at it for a moment and tell me if anything is wrong.

Minus four years old. And two hundred and ten years old.

Now think about what just happened. I did not write a loop. I did not write a filter. I did not inspect two hundred rows by hand. I typed one command on a file I had never opened and it handed me two data quality problems.

That is why you look before you analyse. If you had gone straight to computing the average age, you would have got a number, it would have looked fine, and it would have been wrong — and nothing would have told you.

Same column. Same hundred and ninety-nine people. And two completely different answers about what a normal salary looks like here.

Ninety-five thousand, or sixty-eight thousand. The gap is twenty-seven thousand rupees — the first number is thirty-nine per cent higher than the second.

So if your manager asks what people earn at this company, which one do you say? Because one of these two numbers is lying to you, and you need to know which.

I am not going to tell you today. Look at the maximum salary in that output though — fifty-two lakh. Keep that in the back of your mind and we will come back to it.

Everything so far worked on the whole table. Most of the time you want one column. Here is how you pull one out.

Square brackets, and the column name in quotes. That is the pattern, and you are going to type it ten thousand times.

What comes back is one column with the row labels attached. Pandas calls that a Series. A DataFrame is the whole table; a Series is a single column out of it. That is the only distinction you need today.

And once you have a column, you hang a dot on the end and ask it for things.

Notice these are the same numbers describe already gave us. Nothing new has been computed. describe is a convenience that runs all of them at once — but when you want one number to put in a variable, this is how you get it.

The pattern is always the same. Table, square brackets, column name, dot, what you want. Say that with me once.

Quantile takes a list of positions between zero and one. Nought point five is the median. Nought point seven five means the value that seventy-five per cent of people are below.

Read those three numbers as a sentence: half this company earns between fifty-nine thousand and eighty thousand six hundred. That is a much more useful statement than any single average, and it is one line of code.

For text you cannot ask for a mean. What you ask for is how often each value appears, and that is value_counts.

Sixty-five people in Sales, forty-two in Marketing. One line, and you know the shape of the organisation.

Now watch what happens if I forget which kind of column I am dealing with.

There is the error. Pandas refuses, because you cannot average the words Sales and Tech.

But hold on — twenty minutes ago I averaged dept_id and it worked perfectly. Same information. Departments. One version errors and one version hands me two point three two.

The only difference is that one was stored as words and one was stored as numbers. When it is words, Pandas protects you. When somebody has coded it as one two three four, nothing protects you at all.

That is the trap, and tomorrow we learn the rule that gets you out of it.

Almost every real question is about a subset. Not "what is the average salary" but "what is the average salary in Tech". So you need to be able to keep some rows and throw away the rest.

I am going to show you this in two steps, because if I do it in one line it looks like magic.

Look carefully at what came back. Not a number. Not a list of salaries. Two hundred and one Trues and Falses.

I asked one question — is the salary above eighty thousand — and Pandas asked it of every single row at once and gave me back every answer. Row zero, no. Row one, yes.

Now I take that entire column of yes-and-no answers and put it inside the table's square brackets. And Pandas keeps only the rows where the answer was yes.

Read the line out loud from the inside out: salary greater than eighty thousand — that is the question. Wrap it in df square brackets — that means keep those rows.

It looks strange for about two days and then it becomes the most natural thing in the world. You will write this line more than any other line in your career.

Twenty-nine people in Tech earning over eighty thousand.

Two rules here, and you are going to break both of them this week, so let me say them clearly.

One. Use the ampersand, not the word "and". The word "and" expects one true-or-false, and you are handing it two hundred and one of them. It will throw an error about the truth value being ambiguous, and now you know what that message means.

Two. Brackets around every single condition. Not optional. The ampersand binds tighter than the greater-than sign, so without brackets Python groups it wrongly and the whole thing breaks.

Ampersand instead of "and". Brackets around each condition. Write both down.

One line, and I have four answers instead of one. Read it left to right: group by department, take the salary column, give me the mean. Pandas splits the rows into four piles, computes the average inside each pile, and stacks the results.

Now look at the numbers. HR fifty-six thousand. Sales sixty-six. Marketing seventy-three. Tech one lakh eighty-nine thousand.

Last ten minutes. Pair up — if you have done Pandas before, sit with somebody who has not.

And here are the rules. The person who has not done Pandas is the one typing. If you already know this, your job is to talk, not to grab the laptop. Explaining it is going to do more for your revision than watching me would have.

Four tasks on the screen. Get through what you can.

Question four is a trap and I will tell you now so you are ready for it. The highest paying department is Tech, on paper. Anybody who tells me "Tech, but I do not trust that number because one person is distorting it" has understood everything I taught today.

Two things before you go.

First — every single thing we typed today was descriptive statistics. Averages, spreads, counts, groupings. You were doing statistics for the last hour and I only used the word twice. It is not a new subject. It is a name for what you were already doing.

Second — I left two questions unanswered on purpose, and they are both on the board. Why is the average department two point three two? And why are the mean and median salary twenty-seven thousand rupees apart?

Tomorrow we answer the first one. And the answer turns out to be a rule that decides, for every column you will ever work with, which statistics you are actually allowed to compute. See you then.

One correction from last session before we start.

I described descriptive statistics using an example that sounded like prediction. Let me make the line sharper, because everything we do for the next two weeks sits on it.

Descriptive statistics describes the data you have.199 employees in this file, average salary ninety-five thousand six hundred and forty-eight. Full stop. No claim about anybody outside this file.

Inferential statistics makes a claim about people you never measured. That is a completely different act, and it needs machinery we do not have yet.

And one thing that catches people: a bigger file does not turn descriptive into inferential. A million rows described is still description. What changes it is the claim you make, not the size of the data.

Everything until day eight is describing. After that we start generalising, carefully.

The average department in this company is two point three two. The average employee is one thousand one hundred point one.

Both of those are real numbers. Neither of them means anything at all.

And I said yesterday that Python did nothing wrong — it was handed a column of numbers and asked for an average, and it computed one perfectly. The mistake was ours.

So today: how do you know, before you type it, whether a statistic is legal on a column? There is a rule, it takes about twenty minutes to learn, and you will use it for the rest of your career.

Before we get to four categories there are only two. Look at any column and ask: is this describing a quality — what kind of thing it is? Or a quantity — how much of something there is?

Department is a quality. Salary is a quantity. That is the first split, and it is usually obvious.

Now, if it is a quantity, one more question. Can you count it, or must you measure it? Number of children you count — there is nothing between three and four. Height you measure, and between 168 and 169 centimetres there are infinitely many possible values.

Counted is called discrete. Measured is continuous.

Now the finer classification, and there are four. They are cumulative — each one can do everything the one above it can, plus one more thing.

Sales is not more than Tech. Bangalore is not greater than Chennai. There is no order here at all, so the only things you can do are count them and name the most common one.

So the order exists — you all know Poor is worse than Good — but it exists in your head, not in the data. That is exactly what makes this column ordinal rather than nominal. There is a rank, but the computer cannot see it.

And because there is a rank, you can do one more thing than with nominal data: you can take a median. There is no median department. There is a median rating.

The last two are both proper numbers with equal gaps. The difference between them is a single question: does zero mean none?

Zero rupees means no salary. So salary is ratio, and I can say one person earns twice another.

Zero degrees does not mean no temperature. It is just a point somebody chose. So temperature is interval, and twenty degrees is not twice as warm as ten.

Nominal gets counts and mode. Ordinal adds median and percentiles. Interval adds mean and standard deviation. Ratio gets everything.

Notice it is cumulative. Move down the list and you unlock more statistics.

There it is again. The third line runs. No error, no warning. Python cannot stop you.

The scale lives in your understanding of what the data represents, not in the file.

Interesting. Here Pandas does refuse. Because the ratings are stored as words, it cannot rank them — it does not know the order.

So notice the pattern. When your categories are stored as text, Pandas protects you. When somebody has helpfully coded them as one, two, three, four, nothing protects you at all.

That is the whole danger of dept_id. The information is identical. Only the storage changed, and with it, whether the computer will catch your mistake.

Right. Completely different question for the second half.

Everything we have done so far is about a file that is sitting in front of us. Now: what is the average salary of every data analyst in India?

Not in this file. In the country. Take a moment — can anyone compute that?

Nobody can. That file does not exist and never will. And yet companies decide what to pay you based on that number every year.

So how do they get it? They ask a few thousand people, and from those few thousand they make a careful claim about the whole country.

Four words for that, and interviews test them. The group you care about is the population. The part you actually measured is the sample. A number describing the population is a parameter. The same number from a sample is a statistic.

P with P, S with S.

Ninety-five thousand six hundred and forty-eight. Remember that number.

In real life you would never know it. Today we do, and that is what makes the next ten minutes work — we can check every estimate against the truth.

There are a lot of named sampling methods. Before any of the names, one division that matters more than all of them put together.

In some methods, every person has a known, non-zero chance of being picked. Those are called probability sampling. In the rest, they do not.

Here is why you should care. Everything we do from day eight onward — every confidence interval, every p-value — assumes the first kind. Sample the second way and you can still learn things, but you cannot put a confidence interval on it and call it valid. The mathematics is assuming a randomness that was never there.

The one line to remember: you use every stratum, but only some clusters.

Stratified — this company is twenty-four percent Tech, so I make my sample twenty-four percent Tech deliberately instead of hoping it works out. And I get a more accurate answer than plain random sampling for the same number of people.

Cluster — I need to survey students across the whole state and I cannot visit every school. So I pick thirty schools at random and survey those. Far cheaper, and less accurate, because students inside one school resemble each other.

Same company. Same method. Thirty people drawn honestly at random, five separate times.

Answers from sixty-five thousand to two hundred and forty-four thousand. Not one of them is right.

And nobody made a mistake. No bug. This is simply what sampling is. An estimate moves.

Remember what I said earlier — a parameter is a fixed constant that happens to be unknown. A statistic is a random variable. You are watching that sentence happen.

Forty-five thousand, against a true average of ninety-six. Off by more than half.

And every single salary in that sample is real. Nobody faked anything. The data is perfect. The method was broken.

More data fixes noise. It never fixes bias. A bigger broken sample just makes you more confident about a wrong number.

So the first question about any survey is never how many people they asked. It is how they chose them.

Nineteen thirty-six, American presidential election. A magazine called the Literary Digest mails out ten million surveys and gets back two point four million replies.

Two point four million. That is not a poll, that is practically a census. They predict Landon wins comfortably.

Roosevelt won forty-six of the forty-eight states.

And in the same election, George Gallup polled about fifty thousand people using proper methods, and got it right.

So: a biased sample of two point four million lost to an unbiased sample of fifty thousand.

That is the two targets, as a historical fact.

Second World War. Engineers map the bullet holes on bombers coming back from missions — wings and tail. Obvious conclusion: armour the wings and the tail.

A statistician called Abraham Wald says the opposite. Armour the engines — where there are no holes at all.

Why? Because those are the planes that came back. The ones hit in the engine are at the bottom of the sea and were never in the data.

Two things to take away, and they are the two halves of today.

First — every dataset you ever open, the first question is now: what is each column, and what am I allowed to compute on it? Python will not ask you that. You have to.

Second — an estimate from a sample moves. Five samples, five answers, and that is normal rather than broken.

And one number before you go. For ninety-five percent confidence and a three percent margin of error, you need about one thousand and sixty-eight people.

Now — how much bigger does that get for a country of one point four billion instead of thirty million?

Essentially not at all. Population size barely enters the formula. That is why national opinion polls sample around a thousand people whatever the country. Which is the whole session in one fact: it was never about how many. It was about how you chose them.

Which leaves an obvious question hanging. If every sample gives a different answer, how does anybody ever report a number honestly? Hold that. The answer is day nine, and it is called a confidence interval.

Tomorrow is simpler. You have seen the mean and median of salary sitting twenty-seven thousand rupees apart since day one. Tomorrow we finally find out which one is lying.

Last time we finished on two words. Population and sample. Somebody remind me — what is the population?

Right. Everyone you care about. And the sample?

The part you actually measured. Good. And I said the whole of statistics is one move: you use the sample to say something about the population you can never fully see.

Now. I left something out, and I left it out on purpose.

I told you what a sample is. I never told you how you build one.

Think about that for a second. If I ask you to survey a thousand people in Bangalore, who exactly do you ask? The first thousand you meet? Your friends? People outside a mall? People who answer their phone?

Every one of those gives you a thousand people. Every one of them is a sample. And they will give you completely different answers.

So today is one question: how do you choose who goes into the sample, and does it actually matter?

Let me show you that it matters, before I tell you anything.

Same company. Same file. Thirty people picked completely at random, five separate times.

Sixty-five thousand. Two hundred and forty-four thousand. Two hundred and thirty-nine thousand. Not one of them is the real answer of ninety-five thousand.

And listen — nobody made a mistake here. There is no bug. I did nothing wrong five times in a row. This is just what sampling is. Every time you take a sample you get a slightly different answer, and sometimes a very different one.

Now watch this one. This time I am not going to pick at random. I am going to pick the thirty lowest paid people in the company, on purpose.

Before I run it — what do you think the average will be?

Forty-five thousand, against a real answer of ninety-five. Off by more than half.

And here is the thing I want you to notice. Every single salary in that sample is real. Nobody faked anything. The data is perfect. The arithmetic is perfect.

The method was broken. That is the difference between these two demos, and it is what the next forty minutes is about.

There are a lot of named sampling methods, and I am going to give you several. But before any of the names, there is one division that matters more than all of them put together.

In some methods, every person has a known chance of being picked. I can tell you, before I start, that Rahul has a one in ten chance. Those are called probability sampling methods.

In the others, I cannot. I have no idea what Rahul's chance was, because it depended on whether he happened to walk past me.

Why does this matter so much? Let me tell you where this comes back.

On day nine I am going to teach you confidence intervals — how to say “the average is ninety-five thousand, give or take four thousand.” That give-or-take is calculated. And the calculation only works if it knows what chance each person had of being in your sample.

Take that away and the formula has nothing to stand on. You can still do the survey. You just cannot put statistics on top of it and pretend.

So this is not a small classification. Everything we do from day eight onwards silently assumes you sampled the first way.

Same job every time, so you can see what actually changes. Two thousand employees. You need two hundred of them. Only the way you pick changes.

Simple random sampling. Every name goes into the computer, it picks two hundred. That is the whole method. The old way to picture it is two thousand slips of paper in a drum — shake it, pull out two hundred.

Everybody had exactly the same one in ten chance. Not because I was careful and fair-minded, but because the method makes it impossible for me to favour anybody. That is the important part. Fairness is built in, not promised.

This is the baseline. Every formula in this course assumes you did it this way, and every other method gets compared back to it.

What does it cost you? You need a complete list of all two thousand people before you can start. And bad luck is still possible — you could draw two hundred people who all happen to sit in Sales.

Systematic sampling. Line everybody up. Two thousand divided by two hundred is ten, so ten is my step. I pick a random starting point between one and ten — say seven — and then I take person seven, seventeen, twenty-seven, and keep going.

Why would you do this instead of simple random? Because you do not need the list up front. You can stand at a factory line and take every tenth item as it goes past. You can stand at a shop door and take every tenth customer. Simple random sampling cannot do that — it needs everybody listed before it starts.

Stratified sampling. Now I split the company into groups first — here, by department.

Tech is sixty percent of the staff, so I decide up front that sixty percent of my sample will be Tech. That is a hundred and twenty people. Sales is twenty-five percent, so fifty. HR is fifteen percent, so thirty. Then I do a simple random draw inside each department separately.

Why bother? Think back to simple random. If I draw two hundred at random, I might get only eight people from HR just by bad luck. And then I cannot say anything at all about HR — which was probably one of the things I was asked about.

But there is something better here than just guaranteed representation, and this is the bit people skip.

Stratified sampling gives you a more accurate answer than simple random sampling, using exactly the same two hundred people.

Same effort. Same cost. Better estimate. Why? Because I removed a source of randomness. With simple random, part of the wobble in my answer comes from not knowing how many Tech people I happened to draw. I have deleted that uncertainty by fixing it in advance.

What does it cost? I have to already know which department all two thousand people are in, before I start.

Cluster sampling. Change the situation slightly. Those two thousand people are now spread across forty branch offices, fifty people in each, all over India. And you have to go in person.

Simple random sampling would give you two hundred people scattered across all forty cities. That is forty trips to interview a handful of people each. Nobody is paying for that.

So instead: pick four offices at random, and survey everyone in them. Four times fifty is two hundred. Same sample size, four cities instead of forty.

And you only needed a list of offices to start, not a list of every employee.

What does it cost you? Accuracy. People in one office share a manager, share a local culture, share office conditions. So two hundred people from four offices genuinely tells you less than two hundred people from everywhere. You are trading precision for affordability — and that is a reasonable trade, as long as you know you are making it.

Both of these start the same way — divide everyone into groups. That is why people mix them up. The difference is entirely in what you do next.

One line. Remember this one line and you will never confuse them again:

You use every stratum, but only some clusters.

Every department appears in a stratified sample. Only four of the forty offices appear in a cluster sample.

Four more, quickly, and these are the other family — the ones where you do not know anybody's chance.

Convenience sampling is asking whoever is easiest. Standing outside your own office. A pop-up on your website. This is what most real surveys actually are, if we are honest.

Quota sampling looks like stratified — get me fifty men and fifty women — but which fifty is left to the interviewer. And interviewers approach people who look approachable. So the groups are right and the people inside them are not random.

Judgmental is when an expert picks. Snowball is when the people you find introduce you to more, which is often the only way to reach a hidden group.

All four are legitimate ways to learn something. None of them lets you compute a confidence interval and call it valid. That is the only distinction I want you to hold on to.

Two targets. Two archers.

The first archer's arrows are all over the place — above the bullseye, below it, left, right. None of them are on it, but they are scattered evenly all the way round.

The second archer's arrows are in a tight little group. Beautiful grouping. All of them up and to the left, nowhere near the centre.

Which archer is better?

The first archer has a true aim and a shaky hand. Their mistakes go in every direction and cancel each other out.

The second archer has a perfectly steady hand and a crooked sight. Every single shot is wrong in the same direction.

Now the question that matters. Which archer does firing a hundred more arrows help?

The first one. Fire enough arrows and their average lands right on the centre, because the errors cancel.

The second one just gets a tighter bunch, in the same wrong place. More arrows makes them more confident and no more correct.

The first archer is what we call noise. The second is bias. And this is the whole lesson of today.

Remember the thirty lowest paid people? That gave us forty-five thousand instead of ninety-five. That is archer number two — wrong in the same direction every time.

So here is my question. If I take a hundred of them instead of thirty, does it get better?

Forty-five thousand became fifty-six thousand. It moved a little. And it is still nowhere near ninety-five.

I more than tripled the sample and I am still wrong by forty thousand rupees. More data did not fix it.

A bigger broken sample does not get you closer to the truth. It just makes you more confident about a wrong number, which is worse than being unsure.

So when somebody shows you a survey, the first question is never “how many people did you ask?” It is “how did you choose them?”

Nineteen thirty-six. American presidential election. A magazine called the Literary Digest runs the biggest poll anyone has ever attempted.

They mail out ten million surveys. They get back two point four million replies.

Two point four million people. That is not a poll, that is nearly a census. They predict Landon wins comfortably.

Roosevelt won forty-six of the forty-eight states.

And in the same election, a man called George Gallup asked about fifty thousand people using proper random methods. He got it right.

So: a biased sample of two point four million lost to an unbiased sample of fifty thousand.

That is the two archers, as a real historical fact.

Which brings me to the question you are all going to ask eventually. How many people do I actually need?

The formula is day nine. But let me give you the answer now, because it will surprise you.

For ninety-five percent confidence and a three percent margin of error, you need about one thousand and sixty-eight people.

Now — how much bigger does that number get for a country of one point four billion instead of thirty million?

It barely changes at all. Still around a thousand.

That is why national opinion polls ask about a thousand people, whatever the size of the country. The accuracy depends on how many you asked, not on what fraction of the population they were.

Which is the whole of the last hour in one sentence: it was never about how many. It was about how you chose them.

Right. Completely different thing for the last part of today.

You now know how to get a sample. Now: once you have your data, how do you describe it with a single number?

There are three answers, and choosing the wrong one is how people mislead without lying.

Five numbers. Small enough to do in your head.

The mean is the one you already call the average. Add everything up — that is a hundred and fifty — and divide by how many there are, which is five. Thirty.

The median is different. Sort the numbers, then take the middle one. They are already sorted, and the middle one is thirty.

Same answer here. Hold on to that, because it is about to stop being true.

And the third one is the mode. That is just the value that appears most often.

Now, mode looks like the boring one, but it is the only one of the three that works on words. There is no mean department. There is no median department. But there is definitely a most common one.

Remember day two, when I said you cannot take the mean of a department? This is what you do instead.

Top row. Five salaries in thousands. Mean is thirty, median is thirty. They agree.

Now the bottom row. I have changed exactly one number. The forty became four hundred — one person got a huge raise. Everything else is identical.

Work out the new mean. Add them up: twenty, forty-five, seventy-five, a hundred and ten, five hundred and ten. Divided by five is a hundred and two.

Now the median. Sort them — already sorted. Middle one is thirty.

The mean jumped from thirty to a hundred and two. The median did not move at all.

Why? Because the mean uses every single value, so one enormous number gets a vote. The median only cares about position. Thirty is still the number sitting in the middle, whether the largest value is forty or four hundred or four million.

And look at what the mean is now telling you. It says the typical salary here is a hundred and two thousand. Four of the five people earn less than half that. The number is arithmetically perfect and it describes nobody.

There it is in our own file. The mean says ninety-five thousand. The median says sixty-eight. Twenty-seven thousand rupees apart.

And there is the reason, on the last line. One person in this company earns fifty-two lakh.

So if your manager asks what a typical employee earns and you say ninety-five thousand, you have just been wrong about almost everybody in the building — using a number that is completely correct.

Three rules. That is the whole decision.

And this is not a classroom rule. It is why house prices in the news are always reported as a median, never an average. A handful of enormous properties would drag the mean somewhere meaningless.

One word before you go. That gap between the mean and the median has a name. When the mean sits well above the median, we say the data is skewed.

All it means is: lopsided. A few very large values pulling one side out into a long tail.

That second number, fourteen, is a measurement of how lopsided. You do not need the formula today. Just know that big and positive means a few very large values.

And that leaves an obvious question hanging. I have shown you that the fifty-two lakh salary wrecks the average. So — should we remove it?

Hold that. It is a genuinely hard question and it is the whole of next session.

Yesterday we finished on one sentence. More data fixes noise, it never fixes bias.

Today I want to start by showing you that this is not a classroom idea. It has actually happened, on the largest possible scale, and it is one of the most famous mistakes in the history of statistics.

Nineteen thirty-six. American presidential election.

A magazine called the Literary Digest decides to run the biggest opinion poll anybody has ever attempted. They mail out ten million surveys.

They get back two point four million replies.

Two point four million people. That is not a poll any more, that is close to a census.

Based on that, they confidently predict that a man called Landon will win comfortably.

Roosevelt won forty-six of the forty-eight states.

And here is the part that finishes it. In the same election, a man called George Gallup asked about fifty thousand people, using proper random methods.

He got it right.

So a biased sample of two point four million lost to an unbiased sample of fifty thousand.

That is the two archers from yesterday, as a real historical fact. The Digest had a beautifully steady hand and a crooked sight.

Which brings me to the question all of you are going to ask eventually. Fine — so how many people do I need?

The actual formula is day nine. But let me give you the answer now, because it will surprise you.

For ninety-five percent confidence and a three percent margin of error — which is what most professional surveys aim for — you need about one thousand and sixty-eight people.

It barely changes. Still about a thousand people.

Why? Because the accuracy of your estimate depends on how many people you asked, not on what fraction of the population they represent. Once the population is big, making it bigger does not change how much a sample of a thousand wobbles.

That is why national opinion polls ask about a thousand people, whatever the size of the country. It is not laziness. It is the maths.

So that is the whole of the last two sessions in one sentence.

It was never about how many. It was about how you chose them.

Completely different question now.

You know how to get a sample. Now: you have your data in front of you. How do you describe the whole thing with a single number?

There are three sensible answers. And picking the wrong one is how people mislead other people without ever telling a lie.

Five numbers. Small enough that you can all do this in your head.

The mean is the one you already call the average. Add everything up — that is a hundred and fifty. Divide by how many there are, which is five. Thirty.

The median is a different idea. Sort the numbers first, then take the middle one. They are already sorted, and the middle one is thirty.

Same answer. Remember that they agreed here, because in about ten minutes they are going to stop agreeing very dramatically.

And the third one is the mode. That is simply the value that appears most often.

Now, mode looks like the boring one of the three. It is not, and here is why.

It is the only one of the three that works on words.

There is no mean department. There is no median department. But there is definitely a most common one, and it is Sales.

Remember day two, when I showed you that the average department ID was two point three two and it meant nothing? This is what you do instead.

Top row. Five salaries, in thousands. Mean is thirty, median is thirty. They agree, like before.

Bottom row. I have changed exactly one number. The forty has become four hundred — one person got an enormous raise. Everything else is identical.

Work out the new mean with me. Twenty. Forty-five. Seventy-five. A hundred and ten. Five hundred and ten.

Five hundred and ten divided by five is a hundred and two.

Now the median. Sort them — they are already sorted. The middle one is thirty.

The mean jumped from thirty to a hundred and two. The median did not move at all.

Why? Two different reasons, and this is the whole idea.

The mean uses every single value. So one enormous number gets a vote, and it is a very loud vote.

The median only cares about position. Thirty is still the number sitting in the middle — whether the biggest value is forty, or four hundred, or four million. The median never even looks at how big it is.

Now look at what the mean is actually claiming. It says the typical salary here is a hundred and two thousand.

Count how many of those five people earn less than half of that. Four out of five.

So the mean is arithmetically perfect and it describes nobody in the room. Nobody earns a hundred and two. Four people earn far less and one earns far more.

There it is in our own file. The mean says ninety-five thousand. The median says sixty-eight thousand. Twenty-seven thousand rupees apart.

And the last line is the reason. One person in this company earns fifty-two lakh.

One row. Out of two hundred. Moving the answer by twenty-seven thousand rupees.

So here is the situation you will actually be in. Your manager asks you: what does a typical employee here earn?

You run .mean(), because that is what everybody does. You say ninety-five thousand.

You have just been wrong about almost everybody in the building — using a number that is completely, provably correct.

That is the difference between computing something and understanding it. And it is roughly the whole job.

Three rules. That is the entire decision, every time.

And this is not a classroom rule. It is why house prices in the news are always reported as a median, never an average. A handful of enormous properties would drag the mean somewhere that describes no house anybody could buy.

Two teams. Five people each, and these are their test scores.

Work out the average for team A. Two hundred and fifty divided by five. Fifty.

Now team B. Ten, forty, ninety, a hundred and sixty, two hundred and fifty. Divided by five. Fifty.

Same average. Exactly the same. Fifty and fifty.

Now tell me honestly — are these the same team?

Team A is uniform. Everybody is doing the same thing.

Team B has somebody drowning at ten and somebody excellent at ninety, and the average has quietly erased both of them.

So here is the rule that the rest of today and all of tomorrow is built on.

The average tells you where the middle is. The second number tells you how far people are from that middle. You need both, always.

So for the rest of today: how do we measure spread?

Biggest minus smallest. That is called the range, and you just invented it, which is a good sign.

Team A: fifty-two minus forty-eight is four. Team B: ninety minus ten is eighty.

And look at that — it works. Same average of fifty, but ranges of four and eighty. The second number has told you something the first one hid completely.

So let us use it on our salary column. What do you think the range of salaries is?

Fifty-one lakh, sixty-six thousand. That is the range of salaries in this company.

Now — is that useful? Does that number tell you anything at all about what a normal employee here earns?

No. And here is exactly why.

The range uses two numbers out of two hundred. And not two random ones — by definition, the two most extreme values you have.

So one person, or one typing mistake, decides the entire answer. Everybody else in the file might as well not exist.

Notice this is the same problem the mean had. The range is to the mean what we need next is to the median. It gets destroyed by extremes.

You have been looking at this output since day one. Look at those three rows in the middle — twenty-five percent, fifty percent, seventy-five percent.

You never asked me what they were. Today they become the most useful thing on the screen.

Line everybody up in order of salary and cut the line into four equal groups.

The three cut points are called quartiles. Q1 is the value that twenty-five percent of people fall below. Q3 is the value seventy-five percent fall below.

And Q2, in the middle — what is that the same as?

So a quarter of this company earns below fifty-nine thousand. Three quarters earn below eighty thousand six hundred.

Now — the middle half of the company, everyone between Q1 and Q3, sits between fifty-nine thousand and eighty thousand. How wide is that?

Twenty-one thousand five hundred. That is called the interquartile range, or IQR. Q3 minus Q1. The spread of the middle half.

Now put the two numbers side by side.

Same column. Same two hundred people. Five million against twenty-one thousand.

Which of those two numbers actually describes this company?

The IQR. And the reason is simple: it throws away the top twenty-five percent and the bottom twenty-five percent before it measures anything. The fifty-two lakh salary is in the discarded part. It cannot touch the answer.

One warning before you go and try this yourselves.

If you look up how to find Q1 by hand, you will be told: take the median, then take the median of the lower half. For the numbers ten to seventy that gives you Q1 of twenty and Q3 of sixty, so an IQR of forty.

Pandas will give you twenty-five and fifty-five, so an IQR of thirty. On the same numbers.

Neither is wrong. There are several accepted ways to cut a small dataset into quarters, and pandas slides between the two nearest values instead of picking one. On seven numbers you can see the difference. On two hundred it is invisible.

So do not panic when the textbook and the computer disagree by a little. They are both right.

One word before we finish. That gap between the mean and the median has a name.

When the mean sits well above the median, we say the data is skewed. All that means is lopsided — a few very large values pulling one side out into a long tail.

The second number, fourteen, is a measurement of how lopsided. You do not need the formula today. Just know that big and positive means a few very large values.

And you already know exactly which value is doing it.

This one person. Fifty-two lakh.

They have wrecked the mean. They have wrecked the range. They are the entire reason the skew is fourteen instead of nearly zero.

So the obvious question, and I want you to actually think about it before tomorrow:

Should we just remove them?

Think about it tonight. It sounds like an easy yes. It is not, and there is a name for what that person is — an outlier — and a proper method for deciding what to do about them.

That is tomorrow.

Yesterday I left you with one question, and I asked you to actually think about it.

One person in this company earns fifty-two lakh. They pull the mean up by twenty-seven thousand. They make the range five million. They are the reason the data is lopsided.

So — do we delete them?

Good. You are all partly right, and that is exactly why this is worth a whole session.

Here is the thing I want to fix first. Almost everyone assumes that a strange value means a mistake. It does not.

A value can be strange for three completely different reasons.

Number three is worth sitting with. If you work in fraud detection, the unusual transaction is not noise you clean away — it is the entire thing you were hired to find. An analyst who automatically deletes outliers there would delete the answer every single time.

So there is no universal rule for what to do with them. There is only a rule for finding them.

Write that down. That is the session in five words. I am going to give you a proper method for flagging, and then the decision is always yours, and you must be able to defend it.

And there is one professional standard that goes with it. You are allowed to remove a value. You are not allowed to do it quietly. You say that you removed something, how many, and why.

An analysis that silently dropped six rows is not trustworthy, even when those six rows deserved to go.

Before I can give you the rule, I need five numbers. And you already have all five — you just have never seen them named as a set.

Minimum, Q1, median, Q3, maximum. Those five numbers together are called the five-number summary, and between them they describe the whole shape of a column.

Four of those five you learned yesterday. This is not new material — it is a name for something you already have.

Look at the gaps between them. Thirty-three to fifty-nine. Fifty-nine to sixty-eight. Sixty-eight to eighty. And then eighty thousand to five million.

Four of the numbers are huddled together and the fifth one has left the building.

That is what skew looks like when you read it as numbers. Yesterday I gave you the word. Today you can see it.

Five numbers is fine, but comparing five numbers across ten departments is horrible. So somebody worked out how to draw them.

Watch what I build. Every single piece is one of the five numbers you just calculated.

The box goes from Q1 to Q3. So the box is the IQR — the middle half of your data. You learned that yesterday, this is the same thing drawn.

The line inside the box is the median. Not the mean. That catches people constantly, so remember it — there is no mean anywhere on a box plot.

The whiskers reach out to the furthest value that is still normal.

And anything past the whiskers gets its own dot. Those are your flagged outliers.

And this is why the box plot is usually the first chart an analyst draws. Not because it is pretty — it is not — but because in about one second it tells you the middle, the spread, whether the data is lopsided, and whether there is anything strange in it.

A table of numbers hides all four of those.

I have been saying “flagged as an outlier” without telling you what decides that. Here is the actual rule, and it is only possible because of what you learned yesterday.

You build two boundaries. They are called fences.

That is it. Take the box — Q1 to Q3 — and extend it out by one and a half times its own width in each direction. Anything still outside is unusual enough to look at.

Notice what this is made of. Q1, Q3 and the IQR. All three of those you learned yesterday. I could not have taught you this rule before today.

Eight numbers. Seven of them sit between ten and twenty. One is ninety.

Work it through. The median falls between fifteen and sixteen, so fifteen and a half. The lower half gives Q1 of thirteen. The upper half gives Q3 of nineteen.

IQR is six. One and a half times six is nine.

So my fences are thirteen minus nine, which is four, and nineteen plus nine, which is twenty-eight.

Now — is anything below four or above twenty-eight? Ninety. And the rule found it without me telling it anything about what a normal value looks like.

So for salary, anything below twenty-six thousand or above one lakh thirteen thousand gets flagged.

Let us see who that is.

Two people out of a hundred and ninety-nine. One earning about one lakh seventeen, and our fifty-two lakh.

And here is the important thing. Both of those are real salaries. Nobody typed them wrong. The rule has not found any mistakes here — it has found two people who genuinely earn a lot.

Now watch what happens when I run the exact same rule on a different column. Same arithmetic, nothing changed except which column I point it at.

Minus four years old. And two hundred and ten.

Now. Same rule. Same formula. Same three lines of code.

But tell me — is the right answer here the same as it was for salary?

No. And this is the whole point of today.

The fifty-two lakh is real. Somebody genuinely earns it. If I delete them, my analysis is now a lie — I am describing a company that does not exist, one with no highly paid people in it.

Minus four is not real. There is no employee who is minus four years old. That is somebody's typing mistake, or a broken import, and if I leave it in it poisons the average age of the entire company.

Same rule found both. Completely opposite correct responses.

This is the difference between someone who runs code and an analyst. The code found four unusual values today. Deciding what each one is required knowing that people cannot be minus four years old and that companies do have CEOs.

No library will ever do that part for you. That part is your job, and it is why you have a job.

Let me show you why people delete outliers, because the temptation is genuine and I want you to feel it.

Here is our salary column with the two flagged values, and without them.

Look what happened.

The mean fell from ninety-five thousand to sixty-nine thousand — which is now almost exactly the median. The two agree. The data has stopped being lopsided.

And look at the last number. The spread went from three hundred and sixty-four thousand to sixteen thousand. It fell by more than twenty times.

The data is now beautiful. Everything agrees with everything. Every chart you draw will look tidy.

And that is exactly why you have to be disciplined about this.

The reward for deleting inconvenient rows is immediate and visible. Your numbers get prettier the moment you do it. Nobody looking at your final chart can tell that you did.

Which is precisely what makes it dangerous. I have just deleted the CEO and produced a lovely, tidy, confident description of a company that does not exist.

You are allowed to remove things. You are not allowed to remove them silently.

Say what you removed, how many rows that was, and why you did it. One sentence in your report.

An analysis that quietly dropped six rows is not trustworthy — even when those six rows genuinely deserved to go. Because nobody can check you.

So for our file: fix the ages, keep the salaries.

The ages are impossible, so they go — and I will say that I removed them.

The salaries stay, because they are true. And to describe this company honestly I report the median, not the mean.

Which is exactly what I taught you the median for, yesterday. It exists so that you do not have to delete real people to get an honest number.

Two things to take away.

First — the rule flags, you decide. One and a half times IQR will find you unusual values in any column in about three lines of code. It will never tell you whether they are mistakes, or the CEO, or the fraud you were hired to catch. That judgement is yours and you must be able to defend it out loud.

Second — whatever you decide, say it. Removing rows quietly is the difference between an analysis and a story.

One last thing. Look back at that number from ten minutes ago — the spread of our salary column. Three hundred and sixty-four thousand.

I have used that number twice today and I have never told you what it is or where it comes from. You have seen it in describe() since day one, sitting there, labelled std.

Tomorrow you find out what it actually measures — and why one strange value can make it twenty times bigger than it should be.

Yesterday I finished on this number and told you I would explain it today.

You have actually been looking at it since day one. Every time you ran describe, it was sitting there in a row labelled std, and nobody ever asked me what it meant.

Today you find out. And more importantly, you find out why one person earning fifty-two lakh made it twenty-two times bigger than it should be.

But first — you already know two ways to measure spread. Why do we need a third?

You have the range. Biggest minus smallest. Its problem is that it uses two numbers out of two hundred, and both of them are the most extreme ones you have.

And you have the IQR. That one is solid, it ignores extremes. But look at how it does that — it deliberately throws away the top quarter and the bottom quarter of your data. Half your data never gets a say.

So neither one uses every value and stays sensible. That is the hole, and that is what today fills.

Right. If you want a measure of spread that uses every value, what is the obvious thing to do?

Work out how far each value is from the mean, then average those distances. That sounds completely reasonable. Let us try it.

Eight numbers. They add to forty, so the mean is five.

Now, how far is each one from five? Two is three below. Four is one below, three times. Five is exactly on it, twice. Seven is two above. Nine is four above.

Zero.

And before anyone checks their arithmetic — you did not make a mistake. Try it with any set of numbers you like. It will always be zero.

Why? Because that is what the mean is.

The mean is exactly the point where everything above it and everything below it cancel out perfectly. That is the definition of where it sits. So the distances have to sum to zero — if they did not, it would not be the mean.

Which means averaging raw distances can never measure anything. The answer is always zero, for every dataset in the world.

So we need to get rid of the minus signs. Any ideas?

Ignore the signs and average. One point five. And that works — it is a real measure, it has a name, it is called the mean absolute deviation.

Almost nobody uses it. I will tell you why in a minute. First let us try the other idea.

Square each distance. Minus three squared is nine — and notice the minus sign has gone, which is the whole point. Minus one squared is one. Four squared is sixteen.

They add up to thirty-two. Divide by eight, and we get four.

That number has a name. It is called the variance.

Two reasons, and I want to be straight with you about both.

First, squaring punishes big distances much harder. Look at the numbers — a distance of four became sixteen, while a distance of one stayed one. Being badly wrong counts for far more than being slightly wrong. Most of the time that is exactly what you want.

Second, and honestly, squares are much easier to work with mathematically than absolute values, and almost everything you learn after this is built on top of variance. So it won partly on convenience.

Absolute deviation is not wrong. It just lost.

So we have variance. Four.

Four what, though? Our data was in whatever units — say these were salaries in thousands. What are the units of the variance?

Rupees squared. And nobody has any idea what a rupee squared is.

So there is one final step, and it is the most obvious one in statistics. We squared everything to get rid of the minus signs. Now we take the square root to undo it and get back to normal units.

Square root of four is two. And that is the standard deviation.

Four steps. That is the entire calculation.

And now the sentence I actually want you to remember, because you will never be asked to compute this by hand again:

Our numbers averaged five, and the standard deviation is two. So a typical value sits about two away from five. Look back at the data — two, four, four, four, five, five, seven, nine. Is that about right?

Yes. Most of them are within two of five. That is what the number is telling you.

Variance is the calculation. Standard deviation is the version you can say out loud to another human being.

Remember these two teams from a couple of sessions ago?

Team A: one point five eight. Team B: thirty-one point six.

Same average of fifty. And now the second number tells you they are completely different teams — which is exactly what I said an average could never do on its own.

Team A's typical person sits about one and a half marks from fifty. Team B's typical person sits over thirty marks away.

There it is. Three hundred and sixty-four thousand.

Now read it the way I taught you. The standard deviation is roughly the typical distance from the mean. So this is claiming a typical employee's salary is three hundred and sixty-four thousand rupees away from the average.

Is that true? Look at the actual salaries. Most people are within twenty thousand of each other.

So the number is wrong. Not arithmetically — it is computed correctly. But as a description of this company it is nonsense. Why?

Sixteen thousand. That is the honest answer, and it matches what you can see in the data.

Three hundred and sixty-four thousand against sixteen thousand. Twenty-two times bigger, because of two rows out of a hundred and ninety-nine.

And now you know why. We squared the distances. That salary is five million away from the mean. Square five million and it dwarfs everything else in the sum before we ever divide.

The same squaring that made variance useful is what makes it fragile.

One thing I glossed over. When we divided the total of thirty-two, I divided by eight — the number of values.

If those eight numbers were a sample rather than the whole population, the convention is to divide by seven. By n minus one.

Short version: when you measure spread from a sample, you use the sample's own average. And the sample's average sits closer to its own values than the real population average would.

So your answer comes out slightly too small, every single time. Dividing by a smaller number pushes it back up. That is all it is.

Now here is the part that will actually cost you an evening if I do not warn you.

Same column. Same data. Two different answers.

Neither library is broken. Pandas assumes you have a sample and divides by n minus one. NumPy assumes you have the whole population and divides by n.

They just made different default choices, and nobody warns you.

ddof stands for delta degrees of freedom. Setting it to one tells NumPy to divide by n minus one as well.

Here the gap is about nine hundred rupees out of three hundred and sixty thousand — a quarter of a percent. On two hundred rows it barely matters. On ten rows it would matter a lot.

The point is not the size of the gap. The point is that when two libraries disagree, neither of them is broken — and you should know which assumption you want.

So far standard deviation is just a number. Here is what it actually buys you.

For data with a symmetric, bell-like shape, there is a rule that holds remarkably well.

Sixty-eight, ninety-five, ninety-nine point seven. It gets called the empirical rule, and it turns an abstract number into a statement about where people actually are.

If I tell you the average salary is fifty thousand and the standard deviation is five thousand, you now know that roughly two thirds of people earn between forty-five and fifty-five, and ninety-five percent earn between forty and sixty. That is genuinely useful.

Let us test it.

Ninety-nine point five, ninety-nine point five, ninety-nine point five.

The rule says sixty-eight, ninety-five, ninety-nine point seven. We got the same number three times. It has completely failed.

Why?

Because the standard deviation is three hundred and sixty-four thousand.

So one standard deviation either side of the mean is a band stretching from minus two hundred and sixty-eight thousand to plus four hundred and fifty-nine thousand. Everybody is already inside it. Going out further adds nobody, because there is nobody left.

The outlier stretched the ruler so far that the ruler stopped measuring anything.

Now watch what happens if I take those two rows out.

Sixty-six. Ninety-five. A hundred.

The rule says sixty-eight, ninety-five, ninety-nine point seven. That is almost exactly it.

Same column. Same rule. Two rows removed.

And notice this is day five's lesson again, wearing different clothes. One extreme value did not just distort the average — it distorted the measure of spread, and then it broke a rule built on top of that measure.

One row. Three things wrong.

One last thing, and it closes something I have been dodging.

Twice now I have shown you this number and told you not to worry about it.

Fourteen point zero six. I could not explain it before today, because skew is built out of the standard deviation — and until ten minutes ago you did not have that.

Now you do. So here is the whole thing, in four steps.

Look at step three.

All hundred and ninety-nine rows together give two thousand seven hundred and fifty-six. The fifty-two lakh row on its own gives two thousand seven hundred and fifty-seven.

The other hundred and ninety-eight employees contribute minus nought point two between them.

One row is the entire number.

And that is why cubing is in there. Squaring kills the minus sign, which is what variance wanted. Cubing keeps it — which is how skew knows whether the tail is on the left or the right — and it blows up big distances even harder than squaring does.

So you do not need to memorise this. You need one sentence: big and positive means a few very large values. Which is exactly what I told you two sessions ago.

You now have every tool for describing one column. Where the middle is — mean, median, mode. How spread out it is — range, IQR, standard deviation. What shape it is — skew. And whether anything strange is in it — the IQR rule.

One column. Completely covered.

But nobody was ever hired to describe one column.

The real questions all have two columns in them. Do people with more experience earn more? Does the department affect the salary? Does one thing move with another?

Nothing you have learned so far can answer that, because everything so far looks at one column at a time.

That is next session.

Let us take stock. In six sessions you have learned to describe a column completely.

Where the middle is — mean, median, mode. How spread out it is — range, IQR, standard deviation. What shape it has — skew. Whether anything strange is in it — the IQR rule.

That is genuinely a lot. And every single one of those looks at one column at a time.

So let me ask you a question none of it can answer.

Do people with more experience earn more?

Think about what you would do. You could take the mean salary. You could take the mean experience. You could get the standard deviation of both. And none of that helps at all, because the question is not about either column on its own — it is about whether they move together.

And notice — nearly every question a business actually asks has two columns in it. Does this affect that. Does this move with that.

So today we finally leave one column behind.

Here is the reasoning, and it is nicer than the formula.

Take one person. Ask two questions about them. Is their experience above or below the average experience? And is their salary above or below the average salary?

If somebody is above average on both, that is evidence the two go together. If they are below average on both — that is also evidence they go together. And if they are above on one and below on the other, that is evidence against.

Now the trick. We turn that into arithmetic by multiplying.

Look at the second line, because that is the clever part. Two negatives multiply to a positive. So somebody who is below average on both counts as evidence that the two move together — which is exactly right.

So: do that multiplication for every person, add them all up, and average. If the answer is strongly positive, the two things rise together.

That number is called the covariance.

Five people. Take the distances from each mean, multiply them pair by pair, add them up.

Notice the first column. Minus two times minus four is plus eight. That person is below average on both, and it counts as positive evidence.

Twenty in total. Divide by four — n minus one, same correction as last session — and the covariance is five.

One hundred and twenty-eight thousand.

So — is that a strong relationship?

Nobody knows. And there are two reasons why not.

First, the units. What are the units of that number? We multiplied a distance in years by a distance in rupees. So it is in years-times-rupees. That is not a thing that exists.

Second, there is no ceiling. Covariance can be any number at all. Without a maximum, you have nothing to compare against.

Experience against age gives four point one. Height against salary gives a hundred and seventy-one thousand.

Does that mean height and salary are forty thousand times more related than experience and age? Obviously not. The numbers are just in different units.

And it gets worse. If I measured salary in thousands instead of rupees, the covariance would change completely — while the actual relationship stayed exactly the same.

So it is half a tool. We need to strip the units out and put it on a fixed scale.

The fix is one line, and you already have both pieces.

Take the covariance. Divide it by the standard deviation of x, and by the standard deviation of y.

The units on the top are years-times-rupees. The units on the bottom are years-times-rupees. They cancel. The answer has no units at all.

And it does something else. It forces the answer onto a fixed scale from minus one to plus one, no matter what you measure or in what units.

This is called the Pearson correlation coefficient, and it is the number everybody means when they say “correlation”.

Those bands are rough, not law. What counts as a strong correlation in physics would be laughable in psychology. Use them as a starting point, not a rule.

And hold on to one word in that definition: straight-line. Correlation measures how well the points sit on a straight line. Remember that — we come back to it at the end and it matters.

One. Perfect. And look at why — y is exactly two times x, every single time. A perfectly straight line.

Now let us try it on something real.

Right. The question I opened with. Do people with more experience earn more?

Before I run it — what do you expect? Strong positive? Weak? Give me a number between minus one and plus one.

Nought point zero seven nine.

Read that with the scale I just gave you. Below nought point three is weak. This is barely above zero.

So what does this number say? It says experience has essentially nothing to do with salary. Twenty years in the job, or two — makes no difference to your pay.

Do you believe that?

Neither do I. So let us think about what could do this.

We have spent two sessions on two particular rows. They wrecked the mean. They wrecked the range. They made the standard deviation twenty-two times too big. They made the skew fourteen.

Any guesses?

Nought point zero seven nine becomes nought point six zero seven.

From “no relationship at all” to “a solid, moderate relationship”. Because of two rows out of a hundred and ninety-nine.

Now think about what that means practically. If you had run that first line, seen nought point zero seven nine, and put it in a report — you would have told your company that experience does not affect pay. Which is completely false, and you would have had a correctly computed number to back it up.

So we have a measure that gets destroyed by extreme values. Where have you seen that before?

The mean. And what did we use instead? The median — because the median does not care how big the biggest value is. It only cares that it is the biggest. Position, not value.

There is a version of correlation that does exactly the same thing. It is called Spearman.

Spearman throws away the actual numbers and replaces them with ranks. First, second, third.

That is the whole trick. The fifty-two lakh salary becomes “the highest one”. Not five point two million. Just highest.

So an extreme value cannot distort anything, because its extremeness has been thrown away before the calculation starts.

Let us see what it says.

Look at that table properly, because there are two things in it.

The first one you already know. Pearson went from nought point zero seven nine to nought point six zero seven once we cleaned.

But look at the Spearman row. Nought point six one two, and nought point six zero five. Almost identical.

Spearman gave the right answer before anybody found the outliers. It never needed the cleaning. It was correct on the raw file.

Practical advice, and this is what I actually want you to do at work: run both.

If they agree, you are fine. If they disagree badly — like nought point zero eight against nought point six one — that gap is not a nuisance. It is a warning that something extreme is in your data, and you should go and look at it before you write a single sentence about the result.

You have all heard this sentence before. Correlation is not causation. Everybody can say it.

And almost nobody catches themselves doing it, because in the moment it never feels like a mistake. It feels like the obvious interpretation.

So instead of the slogan, I am going to give you something you can actually use.

When you find that two things move together, there are exactly three explanations. And the correlation number cannot tell you which one you have.

Number two catches people constantly. Do hospitals make people ill? There is a strong correlation between going to hospital and being unwell. The arrow points the other way.

Number three is the one that will actually get you. Ice cream sales and drowning deaths rise together, strongly, every year. Does ice cream cause drowning?

Summer. Hot weather sells ice cream and puts people in the water. The two numbers move together beautifully and neither causes the other.

The thing you did not measure — the weather — is driving both things you did measure. That is called a confounding variable, and it is the most common reason a correlation misleads you.

Let me show you one hiding in our own data.

We just found that experience and salary correlate at nought point six zero seven across the company. Watch what happens when I look inside each department separately.

Inside every department the relationship is between nought point seven four and nought point eight. Across the whole company it is nought point six.

Mixing the departments together made the relationship look weaker. Why?

Because departments pay differently. A junior person in a well-paid department can earn more than a senior person in a lower-paid one. So when you throw them all in together, that difference blurs the picture.

Department is a variable sitting underneath your number, changing it. And nothing in the number nought point six zero seven tells you it is there.

So here is the discipline. Whenever you report a correlation, before you write the sentence, run through the three possibilities out loud. Could it be the other way round? Is there something I did not measure that could be driving both?

If you cannot rule those out — and usually you cannot — then say “these are associated”, not “this causes that”. That one word change is the difference between an analyst people trust and one they do not.

For seven days we did one thing. We looked at data we already had.

We found the average salary. We found how spread out the heights were. We checked if pay goes up with experience.

And every answer was sure. When I said the average salary is sixty-nine thousand six hundred and thirty, I was not guessing. I had everyone's salary. I added them up.

From today the questions change.

What is the chance the next person we hire works in Tech?

If I ask only thirty people, how close will their answer be?

Tech is paid more than HR. Is that real, or is it just luck?

Look at those three questions. They are not about people I measured. They are about people I did not measure.

So I cannot be sure any more. I can only say how likely something is.

And that is what probability is. A number that tells you how likely something is. That is the whole idea.

One more thing before we start.

Everything left in this course uses probability. Day ten uses it. Day eleven uses it. Day twelve and thirteen use it.

So this is not a side topic. If today does not go in, nothing after this will make sense.

And here is some good news. You have already been doing this for seven days. You just did not know the name.

Let us start with a coin.

I flip a coin. What is the chance I get heads?

One out of two. Everyone knows that.

But stop and see what you just did. You did not flip the coin. You looked at the coin, saw it has two sides, and worked it out in your head.

Now a die. Six faces. What is the chance of a four?

One out of six. Again, you did not roll it.

Now a pack of cards. Fifty-two cards. How many aces?

Four. So the chance of an ace is four out of fifty-two. That is about eight percent.

How many hearts in a pack? Thirteen. So the chance of a heart is thirteen out of fifty-two. That is one out of four.

Every single answer came from thinking. You did not touch a card.

The book has a name for this. It is called theoretical probability. Do not worry about the word. Just remember what it means: you worked it out by thinking.

Now I will ask a different kind of question.

I pick one employee from our table at random. What is the chance that person works in Tech?

One out of four, because there are four departments?

That would be right if this were a die. A die has six faces and they are all the same size.

But are our four departments the same size? No. Sales has sixty-five people. Marketing has forty-two.

So I cannot think my way to the answer. I have to open the file and count.

Forty-nine out of two hundred and one. That is zero point two four four.

Not zero point two five. Close, but not the same.

This one also has a book name. It is called empirical probability. Again, do not worry about the word. It means: you counted what actually happened.

Now look at that second line of code. normalize equals True.

You have been running that since week one. Every time you ran it, you were finding a probability. Nobody told you that is what it was called.

So we have two ways. Think, or count.

Coins, dice and cards are the thinking kind. That is why every textbook uses them. They are easy to explain.

But at work, almost nothing is like that. A customer is not a coin. A sale is not a die.

At work you count. So the counting kind is the one you will use every single day.

That is the last time we will look at a coin.

Every probability is a number between zero and one.

Zero means it never happens. One means it always happens. Nothing can be outside that.

So if someone says “there is a one point five chance”, they made a mistake.

And all the pieces together must add up to one. Let us check.

One point zero. Exactly one.

It has to be. Every employee is in one department. So all the pieces together are all the people.

Now a question. What is the chance a person is not in Tech?

You can add the other three. That gives zero point seven five six.

Or you can just do one minus zero point two four four. Same answer.

That is a small rule but you will use it a lot. The chance of something not happening is one minus the chance it happens.

It sounds too easy to be useful. Here is when it saves you.

Say a machine has ten parts. Someone asks: what is the chance at least one part fails?

Doing that directly is very long. One part fails, or two, or three, all the way to ten.

But “at least one fails” is just the opposite of “none of them fail”. So work out the chance none fail, and take it away from one. One line instead of ten.

Two rules now. One for or, one for and.

First, or. What is the chance a person is in Tech or rated Excellent?

The chance of Tech is zero point two four four. The chance of Excellent is also zero point two four four.

So do we just add them? That would give zero point four eight eight.

Let us check.

Before the answer, look at that last line of code. There is a straight up-and-down line in it. t pipe e.

In Python, that line means or. It is how you write “in Tech or rated Excellent”. Remember it. We will see this symbol again later today.

Adding gave zero point four eight eight. The real answer is zero point four two three.

So adding gave us too much. Why?

Because thirteen people are in Tech and rated Excellent.

When I counted the Tech group, those thirteen were in it. When I counted the Excellent group, those same thirteen were in it again.

I counted them two times. So I have to take them off once.

That is the or rule. Add the two, then take away the ones in both.

Now the and rule. For and, you multiply.

Chance of Tech times chance of Excellent. Let us see if it works.

Real answer zero point zero six five. Multiplying gave zero point zero five nine.

Very close. So multiplying worked here.

But it does not always work. Later today I will show you a pair where multiplying is very wrong. Remember that I said this.

Two words now. They sound similar. They are not.

First word: mutually exclusive. It means both cannot happen together.

Can one person be in HR and in Tech at the same time? No. So HR and Tech are mutually exclusive. The chance of both is zero.

Second word: independent. It means one tells you nothing about the other.

If I tell you a person works in Tech, do you now know their height? No. So department and height are independent.

Now here is where students get confused. Both words sound like “these two things are separate”.

So let me ask you something. I tell you a person works in HR. Do you now know anything about whether they work in Tech?

Yes. You know for sure they do not work in Tech. You are completely certain.

So that is a lot of information, not zero information.

But independent means zero information.

So mutually exclusive and independent are not the same thing. They are opposite things. Write that down.

Last big idea, and this is the useful one.

Every question so far was about a random person from the whole table.

But real questions are not like that. Real questions have a condition in them.

Not “will this customer buy”. It is “will this customer buy, if they clicked the ad”.

Not “will this person repay the loan”. It is “will this person repay, if they earn less than thirty thousand”.

See the difference? There is an extra piece of information. We write it like this.

Now wait. Some of you are looking at that line and thinking something is wrong.

An hour ago I wrote t pipe e and I told you that line means or.

Now I have written P of A pipe B, and I am telling you the same line means given.

Same symbol. Two completely different meanings. So let me clear that up, because it confuses everybody.

Here is the easy way to tell them apart. Look at whether the line is inside quote marks.

In t pipe e, the pipe is not in quotes. So Python reads it and does something. It means or.

In print("P(Excellent | Tech)"), the pipe is inside the quotes. Python does not read it at all. It is just text. It is a label I typed for myself so I know what the number means.

So the rule is simple. Inside quotes, it is a label for us. Outside quotes, it is an instruction for Python.

And you will never write P of A given B as real Python code. There is no such thing. We only write it on the board and inside labels.

Right. Now, doing it is easy. You already know how.

“Given Tech” just means: delete every row that is not Tech. Then count normally.

That is a filter. You have been filtering since week one.

Zero point two four four for everyone. Zero point two six five for Tech only.

Those are almost the same number.

So knowing the person is in Tech told me nothing new about their rating.

When the number does not change, the two things are independent. That is the word from ten minutes ago. Now you can actually see it in a number.

Now let me try a different condition.

First let me find the middle salary. Half the people earn more than that, half earn less.

Then I will ask: out of the people who earn more than the middle, how many are in Tech?

Look at the difference. Zero point two four four for everyone. Zero point three eight four for the well paid people.

That is a big jump.

So this time, the condition did tell me something. If I know someone earns well, they are much more likely to be in Tech.

These two things are not independent.

And you already knew this. On day seven we saw Tech is paid more than the other departments. This is the same thing. We are just saying it a different way.

Remember I said multiplying does not always work? Here is where it breaks.

The real answer is zero point one eight nine. Multiplying gave zero point one two zero.

That is not close. That is about sixty percent off.

Earlier, multiplying worked for Tech and Excellent. Now it fails for Tech and salary.

What is the difference? Tech and Excellent do not affect each other. Tech and salary do.

So here is the full rule. You can multiply only when the two things do not affect each other.

And how do you know? You do what we just did. Multiply, then compare with the real count. If they match, you were fine. If they do not match, they affect each other.

One more thing, and this is the one I really want you to remember.

A few minutes ago we found this. Out of the well paid people, thirty-eight point four percent are in Tech.

Now I will turn the question around. Out of the Tech people, how many are well paid?

Before I run it — what do you think? Same number? Bigger? Smaller?

Zero point three eight four one way. Zero point seven seven six the other way.

That is double. Same table, same two columns, and the answer doubled.

Now here is the strange part. Look at where these numbers come from.

There are thirty-eight people who are in Tech and well paid. Thirty-eight, both times. The same thirty-eight people.

First answer: thirty-eight out of ninety-nine well paid people. That is zero point three eight four.

Second answer: thirty-eight out of forty-nine Tech people. That is zero point seven seven six.

The top number never changed. Only the bottom number changed. And that is enough to double the answer.

Let me say it without any data.

Most people who get food poisoning had eaten at a restaurant. That is true.

Now turn it around. Most people who eat at restaurants get food poisoning.

Is that true? No. That is silly. Millions of people eat out and are perfectly fine.

Same two facts. I just turned them around. And one of them is nonsense.

Now one you will actually meet at work.

Someone tells you: ninety percent of the customers who left us had complained first.

So should we worry about every customer who complains?

Think about it. That sentence is about the people who left. It says nothing about the people who complained.

Maybe five thousand people complained last year. Maybe only one hundred of them left. That is two percent, not ninety.

If you get this backwards, you will spend the whole budget on the wrong customers.

So whenever you hear “ninety percent of the people who…”, stop and ask one question. Ninety percent of which group?

Quick recap.

Probability is a number between zero and one that says how likely something is.

You get it two ways. By thinking, like a coin. Or by counting, like our table. At work it is almost always counting.

For or, add the two and take away the ones in both.

For and, multiply — but only if the two things do not affect each other. Check it first.

And for a question with a condition, filter the rows and then count.

And the last one. Turning a question around gives you a different answer. Always check which group the percentage is about.

One question before you go. I am not going to answer it today.

Everything today was about groups you can count. Departments. Ratings. You can count how many people are in each one.

But look at the height column. Heights are not groups. They are measurements. One person is 168.2, another is 171.9.

So here is my question. If I pick one employee at random, what is the chance they are exactly one hundred and seventy centimetres tall?

Keep thinking about that one.

Counting worked for everything today. Tomorrow I will show you that for heights, counting stops working completely.

And fixing that is where the famous bell curve comes from.

Before we start I want to put two questions back on the board. One I will answer today. One I still will not.

Here is the old one. Way back on day three I took five random samples. Thirty people in each one, all from the same salary column.

Look at what I got. Seventy thousand. Then two hundred and forty-four thousand. Then two hundred and thirty-nine thousand. Then sixty-five thousand. Then two hundred and thirty-five thousand.

The biggest one is nearly four times the smallest one.

And the true answer was ninety-five thousand six hundred and forty-eight. That is not close to any of the five.

So I asked you this: if every sample gives a different answer, how does anybody report a number honestly?

There is an answer. It is the most important idea in this module. And it is next session, not today.

And there is a second question, from yesterday. I asked what the chance is that somebody is exactly one hundred and seventy centimetres tall.

That one I am going to answer today. And answering it is what makes the first question answerable next session.

So why not today? Because the answer is one sentence, and I cannot say it yet.

The sentence is: sample means follow a normal distribution.

And that sentence is useless to you, because you do not yet know what a normal distribution is.

So today is one shape and one piece of arithmetic. Neither is difficult. But everything from here to the end of the module — confidence intervals, every test, the whole second half — is built on them.

Last session we did probability by counting. A die has six outcomes, one of them is a four, so one in six.

Now let me break that.

Look at the height column in our file. What is the probability that somebody is exactly one hundred and seventy centimetres?

Not a hundred and seventy point zero zero one. Not a hundred and sixty-nine point nine nine nine. Exactly one seventy.

The answer is zero. And that sounds absurd, but think about why.

Height does not come in steps. Between a hundred and sixty-nine and a hundred and seventy there are infinitely many possible heights. No single one of them gets any share at all.

So for anything continuous — height, weight, time, salary — we never ask about a point. We ask about a range.

Not “what is the chance somebody is exactly one seventy”, but “what is the chance somebody is between one sixty-five and one seventy-five”. That question has a real answer.

Write that down, because everything for the rest of this module is one version of it.

Every confidence interval is an area under a curve. Every p-value you will ever calculate is an area under a curve. This picture is the whole second half of the course.

And the technical name for that curve is a probability density function. You will see the letters PDF. You do not need the term to do the work, but you should recognise it.

Now, curves can be any shape at all. But there is one particular shape that turns up so often it gets its own name.

It is called the normal distribution, or the bell curve. And rather than draw you an idealised one, let me show you one sitting in our own file.

Look at that skew. Minus nought point one nine. Remember what skew means — near zero is symmetric.

Compare that to salary, which was fourteen. This column is not lopsided. It is balanced around its middle.

Now let us test it against the rule I gave you on day six.

Sixty-eight, ninety-five, ninety-nine point seven is the rule. We got sixty-nine point seven, ninety-five point zero, a hundred.

That is real data, two hundred and one actual people, landing almost exactly on the rule. This column is normally distributed.

Remember day six, when I ran the same test on salary and got ninety-nine point five three times? That column was not normal. This one is. Now you can see the difference the rule was talking about.

Point two is the one worth pausing on. Tell me the mean and the standard deviation, and I can draw the entire curve. Nothing else is needed. No other shape in statistics is that obliging.

And why does this shape keep appearing? Because anything produced by lots of small independent effects adding up tends to end up here. Height is genetics plus nutrition plus a hundred other small things. Measurement errors. Exam marks. Manufacturing tolerances.

And — this is the one that matters for next session — sample means.

Somebody scored 78 in a test. Is that good?

You cannot answer that. Seventy-eight where everybody got eighty-five is poor. Seventy-eight where the average was forty is outstanding.

A raw number on its own tells you nothing. You need to know where it sits relative to everybody else.

So here is a single number that answers it.

That is it. Subtract the mean, divide by the standard deviation.

Positive means above average. Negative means below. And the size tells you how unusual.

It is called a z-score, or a standard score.

And you have already used this twice without knowing its name.

Day six, the sixty-eight ninety-five rule. “Within one standard deviation of the mean” — that is a z-score between minus one and plus one.

And at the very end of day six, when I finally explained the skew of fourteen, step one was: how many standard deviations is the fifty-two lakh salary from the mean? The answer was fourteen point zero two. That was a z-score.

One seventy is barely above average — z of nought point two.

One seventy-five is nought point eight four above.

One eighty-five is two point one four standard deviations above. And remember the rule — ninety-five percent of people are within two. So this person is outside that. Genuinely tall.

One sixty is minus one point one. Below average, but not unusually so.

Two students. Two different subjects. Who did better?

Most of you said A. Seventy-eight against fifty-five, it is not even close.

Let us do it properly.

Student A is one standard deviation above their class.

Student B is two and a half.

Student B did considerably better — with a much lower raw mark.

Why? Look at the two standard deviations. The maths class was spread out — eight. Scoring eight above the mean there is ordinary.

The physics class was tightly bunched — four. Almost everybody scored close to forty-five. Getting ten above in that class is genuinely remarkable.

The raw marks were measuring two completely different things. The z-scores put them on the same scale.

The other thing a z-score buys you. If the data is roughly normal, z tells you what fraction of people are below you.

One seventy-five puts you at the eightieth percentile — taller than eighty percent of people.

One eighty-five puts you at ninety-eight.

That is a genuinely useful translation. A raw height means nothing to most people. “Taller than ninety-eight percent” means something to everybody.

Two words you are going to run into, and people use them interchangeably when they should not.

Standardisation is exactly the z-score you just learned. It centres the data on zero and tells you how many standard deviations out each value is.

Normalisation squashes everything between zero and one. The smallest value becomes zero, the largest becomes one, everything else sits in between.

Both put different columns on a comparable footing. They just do it differently.

So today you got two things. A shape, and a way of measuring distance on it.

The normal distribution — symmetric, described completely by a mean and a standard deviation, following sixty-eight ninety-five ninety-nine point seven.

And the z-score — how many standard deviations from the mean, which lets you compare things that were never comparable and turn a value into a percentile.

Now. Back to the thing I have been putting off for two sessions.

Five samples of thirty people. Five different answers. None of them right.

Here is what I could not tell you before today, and can now.

If I took not five samples but five thousand, and wrote down the average from each one, and then looked at the shape of those five thousand averages — they would form a normal distribution.

Think about how strange that is. The salary column itself is wildly lopsided — skew of fourteen, nothing like a bell curve.

But the averages of samples taken from it are normal.

And if they are normal, then everything you learned today applies to them. Mean and standard deviation describe them completely. Sixty-eight ninety-five applies. You can compute z-scores on them.

Which means — and this is the answer to the question — you can say how far wrong your one sample is likely to be.

That is next session. It has a name, it is the most important idea in this module, and I am not saying another word about it today.