Samples, populations and the normal distribution

Alex Tutor

Morning, Sofia. Before we compare any two groups, we need to agree what we're even measuring things about. Grab coffee, this part is short but it trips everyone up.

You

Sure. We're measuring CatChow orders, right? I pulled last month's 500 orders into a sheet already.

Alex Tutor

Those 500 are a sample: the part we actually looked at. The population is every order that exists or could exist — every customer who ever bought, or ever will.

You

So the population is basically infinite and we can never fully measure it?

Alex Tutor

Pretty much — future customers haven't even shown up yet. So we use a sample to make educated guesses about the population. The trick is making sure the sample actually represents it.

You

What could go wrong? I just took the most recent 500 orders.

Alex Tutor

That's actually fine if orders arrive in no particular pattern. But if you'd only sampled orders placed through our app, you'd miss every desktop shopper. That's sampling bias: a systematic reason some units are more likely to be picked.

Alex Tutor

Question 1

CatChow's data team wants to survey a random 10% of everyone who ordered in the last year and use the result to describe satisfaction across all customers who order this year. The average score computed from the replies is a...

Choose one answerA populationA parameterA statisticA sampling bias

Good. Now, two more words you'll hear constantly: a number that describes the whole population (like the true average spend of every CatChow customer) is a parameter. It's usually unknown — nobody has measured every customer.

You

And the number we compute from our 500 orders is...?

Alex Tutor

A statistic. Same idea, different source: computed from a sample, used to estimate the parameter. Statisticians love giving two names to one concept depending on where it came from.

You

So our sample mean of $45 is a statistic, and the 'true' average order value of every CatChow customer ever is a parameter we'll never know exactly.

Alex Tutor

Exactly. Match these four terms to their definitions before we move on — it's the vocabulary the rest of this module leans on.

Alex Tutor

Question 2

Match each term to its definition.

Tap an answer, then tap the row it belongs to. You can also drag.

Population
Sample
Parameter
Statistic

Answers left to place

Alex Tutor

Now, why does everyone keep drawing that bell-shaped curve? Plot the order values of thousands of independent customers and, remarkably often, you get a symmetric hump: the normal distribution.

You

Why would order values just happen to form a bell shape?

Alex Tutor

Loosely: each order value is nudged by many small, independent factors — mood, stock, coupons, cat appetite. Add up enough small random nudges around a typical value and a bell curve tends to emerge.

In a perfect normal curve, mean, median and mode all sit at the same point, right in the middle, and the curve is symmetric on both sides.

Alex Tutor

The standard normal curveA normal distribution with mean 0 and standard deviation 1. About 95% of the area sits between z = -2 and z = 2.

Chart data

z-score	Density
-3	0.004
-2.5	0.018
-2	0.054
-1.5	0.13
-1	0.242

Show 8 more rows

-0.5	0.352
0	0.399
0.5	0.352
1	0.242
1.5	0.13
2	0.054
2.5	0.018
3	0.004

You

Is that the 68-95-99.7 thing I've seen in a footnote somewhere?

Alex Tutor

True or false time, then we'll put a number on 'how many standard deviations away'.

Alex Tutor

Question 3

The empirical rule says that in a normal distribution, about 95% of values fall within 2 standard deviations of the mean.

TrueFalse

You

Which sampling method would actually give us a biased read on customer satisfaction, by the way? I want to make sure I don't repeat my app-only mistake.

Alex Tutor

Fair question — bias sneaks in whenever the chance of being picked correlates with the thing you're measuring. Sometimes it's obvious, sometimes it hides in something as innocent as timing.

You

Give me an example of the hidden kind?

Alex Tutor

Surveying only people who just contacted support: you'll oversample the frustrated and undersample the happily silent majority. Good instinct to double-check. Pick every biased option below — there might be more than one.

Alex Tutor

Question 4

Which of these sampling methods would give CatChow a biased read on customer satisfaction? Select all that apply.

Choose all that applySending the survey only through the mobile app's push notificationsRandomly selecting 10% of all customers from the full customer databaseSurveying only customers who submitted a support ticket in the past weekSelecting every 20th customer from a full, alphabetically sorted customer list

Alex Tutor

Back to the bell curve. A z-score tells you how many standard deviations a single value sits from the mean. It turns any normal variable into the same universal scale.

Alex Tutor

z = \frac{x - \mu}{\sigma}

z-score: subtract the population mean μ from the value x, then divide by the population standard deviation σ.

You

So a z-score of 0 means 'exactly average', and a z-score of 2 means 'two standard deviations above average'?

Alex Tutor

Exactly. Negative z-scores sit below the mean. Combine a z-score with the empirical rule and you instantly know roughly how unusual a value is.

Say CatChow's full population of orders has mean

\mu = \

50

and standard deviation

\sigma = $12

. Where does an order of \

74 sit?

Alex Tutor

Question 5

CatChow's full population of orders has mean $\mu = \$ 50 $and standard deviation$ \sigma = $12 $. What is the z-score of an order worth \$ 74?

Your answer

Units: z (e.g. 1.50)

Alex Tutor

Here's the part that makes hypothesis tests work at all. Imagine drawing thousands of different samples of 40 orders each and computing the mean of every one.

You

Those means would all be a bit different from each other, I guess?

Alex Tutor

Right — that spread of sample means is called the sampling distribution of the mean. And here's the beautiful part: whatever shape the original orders have, that distribution of means tends toward normal as the sample size grows.

You

Even if individual order values are skewed, with a long tail of big bulk orders?

Alex Tutor

Even then. That's the central limit theorem (CLT): sample means become approximately normal as

n

grows, regardless of the population's own shape. It's the reason means are so convenient to work with.

Alex Tutor

Question 6

The central limit theorem says the sampling distribution of the mean becomes approximately normal as sample size grows, even if the underlying population is skewed.

TrueFalse

Alex Tutor

One last piece. The spread of that sampling distribution has its own name: the standard error (SE). Don't confuse it with the standard deviation of the raw data.

Alex Tutor

SE_{\bar{x}} = \frac{s}{\sqrt{n}}

Standard error of the mean: the sample standard deviation s, divided by the square root of the sample size n.

You

So SE shrinks when the sample gets bigger? That feels like 'more data, less uncertainty about the mean'.

Alex Tutor

Precisely — and because of the square root, you need 4x the data to halve the standard error, not 2x. Diminishing returns on sample size, always worth remembering before you ask for 'just a bigger sample'.

One more number, then we wrap up.

Alex Tutor

Question 7

A sample of n = 36 orders has sample standard deviation s = $18. What is the standard error of the mean?

Your answer

Units: USD (e.g. 2.50)

Alex Tutor

Let's recap: population and sample, parameter and statistic, the normal curve, z-scores, and now standard error. That's the full toolkit for the rest of this module.

You

It's a lot of vocabulary for one morning. Can we test it on something real before I forget it all?

Alex Tutor

Happy to. Our head of growth walked past my desk this morning and said something that made me wince.

He said: 'Sample mean is

45, so the true average for every customer is

45.' In your own words, what's wrong with that sentence?

Alex Tutor

Question 8Short answer · AI-checked task

Our head of growth said: 'The sample mean is $45, so the true average order value for every customer is$ 45.' In 2-3 sentences, what's wrong with this, and what should we say instead?

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Nicely put. Sofia, you've got the vocabulary now — let's see you apply it to something messier than a one-line quote.

You

Messier how? Please tell me it's not another one of my own decisions coming back to haunt me.

Alex Tutor

Funny you should mention that. Remember the checkout redesign you shipped last week?

You

Oh no.

Alex Tutor

Question 9Case · AI-checked task

Review Sofia's launch-day claim below.

Sofia redesigned the checkout page. Right after launch she grabbed the order values of the first 8 customers who checked out, computed a mean of $52, and messaged the team: 'Redesign works — average order value jumped from $45 to $52!' Write a short critique before this goes into the board deck: is her conclusion justified? What would you want to see instead?

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Alex Tutor

That’s the lesson. You answered every task — nicely done.

Next lesson: The t-test