How much can you trust an average?

A coffee shop wants to know: how long do customers wait for their order, on average?

They can't time every customer, so they time a sample.

They time 30 customers. Each dot is one customer's wait.

The number of observations in a sample is called n, so here n = 30. Some customers wait a short time, others much longer.

Averaging the 30 waits gives the sample mean, written x:

x = min

It's our best guess for the true average wait, μ (“mu”), which we don't know.

How much should we trust ? If they had timed a different 30 customers, would they get again? Almost certainly not.

x depends on which customers happened to be in the sample. To know how far off our guess might be, we need to know how much x changes from sample to sample.

Another 30 customers: x = min.

Same shop, same process, a different answer. The sample mean is itself random.

Imagine repeating this 20 times: 20 samples of 30 customers, each with its own mean.

(Scroll the strip sideways to see them all.)

Now keep only the means and stack them up.

This pile is the sampling distribution of the sample mean: the values x tends to take.

The means cluster around the true average, μ = 5 min.

Single samples miss, but not consistently too high or too low.

How spread out are the means? Their standard deviation is about min (≈ s).

That's the standard error (SE): how far a typical sample mean lands from the truth.

Compare it with individual customers, whose waits vary by about σ = 1 min (the faint grey bracket). Averages vary much less than individuals.

Statisticians have a shortcut:

SE = σ / √n = 1 / √30 ≈ min

Instead of drawing many samples, it predicts the standard error from just two numbers: σ, how much individual waits vary (1 min), and n, how many customers are in each sample (30). Dividing by √n captures that averaging more customers cancels out more of the ups and downs.

That's close to what we measured (), but not quite the same. Why?

With only 20 samples, our pile is a rough sketch. Let's draw more samples: 50, 100, 200…

The pile fills into a smooth bell, and the measured spread settles at ≈ min, matching the formula.

More samples didn't change the standard error. They gave us a more precise measurement of it.

What if each sample had more customers?

With only n = 5 customers per sample, the means are widely spread: SE ≈ min (≈ s).

With n = 30: SE ≈ min (≈ s).

With n = 100: SE ≈ min (≈ s).

Bigger samples give more reliable means. But because of the square root, timing 4× as many customers only halves the SE.

In real life, the coffee shop has just one sample and one x. It can't see this pile.

But now we know what it looks like: their x is one of these values, usually within about 1 SE of the true average, and almost never more than 2–3 SE away.

So the standard error tells you how much to trust a single sample mean. A small SE means your one number is probably close to the truth.

Your turn. Change the number of samples and n under the sample cards, and watch the standard error change.

Hover a card to follow its mean, or sort the cards by x.