Day 2 · Interactive

Sampling distributions, the standard error, and the central limit theorem

The lecture's fish example, live. A lake holds fish whose lengths have mean 300 mm and standard deviation 50 mm. You catch a sample of n fish and compute their mean length. Do it again and you get a different mean. Do it hundreds of times and the means pile up into a sampling distribution. Its spread is the standard error, and its shape is the central limit theorem at work.

Every population shape has the same mean (300 mm) and standard deviation (50 mm), so only the shape changes. Changing the shape or n starts a fresh set of samples.

1The population: every fish in the lake
2One sample: the 10 fish you just caught
Sample mean $\bar x$—your estimate of the lake's mean, 300
Sample standard deviation $s$—spread of individual fish; estimates $\sigma$ = 50 at any n
Estimated standard error $s/\sqrt{n}$—how far $\bar x$ typically sits from 300, estimated from this one sample
3The sampling distribution: the means of every sample so far
Samples drawn0
Mean of the sample means—lands on the population mean, 300
SD of the sample means = standard error—the theory says $\sigma/\sqrt{n}$ =
Show the sample means as a table

Standard deviation vs standard error

Panels 2 and 3 show two different spreads, and the whole lesson is keeping them apart.

  • The standard deviation of a sample, $s$, measures how much individual fish differ from one another. It estimates the population's $\sigma = 50$ mm, and it does not shrink as you catch more fish: bigger samples just estimate it better.
  • The standard error, $\sigma/\sqrt{n}$, measures how much sample means differ from one sample to the next. It is the standard deviation of the sampling distribution in panel 3, and it does shrink as $n$ grows: 15.8 mm at $n = 10$, 11.2 mm at $n = 20$, 5.0 mm at $n = 100$.

In practice you only ever catch one sample, so you never see panel 3. You estimate the standard error from the one sample you have, as $s/\sqrt{n}$ (the orange box in panel 2). Watch it bounce around the true value as you draw new samples: it is itself an estimate. This is the "Std. Error" column that summary(lm(...)) reports for every coefficient.

$\text{mean of the } \bar x\text{'s} = \mu, \qquad \text{SD of the } \bar x\text{'s} = \text{SE} = \dfrac{\sigma}{\sqrt{n}}$

Things to try

  1. The lecture's example. With the Normal population, press n = 10, then Draw 100 samples. Note the SD of the sample means. Repeat at n = 20 and n = 100. The means stay centred on 300 while their spread shrinks toward 15.8, 11.2 and 5.0 mm.
  2. Quadruple the data, halve the error. Compare n = 25 and n = 100. Four times as many fish only halves the standard error (10 mm to 5 mm), because it shrinks with $\sqrt{n}$, not $n$.
  3. The sample SD does not shrink. Draw single samples at n = 10 and at n = 100 and watch $s$ in panel 2. It hovers around 50 either way, just less erratically at n = 100. Only the standard error shrinks.
  4. The central limit theorem. Choose Right-skewed, set n = 2, and keep drawing: the sample means are still skewed. Move to n = 5, then n = 30, tick Zoom, and watch the histogram become the Normal curve even though the population is nothing like Normal. Try it with Two groups and Flat too. This is the slide's rgamma exercise: a sum (and so a mean) of many draws from almost any distribution is approximately Normal.
  5. A Normal population needs no CLT. With the Normal population the sample means are Normal even at n = 2, as the last slide on the theorem says.

Next: when the Normal approximates a Binomial or Poisson, and the continuity correction · ← Back to the Day 2 lab