Samples and sample size

Five random heights from N(170, 6.28) can give a standard deviation of 3.4 cm; five thousand give a value within a few hundredths of 6.28.

In practice we almost never see a whole population. We see a sample, and the histogram of a sample is only a rough copy of the population's distribution. How rough depends mostly on one number, the sample size n. This unit uses the lab to draw random samples of any size from a known population, so you can see the copy sharpen as n grows.

Small samples are bumpy

Start the lab with the normal population of heights, N(170, 6.28), and a sample of 20. Press New sample a few times. The histogram jumps around from one sample to the next; one time it leans left, the next it has a gap in the middle, and the sample mean x̄ and standard deviation s move too. In one simulated run made for this page (random seed 7), the same population gave x̄ = 167.72 and s = 3.37 for n = 5, x̄ = 171.09 and s = 5.74 for n = 20, and x̄ = 170.16 and s = 6.92 for n = 100. None of these is wrong. They are what random samples of that size do.

Large samples settle down

Raise n to 1,000 and then 5,000. The bars hug the red curve, and x̄ and s stop wandering: in the same run, n = 1,000 gave x̄ = 170.10 and s = 6.11, and n = 5,000 gave 169.84 and 6.29. How far x̄ typically sits from μ shrinks like σ ÷ √n. For σ = 6.28 that is about 2.81 cm for n = 5, 1.40 cm for n = 20, 0.63 cm for n = 100 and 0.20 cm for n = 1,000. To halve the typical error you need four times as many values. The course on probability looks at this square-root law in detail with the sampling distribution of the mean.

The 68-95-99.7 rule needs a normal shape

The panel also reports what share of the sample lies within one, two and three standard deviations of x̄. For the normal population and a large sample these settle near 68%, 95% and 99.7%. Now switch the population. For the uniform population from 0 to 10, σ = 10 ÷ √12 ≈ 2.89 and only about 57.7% of values lie within one σ of the mean, while all of them lie within two. For the right-skewed waiting times with mean 5 minutes, the population shares within 1, 2 and 3 σ are about 86.5%, 95.0% and 98.2%. A large sample copies whatever shape the population has; it does not turn a skewed population into a bell.

Why the vertical axis says density

The sample lab draws each bar with height equal to its relative frequency divided by the class width. That makes the total area of the bars equal to 1, the same as the total area under the population curve, so the bars and the curve can sit on one scale whatever the sample size. A count axis would make the bars for n = 5,000 a thousand times taller than for n = 5 and the comparison with the curve would be lost.