A statistic is a number squeezed out of a sample — and a different sample would have given a different number. So the statistic is itself a random quantity, and random quantities have distributions. Sample after sample, plot the statistic you get, and a shape appears: the sampling distribution. It is not the population, and it is not the data from any one sample. Build one below, one dot at a time.
Adult male height — sample mean, n = 9
populationthis sample's valuesone dot = one sample's statisticnewest dotwhat the population says the centre should be
Adult male heightμ = 69 in · σ = 3 in
1 · Pick a population to sample from
2 · Pick the statistic you compute from each sample
n = 9
3 · Draw samples and watch the pile build
Samples drawn0dots in the pile
Latest value—
Centre of pile—
Spread of pile—
Three distributions, and students confuse them constantly.(1) The population distribution — how the individual values are spread out. Its centre is μ and its spread is σ, and they do not budge no matter what you do. (2) The distribution of one sample — the n values you actually collected, sitting in the middle strip. Draw again and you get n different values. (3) The sampling distribution — the spread of the statistic across all the samples you could have taken. Only the third one is new here, and only the third one changes when you change n.
Four things this picture is quietly telling you.A sampling distribution is a theoretical object — the distribution of the statistic over every sample that could be drawn. The dots are a simulation that approximates it; 20 dots barely sketch it, 1,000 dots trace it closely, but no finite pile is it. It belongs to a statistic, not to data — change the statistic and the shape can change entirely, as the maximum shows. It depends on n, which is why the pile clears whenever you move the slider: a new n is a different distribution, not more of the same one. And in real work you get one sample and one dot — the whole reason for this theory is to know what the rest of the pile would have looked like without ever having to draw it.