Three Distributions That Are Easy to Confuse
- Distinguish the population distribution, the distribution of one sample, and the sampling distribution
- Identify which of the three a given statement or graph refers to
- Explain what each distribution is used for in inference
Three different things, three different pictures
The population distribution describes the values of the variable for every individual in the population. The distribution of one sample describes the values in the particular sample you collected — a histogram of your data. The sampling distribution describes something entirely different: the values a statistic would take over all possible samples of a given size. Its individuals are not people; they are samples. Almost every conceptual difficulty in inference comes from sliding between these three without noticing.
What each one is for
The population distribution is what you would like to know and almost never see. The sample distribution is what you actually observe, and it is used to estimate the population's shape, center and spread. The sampling distribution is what makes inference possible: it tells you how much a statistic bounces around from sample to sample, which is exactly what a margin of error or a p-value quantifies. You never observe a sampling distribution either — it is a theoretical object, described by theory rather than collected.
Shape, center and spread of each
For a sample mean drawn from a population with mean μ and standard deviation σ: the sampling distribution of x̄ is centered at μ — the sample mean is an unbiased estimator — with standard deviation σ/√n, which shrinks as n grows. Its shape approaches Normal as n grows regardless of the population's shape, which is the Central Limit Theorem. Note what does not shrink: the standard deviation of the population and of the sample are both about σ no matter how large n is. Only the sampling distribution narrows.
A question that says "the distribution of sample means" or "if we repeated this many times" is about the sampling distribution. A question that says "the heights of the 40 students in the sample" is about the sample. Reading which one is being asked about is often the whole item.
A population of package weights is strongly right-skewed with mean 12.0 kg and standard deviation 3.0 kg. Samples of size 36 are taken repeatedly. Describe the shape, center and spread of (i) the population, (ii) a typical single sample, and (iii) the sampling distribution of x̄.
- 1.Population: strongly right-skewed, mean 12.0 kg, standard deviation 3.0 kg. These are fixed facts about the population.
- 2.A single sample of 36: its histogram should look roughly like the population — right-skewed, mean near 12.0, standard deviation near 3.0. A sample resembles what it came from.
- 3.Sampling distribution of x̄: centered at μ = 12.0 kg, with standard deviation σ/√n = 3.0/√36 = 3.0/6 = 0.50 kg.
- 4.Its shape is approximately Normal by the Central Limit Theorem, since n = 36 is reasonably large — even though the population is skewed.
As the sample size increases, which of the following decreases?
The phrase "standard error" is just the standard deviation of a sampling distribution, estimated from data. When you see s/√n in a t procedure, that is the standard error of the mean — a measure of how much x̄ varies from sample to sample, not how much the data vary.
A histogram of the 40 measurements a researcher collected is a picture of —
Answer the 2 checkpoints as you read.
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