Introduction
A statistic calculated from a sample is not fixed in advance: a different sample can produce a different value. To understand how much a statistic can vary, imagine applying the same sampling procedure again and again, always using the same sample size. The resulting distribution of statistic values describes the statistic’s sampling variability.
This idea is distinct from looking at the population itself or at the measurements in one particular sample. Keeping those three things separate is essential: the population supplies the values, a sample gives one set of observed data, and the sampling distribution describes how a statistic would vary across samples drawn by a specified procedure.
Three Distributions That Should Not Be Confused
A population distribution describes the values of a variable across the population. For example, it might describe the heights of all plants in a greenhouse. A sample distribution describes the observed values in one particular sample: the heights of the plants actually selected. A sampling distribution, by contrast, describes a statistic calculated across all samples that could have been selected using the stated procedure.
The sampling distribution is about a statistic, not about the raw observations. If the statistic is the sample maximum, each possible sample contributes one maximum to the distribution. The same sample could also be used to calculate a different statistic, such as its range, which would have a different sampling distribution.
In mathematical notation, let \(T\) be a statistic calculated from a sample of size \(n\). The sampling distribution is the probability distribution of \(T\) under the sampling procedure. A statistic is a function of sample data; its value can vary from sample to sample even when the population and procedure stay the same. A population parameter, in contrast, describes the population and does not change just because a different sample is selected.
In a finite setting, the distribution can be constructed by listing the possible samples, finding the statistic for each, and assigning probabilities according to the sampling procedure. When different samples are equally likely, a statistic value’s probability is the number of samples producing that value divided by the total number of possible samples. If the procedure gives samples unequal probabilities, those probabilities must be used instead.
Worked Example: The Sampling Distribution of a Sample Maximum
Consider a hypothetical population consisting of four values: 2, 4, 8, and 10. Select a simple random sample of size \(n=2\) without replacement. Let \(M\) be the maximum value in the sample. What is the sampling distribution of \(M\)?
There are six equally likely samples of size 2:
| Sample | Maximum \(M\) |
|---|---|
| {2, 4} | 4 |
| {2, 8} | 8 |
| {2, 10} | 10 |
| {4, 8} | 8 |
| {4, 10} | 10 |
| {8, 10} | 10 |
Because each of the six samples has probability \(1/6\), \(M=4\) occurs for one sample, \(M=8\) for two samples, and \(M=10\) for three samples. Therefore:
These probabilities sum to 1, as a probability distribution must. Notice that \(M\) cannot be 2 in this sampling setup: a sample of size 2 drawn without replacement from these values must contain a value larger than 2. The distribution describes the maximum across possible samples, not the distribution of the four population values and not the values in one particular sample.
How the Sampling Procedure Determines Probabilities
The statistic alone does not determine its sampling distribution. The sample size and the method of selecting observations matter too. Even with the same population and the same statistic, changing from sampling without replacement to sampling with replacement can change the probabilities of possible statistic values. So can changing how likely different units are to be selected.
The probabilities in a sampling distribution refer to the chance of obtaining each statistic value when the stated sampling procedure is carried out. They do not refer to uncertainty about whether a fixed population parameter has a particular value. The procedure is part of the model: the distribution is meaningful only in relation to the sampling plan it describes.
Worked Example: A Different Statistic From the Same Samples
Use the same hypothetical population, {2, 4, 8, 10}, and the same simple random samples of size 2 without replacement. This time let \(R\) be the sample range, defined as the largest observed value minus the smallest observed value. Find the sampling distribution of \(R\).
Calculate the range for each of the six equally likely samples:
| Sample | Range \(R\) |
|---|---|
| {2, 4} | \(4-2=2\) |
| {2, 8} | \(8-2=6\) |
| {2, 10} | \(10-2=8\) |
| {4, 8} | \(8-4=4\) |
| {4, 10} | \(10-4=6\) |
| {8, 10} | \(10-8=2\) |
Two of the six samples have range 2, one has range 4, two have range 6, and one has range 8. Thus:
The possible samples and their selection probabilities have not changed from the previous example, but the statistic has. As a result, the distribution is different. For instance, the sample {2, 10} contributes a maximum of 10 to the earlier distribution and a range of 8 to this one. A sampling distribution is always the distribution of a specified statistic, not a general distribution attached to a population or a sample size alone.
A Justification From the Possible Outcomes
The construction works because every possible sample produces exactly one value of the statistic. If samples form an exhaustive set of mutually exclusive outcomes, and each sample has a probability assigned by the sampling procedure, then the probability of a statistic value is the total probability of the samples that produce that value. In symbols:
Here \(T(s)\) means the statistic calculated from sample \(s\), and \(P(s)\) is the probability that the sampling procedure selects that sample. Each possible sample contributes its probability to exactly one statistic value. Since the probabilities of all possible samples sum to 1, adding the probabilities for all possible values of \(T\) also gives 1. If all samples are equally likely, each \(P(s)\) is the reciprocal of the number of possible samples, which yields the counting method used in the first two examples.
This also explains why the same statistic value can receive probability from several different samples. The statistic may compress or summarize the data, so distinct samples can lead to the same result. The probability of that result includes every such sample, not just one representative sample.
Worked Example: The Minimum of Two Die Rolls
Roll a fair six-sided die twice, independently, and let \(L\) be the smaller of the two results. This is a hypothetical probability model with 36 equally likely ordered outcomes: each roll can be any of 1 through 6. Find the sampling distribution of \(L\), treating the two rolls as a sample of fixed size 2 from the die-roll process.
For \(L\) to equal \(k\), both rolls must be at least \(k\), and they cannot both be at least \(k+1\). The probability that both rolls are at least \(k\) is \(((7-k)/6)^2\), because there are \(7-k\) eligible faces on each independent roll. Similarly, the probability that both are at least \(k+1\) is \(((6-k)/6)^2\). Subtracting the second event leaves precisely the outcomes whose minimum is \(k\):
Substitution gives:
The probabilities sum to \((11+9+7+5+3+1)/36=36/36=1\). A single die roll has a uniform population distribution over its six faces, but the minimum of two rolls does not: small values are more likely to appear as the minimum than large values. This difference illustrates why a statistic’s sampling distribution cannot be substituted for the distribution of the original observations.
What a Sampling Distribution Tells Us
A sampling distribution describes the statistic’s behavior under repeated use of the same sampling plan. It lets us ask questions such as how likely a statistic value is, which values are more or less common, and how much the statistic varies across samples. In later work, these features help explain the precision of estimates and the behavior of statistical procedures.
The phrase “repeated samples” describes the probability model, not a requirement that the same researcher actually collect an enormous number of samples. In some small finite examples, every possible sample can be listed. In other settings, a theoretical model describes the possibilities. In either case, the sampling distribution is defined by the sampling process, not by the number of samples a person happened to observe.
A particular data set gives only one realized sample and one realized statistic. The sampling distribution provides the broader reference for understanding how that statistic could have turned out under the same conditions. Its probabilities are conditional on the specified population model and sampling procedure; if either changes, the distribution may change as well.
Common Mistakes
- Calling the observations in one sample a sampling distribution. A list of measured values describes one sample. A sampling distribution requires considering the statistic across possible samples.
- Confusing the population distribution with a statistic’s distribution. The population distribution concerns individual values; the sampling distribution concerns a statistic computed from a fixed-size sample.
- Leaving out the statistic or sampling plan. “The sampling distribution for samples of size 10” is incomplete unless the statistic and sampling procedure are also specified.
- Thinking that a statistic must have the same value in every sample. The formula for the statistic stays the same, but its value can change when the sample changes.
- Treating a probability as a claim about a fixed parameter. A probability such as \(P(T=t)\) describes the result of the modeled sampling process, not the probability that a fixed population parameter equals \(t\).
Check Your Understanding
Try these questions without looking back.
- In one sentence, define a sampling distribution and name the information needed to specify one.
- A population contains the values 1, 3, 5, and 7. Explain how the population distribution differs from the distribution of sample values in one sample, and from a sampling distribution of a statistic.
- Why can two different samples contribute probability to the same value of a statistic?
- If the statistic and population stay the same but the sampling procedure changes, why might the sampling distribution change?
- Suppose a statistic has probability \(0.2\) of equaling 4 under a stated sampling plan. What does that probability describe, and what does it not describe?