A new sample produces a new mean, but those means do not vary without structure.
A hospital wants to estimate average patient waiting time. Instead of measuring every patient, it samples 50 patients and calculates a mean.
| Sample | Patients | Mean wait |
|---|---|---|
| 1 | 50 | 14.2 min |
| 2 | 50 | 16.1 min |
| 3 | 50 | 15.4 min |
A different sample gives a different mean. The key question is how those means vary across repeated samples.
Key question
What pattern appears when sample means are calculated again and again?
A distribution of statistics
Draw many samples of the same size from the same population and calculate the mean each time. The resulting collection forms the sampling distribution of the mean.
Repeated samples of 50 patients
Sample 1
14.2 min
Sample 2
16.1 min
Sample 3
15.4 min
Sample 4
13.9 min
Sample 5
15.8 min
Sample means cluster around a center
Repeated estimates form a distribution of their own.
This is not the distribution of individual waiting times. It is the distribution of means calculated from repeated samples.
Why does a bell shape appear?
Some samples produce means above the population mean and others below it, but repeated sample means tend to center on the true population mean.
As sample size grows, the sampling distribution of the mean often becomes approximately normal even when the original data are not perfectly normal. This is the central limit theorem.
Central limit theorem
Larger samples make the distribution of sample means more nearly normal and more tightly concentrated around the population mean.
What sample size changes
| Sample size | Sampling distribution | Interpretation |
|---|---|---|
| n = 5 | Wide and irregular | Means differ substantially |
| n = 30 | Smoother and narrower | Means are more stable |
| n = 200 | Very concentrated | Means stay close to the population mean |
The spread of the individual observations does not disappear. What shrinks is the variation of the sample mean.
When does the CLT help?
Symmetric data without severe outliers may behave well with smaller samples. Strongly skewed outcomes such as income, medical cost, or transaction value may require larger samples.
Two different distributions
Data distribution: waiting times for individual patients
Sampling distribution: average waiting times from repeated samples
Draw repeated samples
Change the sample size to 5, 30, or 100 and watch the histogram of sample means.
The underlying data is a heavily right-skewed session length. 1,200 samples are drawn from it, and one statistic is computed each time.
Centre
4.00
Standard error
0.734
Skewness
0.47
At n = 30 the sampling distribution of the mean is far less skewed than the raw data (skewness 0.47). This is the textbook case the central limit theorem covers, and raising n brings it closer to normal still.
Every statistic has a sampling distribution. Not every sampling distribution is normal.
Sampling distributions make it possible to quantify uncertainty from a single observed sample.