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Sampling Distribution — Definition, Formula & Examples

A sampling distribution is the probability distribution of a statistic (such as the sample mean or sample proportion) computed from every possible sample of a given size drawn from a population. It shows how that statistic varies from sample to sample due to random chance.

Let θ\theta be a population parameter and θ^\hat{\theta} be a statistic computed from a random sample of size nn. The sampling distribution of θ^\hat{\theta} is the probability distribution that describes all possible values of θ^\hat{\theta} across every possible sample of size nn from the population, together with their associated probabilities. By the Central Limit Theorem, when nn is sufficiently large, the sampling distribution of the sample mean Xˉ\bar{X} is approximately normal with mean μ\mu and standard deviation σn\dfrac{\sigma}{\sqrt{n}}, regardless of the population's shape.

Key Formula

Xˉ∼N ⁣(μ,  σn)\bar{X} \sim N\!\left(\mu,\; \frac{\sigma}{\sqrt{n}}\right)
Where:
  • Xˉ\bar{X} = Sample mean, the statistic whose sampling distribution is described
  • μ\mu = Population mean (center of the sampling distribution)
  • σ\sigma = Population standard deviation
  • nn = Sample size
  • σn\frac{\sigma}{\sqrt{n}} = Standard error of the sample mean

How It Works

Imagine you could draw every possible sample of size nn from a population, calculate a statistic (like the mean) for each sample, and then plot all those statistics on a histogram. That histogram is the sampling distribution. In practice, you never actually draw every sample — you rely on the Central Limit Theorem to predict the shape, center, and spread of the sampling distribution. The center equals the population parameter (making the statistic unbiased), and the spread shrinks as sample size increases. This predictable behavior is what lets you build confidence intervals and run hypothesis tests from a single sample.

Worked Example

Problem: A factory produces bolts with a mean length of 50 mm and a standard deviation of 4 mm. You take a random sample of 64 bolts. Describe the sampling distribution of the sample mean and find the probability that the sample mean exceeds 51 mm.
Identify the parameters: The population mean is 50 mm, the population standard deviation is 4 mm, and the sample size is 64.
μ=50,σ=4,n=64\mu = 50, \quad \sigma = 4, \quad n = 64
Find the standard error: Divide the population standard deviation by the square root of the sample size.
SE=σn=464=48=0.5\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{64}} = \frac{4}{8} = 0.5
State the sampling distribution: By the Central Limit Theorem (n = 64 is large enough), the sampling distribution of the sample mean is approximately normal.
Xˉ∼N(50,  0.5)\bar{X} \sim N(50,\; 0.5)
Calculate the z-score for 51 mm: Standardize the value 51 using the mean and standard error of the sampling distribution.
z=51−500.5=2.0z = \frac{51 - 50}{0.5} = 2.0
Find the probability: Using the standard normal table, the area to the right of z = 2.0 is approximately 0.0228.
P(Xˉ>51)=P(z>2.0)≈0.0228P(\bar{X} > 51) = P(z > 2.0) \approx 0.0228
Answer: The sampling distribution of the sample mean is approximately N(50,0.5)N(50, 0.5). The probability that the sample mean exceeds 51 mm is about 2.28%.

Another Example

Problem: A poll finds that 60% of voters in a large city favor a proposal. If you randomly sample 100 voters, describe the sampling distribution of the sample proportion p^\hat{p}.
Check conditions: Verify that both npnp and n(1−p)n(1-p) are at least 10 so the normal approximation applies.
np=100(0.6)=60≥10,n(1−p)=100(0.4)=40≥10np = 100(0.6) = 60 \geq 10, \quad n(1-p) = 100(0.4) = 40 \geq 10
Find the mean and standard error: The mean of the sampling distribution of p^\hat{p} is pp, and the standard error uses the proportion formula.
μp^=0.6,SE=p(1−p)n=0.6×0.4100=0.0024≈0.049\mu_{\hat{p}} = 0.6, \quad \text{SE} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.6 \times 0.4}{100}} = \sqrt{0.0024} \approx 0.049
State the sampling distribution: The sample proportion is approximately normally distributed.
p^∼N(0.6,  0.049)\hat{p} \sim N(0.6,\; 0.049)
Answer: The sampling distribution of p^\hat{p} is approximately N(0.6,  0.049)N(0.6,\; 0.049).

Visualization

Why It Matters

Sampling distributions are the backbone of statistical inference in AP Statistics and introductory college courses. Every confidence interval and hypothesis test relies on knowing how a statistic behaves across repeated samples. Data scientists, quality-control engineers, and medical researchers all depend on sampling distribution theory to draw valid conclusions from limited data.

Common Mistakes

Mistake: Confusing the standard deviation of the population (σ\sigma) with the standard error (σ/n\sigma / \sqrt{n}).
Correction: The standard error measures the spread of the sampling distribution, not the spread of individual data values. Always divide σ\sigma by n\sqrt{n} when working with sample means.
Mistake: Applying the normal approximation when sample size conditions are not met.
Correction: Before using the normal model, check that nn is large enough (e.g., n≥30n \geq 30 for means, or np≥10np \geq 10 and n(1−p)≥10n(1-p) \geq 10 for proportions). If conditions fail, the sampling distribution may not be approximately normal.

Related Terms