Module 05 · Sampling distributions
Repeat the sampling process to see why estimates vary, what a standard error measures, and how sample size changes precision.
Suppose a retailer wants the mean amount spent per order. The population distribution describes spending across all orders in the target population. One random sample of 25 orders produces 25 observations and one sample mean. Another sample will usually produce a different mean, even when the sampling procedure is unchanged.
The sampling distribution of the mean describes the values of x̄ across all possible repetitions of that procedure. Each point in it is an entire sample summarized by one number. It is therefore different from a histogram of the individual orders in your actual dataset. In the lab, we approximate this distribution by drawing many independent samples and retaining their means.
Choose a population shape, then compare a few observations per sample with many. All populations have mean 50 and standard deviation 10.
Bars are a simulated density of sample means. The line is the matching normal reference; it is exact for the normal population and an approximation otherwise. The number of repetitions controls simulation noise.
The center remains near 50. Averaging 5 observations reduces the theoretical spread from 10 to 4.472. Compare the bars with the normal reference to assess the approximation at this sample size.
With few observations per sample, the right-skewed population still produces a noticeably skewed sampling distribution.
1000 independent samples of 5 observations from the exponential model give a mean of sample means of 50.130. The theoretical standard error is 4.472; the empirical standard error is 4.609.
The population standard deviation σ describes variation among individual orders. The standard error of x̄ describes variation among sample means. For independent observations with a common finite variance, E(x̄) = μ and SE(x̄) = σ / √n. Averaging reduces random variation without changing the center.
The square root matters. Doubling n divides the standard error by √2, which is about a 29% reduction. Halving the standard error requires four times as many observations. In practice σ is usually unknown, so we estimate the standard error using s / √n. The simulator knows the population model and can show the theoretical value separately from a simulation estimate.
For independent and identically distributed observations with finite variance, the standardized sample mean approaches a standard normal distribution as n grows. The original observations do not have to be normal. A skewed population can produce a much more symmetric distribution of averages.
If the population is normal, the sample mean is exactly normal for every n. Otherwise, the quality of the normal approximation depends on the population shape and sample size. There is no universal cutoff at n = 30. Strong skewness or rare extreme observations may require larger samples. Increasing the number of simulation repetitions draws a smoother histogram; increasing the size of each sample changes the sampling distribution itself.
Consider an illustrative order-value model with population mean $50 and standard deviation $10. For samples of 25 independent orders, the mean remains $50 and the standard error is 10 / √25 = $2. For samples of 100, it becomes 10 / √100 = $1.
If order values follow a normal population model, P(48 ≤ x̄ ≤ 52) is about 68.27% for n = 25 because these limits lie one standard error from the mean. With n = 100 the same limits lie two standard errors away, giving about 95.45%. For a nonnormal population these probabilities are approximations whose quality must be assessed. Neither statement means that 95.45% of individual orders lie between $48 and $52.
Every population model below has μ = 50 and σ = 10. The right-skewed model is a shifted exponential distribution, the uniform model has equally likely values over a finite interval, and the normal model is symmetric. This common center and spread make changes in shape easier to compare.
The bars show simulated sample means; the line shows the normal distribution with mean 50 and standard deviation 10 / √n. The empirical standard error is the sample standard deviation of the simulated means. It should be close to the theoretical standard error, but finite simulation introduces Monte Carlo variation. The seed makes each setting reproducible. A larger n reduces random sampling variation; it does not repair a biased sampling frame, nonresponse, dependence, or incorrect measurement.
Try each question before revealing the solution.
For n = 36, SE = 18 / 6 = 3. For n = 144, SE = 18 / 12 = 1.5. Quadrupling sample size halves the standard error. The population standard deviation stays 18.
The theoretical standard error depends on the size of each sample, n. More repetitions improve our numerical approximation to the same sampling distribution. They do not change how variable one sample mean is.
No. The standard error quantifies random variation under the sampling model. Excluding a potentially different customer group can create selection bias. Additional domestic orders do not correct that coverage problem.
Further reading