Sampling error decreases with the

Concept

Sampling error is the random difference between a sample estimate and the corresponding population value.

For the sample mean, the standard error is SE = σ/√n. Therefore, with population variability held constant, sampling error varies inversely with the square root of sample size.

Application

1. Let n denote the sample size and σ the population standard deviation.

2. Increasing n increases √n in the denominator of SE = σ/√n.

3. Therefore σ/√n decreases, so sampling error decreases with an increase in sample size.

Cross-check and contrast

If the sample size becomes four times as large, √n becomes twice as large and the standard error becomes half as large. This confirms the inverse-square-root relationship.

• A decrease in sample size provides fewer observations and moves the standard error in the opposite direction.

• An increase in sample size provides more observations and reduces the standard error under the same population variability.

• Randomization concerns how units are selected or assigned; it primarily controls bias and design validity rather than the sample-size term in the standard-error formula.

• Analysis concerns how collected observations are summarized or interpreted; it does not by itself alter the sample size.