Sampling Distributions and Standard Error

A single sample gives one estimate, such as one sample mean. Draw another sample, and the mean changes slightly. The sampling distribution is the distribution of that estimate across all the samples you could have drawn.

Standard Error

SE = σ / √n

• SE: the standard error, which is the standard deviation of the sample mean. It shows how much the mean would typically vary from sample to sample.
• σ: the standard deviation of the population. When it is unknown, we use the sample standard deviation, s, instead.
• n: the sample size.

Standard deviation describes how spread out individual values are. Standard error describes the spread of the estimate. Because n sits under a square root, quadrupling the sample size only halves the standard error.

Example: Delivery times have a standard deviation of 12 minutes. Find the standard error of the mean for samples of 36 and 144 deliveries.

1. For n = 36: SE = 12 / √36 = 12 / 6 = 2 minutes
2. For n = 144: SE = 12 / √144 = 12 / 12 = 1 minute

Four times as much data cuts the uncertainty in the average by half. This trade-off helps decide how much data is worth collecting.