Standard Error of the Mean

The standard deviation describes how much individual observations scatter; the standard error describes how much the average of n of them scatters, and it is smaller by a factor of √n. With σ = 15 and a sample of 25, the sample mean typically lands within 15/√25 = 3 units of the truth. The square root is the expensive part: quadrupling your sample only halves your error, which is why a poll of 400 is twice as sharp as one of 100 but a poll of 1,600 is needed to double it again. Every research budget eventually collides with this curve.

The classic mistake is reporting SE when you meant σ, or plotting error bars as ± SE and letting readers assume they cover the data. They do not: they cover the uncertainty in the mean, and they get narrower purely by collecting more data even though the underlying variability never changes. Run the relation backwards to plan a study: to pin a mean down to within 2.5 units when σ ≈ 20, you need n = (20/2.5)² = 64 observations.