Sample Size for Estimating a Mean

How many observations are needed?

The goal is to estimate a population mean within a chosen margin of error. More variable observations require larger samples. Halving the margin requires roughly four times as many observations. The calculation uses n = (zσ/E)² and rounds upward.

Planning limitations

Standard deviation may come from earlier measurements in the same units. Such a planning estimate is not a guarantee. A deviation of 10, margin of 2 and confidence of 95 percent require 97 usable observations. This calculator does not estimate test power or survey proportions. Missing observations, clustering and measurement bias require separate study design decisions.

Defining the margin correctly

The margin is not the permitted error of an individual observation. It is half the width of the confidence interval around the estimated mean. Express standard deviation and margin in the same units. Entering deviation in mm and margin in cm without conversion changes the sample size substantially. A relative percentage error must first be converted into the units of the measured quantity.

Suppose standard deviation is assumed to be 2 mm and the mean is to be estimated within 0.5 mm at 95 percent confidence. The normal distribution gives a two sided critical value of approximately 1.959964. The unrounded size is (1.959964 × 2 / 0.5)² ≈ 61.4633. Rounding upward requires 62 usable observations. Rounding down would not preserve the requested margin. The tool also displays the margin achieved with the rounded count.

The standard deviation here is a planning assumption, not a recommendation for every measurement process. Previous data should represent the same measurement method and similar variability. If actual variability is larger, the planned count may be insufficient. Unusable records do not count toward the required observations. Collecting more data alone does not remove selection bias. This estimate must not be confused with a power calculation for detecting a difference.