13 Research methods, statistics, critical review and evidence-based practice13.8 Descriptive statistics
13.8.6 Sampling variability and standard error
2 min read1 figure
Written by Resident Doctor Committee · Published
Sampling variability
Sampling variability is chance variation between estimates drawn from different samples of the same population.
Two random samples of 25 from patients whose true mean depression score is 24 might give means of 22.6 and 25.9.
The standard error of the mean (SEM) is the standard deviation of the sampling distribution, which is the spread of sample means across repeated samples.
The SEM measures how precisely a sample mean estimates the population mean, and it is usually estimated from a single sample.
Standard deviation versus standard error
Measure | What it describes | Kind of statistic |
|---|---|---|
Standard deviation (SD), variance, range, interquartile range | Scatter of individual observations | Descriptive: speaks only about the observed sample |
Standard error | Precision of the sample estimate | Inferential: speaks about the population through sampling variability |
Calculating the standard error
SEM = SD / √n, where n is the sample size.
Rearranged, SD = SEM × √n, so a reported SEM of 0.8 with n = 100 gives an SD of 8.
The SEM is smaller than the SD whenever n exceeds 1, and the two are equal only at n = 1.
Sample size and precision
Sample size | SD | SEM |
|---|---|---|
25 | 8 | 1.60 |
100 | 8 | 0.80 |
400 | 8 | 0.40 |
The SEM falls in proportion to √n, so a fourfold increase in n halves the SEM.
A larger random sample does not change the SD, the shape of the distribution or systematic error, so more responses gathered through a biased process improve precision without removing bias.
A confidence interval is the estimate plus or minus a critical value times the standard error, so a larger sample narrows it at the same confidence level.
A mean of 24 with an SEM of 0.80 gives an approximate 95% confidence interval of 22.4 to 25.6.
When a trial grows larger with unchanged event rates, the point estimate of treatment effect stays the same and its confidence interval narrows.
In a case-control study where the exposure is rare, few participants are exposed, so estimates are imprecise and a much larger sample is needed for the power to detect a meaningful effect.