Standard Error Calculator

Divide the sample standard deviation by the square root of the sample size: SEM = s / sqrt(n). For a sample of 100 with SD = 20: SEM = 20 / sqrt(100) = 20 / 10 = 2.0. This means the sample mean is expected to be within +/- 2.0 of the true population mean about 68% of the time, or within +/- 3.92 at 95% confidence.

Standard deviation (SD) measures how spread out individual data points are from the sample mean. Standard error (SEM) measures how precisely the sample mean estimates the population mean. SEM is always smaller than SD by a factor of sqrt(n). With n = 100 and SD = 20, the SEM is only 2.0. Use SD to describe data variability; use SEM to describe the precision of the mean.

Use SEM when your goal is to show the precision of the sample mean, which is typical for error bars on group means in scientific figures and clinical trial results. Use SD when your goal is to show how variable the individual measurements are, which is appropriate for descriptive statistics sections. Many journals require you to specify which one you are using.

SEM decreases as sample size increases, but with diminishing returns. Quadrupling n halves SEM. Going from n = 25 (SEM = SD/5) to n = 100 (SEM = SD/10) halves the error. Going from n = 100 to n = 400 halves it again. This is why studies use power analysis to find the minimum sample size needed for a target precision level.

There is no universal "good" SEM because it depends on the scale of your measurements. Instead, evaluate SEM relative to the mean. If the mean is 100 and SEM is 2 (2%), the estimate is precise. If SEM is 25 (25%), it is not. A useful rule: the 95% confidence interval (mean +/- 1.96 x SEM) should be narrow enough to be scientifically meaningful for your specific research question.

95% CI = mean +/- 1.96 x SEM. For a sample mean of 75 with SEM = 3: CI = 75 +/- 5.88, or (69.12, 80.88). The multiplier changes by confidence level: 1.0 for 68%, 1.645 for 90%, 1.96 for 95%, 2.576 for 99%. Wider confidence levels give wider intervals but more certainty that the true mean falls within.

Yes. If a paper reports "mean = 50, 95% CI (47.1, 52.9)", the margin of error is 2.9. Since 95% CI uses Z = 1.96: SEM = margin / 1.96 = 2.9 / 1.96 = 1.48. From SEM you can recover n if you also know SD: n = (SD / SEM)^2.

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