Sample std dev (s)

Sample size (n)

SEM = s / √n. Smaller SEM means the sample mean is a more precise estimate of the population mean.

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How sample size cuts the standard error of the mean — Quadrupling the sample from 36 to 144 halves SEM: 2.0 to 1.0.

Method & verification

How This Tool Works

Calculation Method

Computes Standard error (SEM) from Standard deviation and Sample size.

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BoringToolsKit methodology and verification standard

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Worked example: survey time per task

Suppose 36 workers averaged 45 minutes per task and the sample standard deviation is 12 minutes. The standard error is the standard deviation divided by the square root of the sample size: 12 / sqrt(36) = 12 / 6 = 2.0 minutes. That single number says how much a 36-person sample of this size would typically drift from the true population mean, independent of where this particular sample's mean happened to land.

From there the familiar 95% band is 45 +/- (1.96 x 2.0) = 45 +/- 3.92, or roughly 41.08 to 48.92 minutes. Notice the square root at work: cutting the error in half requires quadrupling the sample. Going from 36 to 144 workers drops the standard error from 2.0 to 12 / sqrt(144) = 1.0, while going from 36 to 72 only drops it to 12 / sqrt(72) = 1.414. Doubling n buys far less than most people expect.

When the normal shortcut understates your uncertainty

The 1.96 multiplier in the result panel assumes the estimate of the standard deviation is itself reliable. With small samples it is not, and the honest multiplier comes from a t distribution. Take n = 5 and s = 20: the standard error is 20 / sqrt(5) = 8.9443, so the calculator shows a 95% margin of 1.96 x 8.9443 = +/- 17.53. Using the t value for 4 degrees of freedom (2.776) gives 2.776 x 8.9443 = +/- 24.82 instead, about 42% wider.

As a rule of thumb the two agree within 1% once n reaches about 2,500; at n = 30 the t multiplier is 2.045 versus 1.96, roughly a 4% underestimate. The calculator is also blind to how the sample was gathered: data from 400 rows in one week and 400 rows spread over a year can produce the same standard error while meaning very different things if there is any trend or clustering in the underlying population. If the data are not a plausible random sample, shrink your confidence in the interval even though the arithmetic printed is exact.