The standard error of the mean from raw data, an SD and n, or a count, plus the reverse: SD from a reported SEM or from a 95% confidence interval, and the sample needed for a target precision.
Standard Error Calculator - SEM, SD and CI Conversions
The standard error of the mean from raw data, an SD and n, or a count, plus the reverse: SD from a reported SEM or from a 95% confidence interval, and the sample needed for a target precision.
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Standard error of the mean, from data, a report or a published interval
Divide the standard deviation by the square root of the sample size and you have the standard error of the mean: 15 / √36 = 2.50. This standard error calculator does that from a pasted list, from a standard deviation and n, or from a count of successes for a proportion. It also works backward, which is what most people searching for SEM actually need: it turns a reported standard error or a published 95% confidence interval into the standard deviation of single values. An optional target tells you how many observations bring the standard error down to the precision you want, and an optional population size applies the finite population correction.
Problem: bars on a chart that nobody labeled
A bar chart shows a mean of 103 with error bars. If the bars are one standard deviation, they reach from 88 to 118 and describe how far single measurements spread. If they are one standard error from 36 measurements, they reach from 100.5 to 105.5 and describe how much the mean itself would wobble between samples. Same data, bars six times apart in length, because √36 = 6. Journals print both, often with nothing more than "±" in the legend, and readers routinely treat a small SEM as a small spread.
The confusion runs the other way too. A meta-analysis or a sample size plan needs the standard deviation, but the paper reports "mean ± SEM" or only a 95% interval. Plugging the standard error in where the standard deviation belongs makes the data look far more consistent than it was, and a study planned on it ends up badly underpowered.
Five entries, and the first one decides which boxes appear
- Pick what you have: raw measurements, a standard deviation with n, successes out of a sample, a standard error from a report, or a confidence interval from a paper.
- Type the numbers in plain digits with a dot for decimals. The sample mean is optional where it appears; with it, the result prints ranges and a relative standard error instead of bare half-widths.
- For an interval, give both ends, the group size and the printed confidence level, so the right t or z divisor is used.
- Optionally add the population size (for a sample that is a large share of a finite group) and a target standard error.
- Read the headline, then the table that sets mean ± SD, mean ± SE and the 95% interval side by side.
Solution: keep three plus-or-minus numbers apart
Every result prints the same sample three ways, so the difference is on the screen rather than in a footnote. For the chart above, with a mean of 103, a standard deviation of 15 and n = 36:
| Written as | Half-width | Range | Describes |
|---|---|---|---|
| mean ± SD | 15.00 | 88.00 to 118.00 | about 68% of single measurements, if roughly normal |
| mean ± SE | 2.50 | 100.50 to 105.50 | the typical wobble of the mean between samples |
| mean ± t × SE | 5.08 | 97.92 to 108.08 | 95% confidence interval for the mean, t = 2.030 on 35 df |
The standard deviation stays near 15 however many people you measure. The standard error falls with the square root of n: at 144 measurements it is 1.25, at 576 it is 0.63. That is why the result also prints a row for two, four, nine and sixteen times your sample, with the standard deviation held fixed next to it.
Moving between SD, SE and a confidence interval
| You have | You want | Formula |
|---|---|---|
| SD and n | SE of the mean | SE = SD / √n |
| SE and n | SD | SD = SE × √n |
| 95% interval, n under 60 | SE | SE = (upper - lower) / (2 × t), df = n - 1 |
| 95% interval, larger n | SE | SE = (upper - lower) / 3.92 |
| x successes of n | SE of the proportion | SE = √(p(1 - p) / n), p = x / n |
| Any of the above, population N | corrected SE | SE × √((N - n) / (N - 1)) |
The split at 60 observations comes from the Cochrane Handbook for Systematic Reviews, which recommends the t divisor for small groups. At n = 25 the two differ by about 5%; at n = 200 by well under 1%.
Six inputs and the standard error each one gives
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