Standard Error Calculator
Calculate the Standard Error of the Mean (SEM) from standard deviation and sample size. Includes margin of error for 90%, 95%, and 99% confidence levels.
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How to use this calculator
Standard Error of the Mean measures how precisely the sample mean estimates the population mean. Larger samples (bigger n) give smaller SE and more precise estimates.
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Enter the standard deviation of your sample (or known population standard deviation).
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Enter the sample size n.
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The Standard Error and the margin of error at 90%, 95%, and 99% confidence are calculated.
Frequently asked questions
What is the standard error of the mean?
The Standard Error of the Mean (SEM) estimates how much the sample mean would vary across repeated samples of the same size from the same population. A small SEM means your sample mean is a precise estimate of the population mean. SEM = σ/√n: larger samples give more precise (smaller SEM) estimates.
What is the difference between standard deviation and standard error?
Standard deviation (SD) describes the spread of individual values in a dataset. Standard error (SE) describes how precisely the mean of the dataset estimates the population mean. SD is a property of the data; SE is a property of the estimate. As sample size increases, SE decreases (better precision) but SD does not systematically change — it reflects true variability in the population.
How do I build a confidence interval using the standard error?
For a 95% confidence interval: CI = x̄ ± 1.96 × SE, where x̄ is the sample mean. This means if you repeated the sampling procedure many times, 95% of intervals constructed this way would contain the true population mean. The ±1.96 × SE values are the margins of error shown in this calculator.
How much does sample size affect the standard error?
SE scales as 1/√n. To halve the SE (double precision), you must quadruple the sample size. Going from n=25 to n=100 halves the SE; n=400 gives half the SE of n=100. This "diminishing returns" relationship means beyond a certain sample size, collecting more data provides decreasing gains in precision.
Standard error calculator — SEM and confidence intervals
SE as the standard deviation of the sampling distribution
The Central Limit Theorem states that if you draw many random samples of size n from a population with mean μ and standard deviation σ, the sample means form a normal distribution with mean μ and standard deviation σ/√n — this is the standard error. The SEM is therefore the standard deviation of the sampling distribution of the mean, not the standard deviation of the original data.
When to use SE vs. SD in graphs and tables
A common confusion in published research is whether error bars represent SD or SE. SD bars show the spread of the data (usually ±1 SD). SE bars show the precision of the mean estimate (usually ±1 SE or ±1.96 SE for a 95% CI). SE bars are narrower than SD bars (by a factor of √n) and do not represent data spread — they represent estimation uncertainty. Most statistical guidelines recommend using 95% confidence intervals rather than SE bars for clarity.
Practical sample size planning
To achieve a target SE, rearrange: n = (σ/SE)². If you want to estimate a population mean with SE ≤ 2 and you know σ ≈ 10, you need n ≥ (10/2)² = 25. In clinical trials, sample size is often calculated to ensure a desired power (probability of detecting a true effect) at a specified significance level — both of which depend on the expected SE. This calculator's output can directly inform those sample size planning calculations.
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