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Standard Error Calculator

Find how precisely a sample estimates the population. Paste raw data or enter a standard deviation and sample size for the standard error of the mean (SEM), or enter a sample proportion for its standard error, with t- or z-based confidence intervals and an optional finite population correction.

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Calculate from

Separate values with commas, spaces or new lines

Add it to get the confidence interval

%

Applies the finite population correction; leave empty if the population is large

Enter your values to see the result.
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How to use the Standard Error Calculator

  1. Choose the input: raw data, an SD and sample size, or a proportion.
  2. Enter your numbers, e.g. paste the data or type s = 15 and n = 36.
  3. Pick a confidence level and, for a small population, enter its size N.
  4. Read the standard error and the confidence interval built from it.
  5. Use the table to see how many observations you need for a smaller SE.

Standard error formulas

StatisticStandard errorConfidence interval
Mean, SD estimated from the sampleSE = s ÷ √nx̄ ± t(α/2, n − 1) × SE
Mean, population σ knownSE = σ ÷ √nx̄ ± z(α/2) × SE
ProportionSE = √(p̂(1 − p̂) ÷ n)p̂ ± z(α/2) × SE (Wald) or the Wilson interval
Finite population (any of the above)SE × √((N − n) ÷ (N − 1))same, with the corrected SE

Example: s = 15 and n = 36 give SE = 15 ÷ 6 = 2.5. With a mean of 100 the 95% interval is 100 ± 2.030 × 2.5 = 94.92 to 105.08 (t with 35 degrees of freedom).

Standard error vs sample size

The SE falls with the square root of n, so gains slow down as the sample grows:

nSE when s = 10SE of a 50% proportion
252.00010.0 points
1001.0005.0 points
4000.5002.5 points
1,6000.2501.25 points

Frequently asked questions

What is the standard error of the mean?

The standard error of the mean (SEM) measures how much the sample mean would vary from sample to sample. It is SE = s ÷ √n, where s is the sample standard deviation and n the sample size. A smaller SE means a more precise estimate of the population mean.

What is the difference between standard deviation and standard error?

The standard deviation describes how spread out the individual values are. The standard error describes how uncertain the mean (or another statistic) is. The SD stays about the same as you collect more data; the SE falls in proportion to 1 ÷ √n.

How do I calculate the standard error of a proportion?

SE = √(p̂(1 − p̂) ÷ n). For 42% in a sample of 500: √(0.42 × 0.58 ÷ 500) ≈ 0.0221, about 2.2 percentage points. It is largest when p̂ = 50%.

How do I get a confidence interval from the standard error?

Multiply the SE by a critical value and add and subtract it from the estimate: x̄ ± t × SE for a mean with an estimated SD (t with n − 1 degrees of freedom), or ± z × SE when σ is known or for a large-sample proportion (z = 1.96 for 95%).

When should I use the finite population correction?

When the sample is more than about 5% of a finite population. The SE is multiplied by √((N − n) ÷ (N − 1)), which shrinks it; if you sample the whole population the SE becomes 0.

How many more observations halve the standard error?

Four times as many. Because SE = s ÷ √n, doubling precision requires quadrupling the sample size.

Are all the tools on FreeToolCabin tested for quality and accuracy?

Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.

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