About the Standard Error Calculator
This standard error calculator tells you how precisely a sample statistic estimates the population value. Paste raw data or enter a sample standard deviation and sample size to get the standard error of the mean (SEM), or enter a count of successes and a sample size to get the standard error of a proportion. Each result comes with the margin of error and confidence interval at 90%, 95% or 99%.
It is useful for students learning sampling distributions, researchers adding error bars to charts or tables, and anyone reading a survey who wants to know how much a reported percentage could move. Because the standard error shrinks with the square root of the sample size, the notes also show how many observations you would need to halve it.
For means the confidence interval uses Student’s t distribution with n − 1 degrees of freedom; for proportions it uses the normal (Wald) approximation, which works well when there are at least about 10 successes and 10 failures. If you sampled a large share of a finite population, enter the population size to apply the finite population correction.
With the default inputs, the standard error is 0.5. Change any value above to recalculate instantly.
How to use the standard error calculator
- 1Choose raw data, standard deviation and n, or a proportion.
- 2Paste your data, or enter s and n, or the number of successes and n.
- 3Pick a confidence level for the margin of error.
- 4Optionally enter the population size if you sampled a large share of it.
- 5Read the standard error, then use the margin of error or interval for error bars and reporting.
Formula and method
The standard error is the standard deviation of a statistic’s sampling distribution — how much the sample mean or proportion would vary from sample to sample. For a mean it is the sample standard deviation s (n − 1 denominator) divided by √n; for a proportion it is √(p̂(1 − p̂)/n), because the standard deviation of a 0/1 variable is √(p(1 − p)).
The margin of error multiplies the standard error by a critical value: Student’s t with n − 1 degrees of freedom for a mean (since σ is estimated) and the standard normal z for a proportion (1.645, 1.960 and 2.576 for 90%, 95% and 99%). When the sample is a sizeable share of a finite population of size N, the standard error is multiplied by the finite population correction √((N − n)/(N − 1)).
- s
- Sample standard deviation
- n
- Sample size
- p̂
- Sample proportion (x ÷ n)
- N
- Population size (for the finite population correction)
Worked examples
SEM from ten measurements
The ten values average 13.5 with a sample SD of 1.581, so SEM = 1.581 ÷ √10 = 0.5. With 9 degrees of freedom the 95% t critical value is 2.262, giving a margin of error of 1.13 and a confidence interval of 12.37 to 14.63.
From a reported mean and standard deviation
A study reports a mean of 72 with SD 8.5 from 50 people. SEM = 8.5 ÷ √50 = 1.202. The 95% t value with 49 df is 2.010, so the mean is 72 ± 2.42 (69.58 to 74.42).
Standard error of a survey proportion
420 of 1,000 respondents (42%) said yes. SE = √(0.42 × 0.58 ÷ 1000) = 0.0156, and the 95% margin of error is 1.96 × 0.0156 = 0.0306 — about ±3.1 percentage points.
Frequently asked questions
What is the difference between standard deviation and standard error?+
Standard deviation describes how spread out individual observations are. Standard error describes how precisely the sample mean (or proportion) estimates the population value. SE = SD ÷ √n, so it shrinks as the sample grows while the SD stays about the same.
How do you calculate the standard error of the mean?+
Compute the sample standard deviation s using n − 1 in the denominator, then divide by the square root of the sample size. For example, s = 10 and n = 25 give SEM = 10 ÷ 5 = 2.
How do you calculate the standard error of a proportion?+
Divide the number of successes by n to get p̂, then compute √(p̂ × (1 − p̂) ÷ n). For 50% in a sample of 1,000 that is √(0.25 ÷ 1000) = 0.0158, which gives the familiar ±3.1-point margin of error at 95% confidence.
Should error bars show standard deviation or standard error?+
Use SD when you want to show the variability of the data, and SE (or a confidence interval) when you want to show how precisely the mean is estimated. Always state which one you plotted, because SE bars are much shorter than SD bars for the same data.
How large a sample do I need to halve the standard error?+
Because the standard error is proportional to 1/√n, you need four times as many observations to halve it, and one hundred times as many to cut it to a tenth. This diminishing return is why very large samples add relatively little precision.