How to use the standard error calculator
- Pick what you have. Choose SD and n if you already know the standard deviation and the sample size, List of values to paste your data, or A proportion for a share such as 0.5 of a sample.
- Enter the numbers. For a list, separate values with commas, spaces or new lines. For a proportion, enter a value from 0 to 1 and the sample size.
- Read the standard error. The result also shows the standard error you would get with four times as many observations.
- If you want a range around the mean, take the standard error to the confidence interval calculator.
Formula
Standard error of the mean = s / sqrt(n)
Standard error of a proportion = sqrt( p x (1 - p) / n )
s = sample standard deviation, n = sample size, p = proportion
From a list, the tool uses the sample standard deviation (divide by n - 1). If you type a standard deviation yourself, use the same kind your source used.
Test scores from 25 students
A class of 25 students has a mean score with a standard deviation of 10 points. The standard error of the mean is 10 / sqrt(25) = 10 / 5 = 2 points. That says a mean from another random sample of 25 students would typically land about 2 points from the true mean.
With 100 students the standard error would be 10 / sqrt(100) = 1. Four times the sample cuts the standard error in half. For a list of 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the sample standard deviation is 2.1381 and the standard error is 2.1381 / sqrt(8) = 0.7559.
Standard error and standard deviation are different things
The standard deviation describes how spread out the individual values are. The standard error describes how much a statistic, usually the sample mean, would vary if you drew many samples of the same size. A large, varied data set has a large standard deviation, but its mean can still have a small standard error if the sample is big.
Because the standard error shrinks with the square root of the sample size, the first extra observations help a lot and later ones help less. Going from 25 to 100 observations halves the standard error, but going from 100 to 175 changes it by much less.
Where the standard error is used
The standard error is the building block for confidence intervals and for tests that compare a sample mean with a target. A rough interval is the mean plus or minus a multiple of the standard error. For small samples the multiple comes from a t table rather than the normal curve, which the confidence interval calculator handles.
If you are deciding how many people to survey, work backward from the margin of error you want with the sample size calculator. To see the standard deviation and other summary figures for the same list, use the descriptive statistics calculator.
Assumptions and limits
- The observations come from a random sample and are independent of each other.
- For a list, the tool uses the sample standard deviation with n - 1 in the denominator.
- The standard error of a proportion uses the sample proportion itself, which is an approximation when the sample is small or the proportion is near 0 or 1.
- No finite population correction is applied. If your sample is a large share of a small population, the true standard error is somewhat smaller.
Frequently asked questions
What is the standard error of the mean?
It is the standard deviation of the sample mean across repeated samples, estimated as the sample standard deviation divided by the square root of the sample size. A smaller value means the sample mean is a more precise estimate.
How is standard error different from standard deviation?
Standard deviation measures the spread of the individual observations. Standard error measures the spread of a statistic, such as the mean, across samples. Standard error equals the standard deviation divided by the square root of n, so it is always smaller for samples of more than one.
Why does a bigger sample give a smaller standard error?
Averages of more observations vary less from sample to sample because unusual values are diluted. Since the standard error contains the square root of n, quadrupling the sample size cuts it in half.
Can I use this for percentages from a survey?
Yes. Choose the proportion option and enter the share as a decimal, for example 0.42 for 42 percent, with the number of respondents. The result is the standard error of that sample proportion.