The standard error tells you how much the average of a sample might differ from the true average of the population it came from. The smaller it is, the more precise your estimate. This standard error calculator takes a list of values and returns the mean, the sample standard deviation, the standard error of the mean and a 95 percent confidence interval around the estimate.
What Is the Standard Error Calculator?
The standard error of the mean is the standard deviation of the sampling distribution, which is a way of describing how much the average of a sample varies from sample to sample. It is the standard deviation divided by the square root of the sample size.
That last part is the important one. Doubling the number of observations does not halve the standard error, it divides it by the square root of two. Getting from an error of 10 to an error of 5 requires four times as much data.
How the Calculation Works
Standard error = sample standard deviation ÷ √(sample size)
The sample standard deviation uses n minus one in the denominator rather than n, because the sample mean is already calculated from the data, which makes the naive version slightly too small. The calculator shows both that figure and the population estimate.
The confidence interval is the mean plus or minus a multiple of the standard error, where the multiple is 1.645 for 90 percent, 1.96 for 95 percent and 2.576 for 99 percent under a normal approximation.
Choosing a Sample Size
Work backwards from the precision you need. If you want the estimate to be within 2 units with 95 percent confidence, and the standard deviation is 10, then n must be at least (1.96 times 10 divided by 2) squared, which is 96 observations.
That is the formula behind every sample size calculation in survey work, and it explains why precision is expensive. Halving the margin of error requires four times the sample.
Assumptions Worth Knowing
The confidence interval above assumes the data are roughly normally distributed, or that the sample is large enough for the central limit theorem to hold. With fewer than about 30 observations and clearly skewed data, a t-distribution is the better choice.
It also assumes the observations are independent. If you have measured the same subject several times, or clustered samples from the same area, the standard error is understated and the interval is too narrow.
Frequently Asked Questions
What is the standard error of the mean?
It is the standard deviation of the sampling distribution of the sample mean, calculated as the sample standard deviation divided by the square root of the sample size. It measures the uncertainty in your estimate of the population mean.
Why is the sample size under the square root?
Because independent observations average out. The variance of the mean is the variance of a single observation divided by n, and taking the square root of that gives the standard deviation of the mean, which is the standard deviation over the square root of n.
How do I calculate a confidence interval from the standard error?
Multiply the standard error by the critical value for your confidence level, 1.96 for 95 percent, then add and subtract that margin from the mean. The calculator does this and returns the lower and upper limits.
What is the difference between standard deviation and standard error?
The standard deviation describes the spread of the individual observations around their mean. The standard error describes the uncertainty in the mean of the sample. A large standard deviation is fine as long as the standard error is small enough for your purpose.
Do outliers affect the standard error?
Yes, indirectly, because they inflate the standard deviation, and the standard error is derived from it. If a few extreme values are genuinely part of the population being studied, they should stay. If they are recording errors, removing them will usually make the estimate much more precise.