Standard deviation and standard error: spread versus precision
Understand why a larger sample can improve a mean’s precision without reducing the spread of individual values.
Method
Standard deviation describes the spread of observations. Standard error of the mean estimates uncertainty in a sample average. For independent observations, divide the sample standard deviation s by the square root of the sample size n. These quantities share the measurement unit but answer different questions. A small standard error does not mean the individual values are tightly clustered.
Worked example
Enter a sample standard deviation of 12 and a sample size of 36 in Standard error of the mean. The result is 2. Keeping the same standard deviation while increasing the count to 144 gives 1. Four times as many independent observations halves the estimated standard error.
s = 12; n = 36; SE = 12 / √36 = 2
s = 12; n = 144; SE = 12 / √144 = 1
s = σ × √(n / (n − 1))
Checks and limits
Use a sample standard deviation, calculated with n−1. The five-value Standard deviation tool instead reports population spread; convert it using the equation below if those five observations form a sample. Repeated or correlated observations may not provide independent information. Standard error alone is not a confidence interval: an appropriate critical value and assumptions are also required.
Related calculators
Source: NIST