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- The standard error (SE) shows how precise a sample mean or proportion is.
- SE of the mean = s ÷ √n. SE of a proportion = √(p̂(1 − p̂) ÷ n).
- Works from the SD and sample size, from raw data, or from a proportion.
- Also shows the 95% and 99% margins of error.
How to use the standard error calculator
Choose what you have
SD and sample size, raw data, or a sample proportion.
Enter the values
For example SD 15 and n = 36, or paste your data.
Read the standard error
The SE appears with the 95% and 99% margins.
Check the steps
See the formula filled in with your numbers.
Standard error formulas
Standard error of the mean
SE = s ÷ √n
s is the sample standard deviation and n the sample size. Quadrupling n halves the standard error.
Standard error of a proportion
SE = √(p̂ × (1 − p̂) ÷ n)
Example 1: from SD and n
s = 15, n = 36.
- √36 6
- SE 15 ÷ 6 = 2.5
- 95% margin ± 4.9
Example 2: from raw data
Data: 10, 12, 23, 23, 16, 23, 21, 16.
- Mean 18
- Sample SD 5.237229
- SE 1.85164
- 95% range for the mean 14.3708 to 21.6292
Example 3: a proportion
52% of 1,000 people.
- SE 0.015799
- 95% margin ± 0.030966 (about 3.1 points)
Where a standard error calculator helps
Statistics homework
Find the SE for confidence intervals and hypothesis tests.
Research reports
Show how precise your averages are with error bars.
Surveys
See the margin of error of a percentage.
Experiments
Compare groups by looking at their means and SEs.
The standard deviation describes how spread out individual values are. The standard error describes how much the sample mean would vary from sample to sample. Get the SD first with the standard deviation calculator, then build a full interval with the confidence interval calculator.
Standard error calculator FAQ
What is the standard error?
It is the standard deviation of a sample statistic, usually the mean. It tells you how precisely the sample estimates the population value.
How do you calculate the standard error of the mean?
Divide the sample standard deviation by the square root of the sample size: SE = s ÷ √n.
What is the difference between standard error and standard deviation?
SD measures spread in the data. SE measures uncertainty in the mean, and it shrinks as the sample gets larger.
How does sample size affect the standard error?
SE falls with the square root of n. Four times as many observations gives half the standard error.
How do I get a margin of error from the SE?
Multiply by a critical value: about 1.96 for 95% or 2.576 for 99% with large samples.
What is the SE of a proportion?
√(p̂(1 − p̂) ÷ n). For 52% of 1,000 it is about 0.0158.