The standard error (SE) tells you how much a sample statistic would bounce around if you repeated the study many times. It is the yardstick behind confidence intervals, t-tests and margins of error. This calculator finds the standard error of a mean (from raw data or from a standard deviation and sample size), of a proportion, and of the difference between two independent means, and shows exactly how the error falls as the sample grows.
How to use the standard error calculator
- Choose what you need the standard error of: a mean from raw data, a mean from an SD and n, a proportion, or a difference between two means.
- Enter the data or the summary numbers. Supplying the mean adds a 95% confidence interval and the relative standard error.
- Read the SE on the tape and use the table to see what larger samples would buy you.
Standard error formulas
Here s is the sample standard deviation (dividing by n − 1) and p̂ the sample proportion. The relative standard error, SE ÷ |mean| × 100%, expresses precision as a percentage; CDC’s National Center for Health Statistics, for example, has traditionally flagged survey estimates with a relative standard error of 30% or more as unreliable.
Worked examples
Mean from data. Twelve apples from an orchard weigh (in ounces) 4.2, 3.9, 5.1, 4.7, 4.4, 5.0, 3.8, 4.6, 4.9, 4.3, 4.1 and 4.8.
- Mean = 4.4833 oz and sample SD s = 0.4324 oz.
- SE = 0.4324 ÷ √12 = 0.4324 ÷ 3.4641 = 0.1248 oz.
- With t* = 2.201 for 11 degrees of freedom, the 95% interval for the orchard’s mean apple weight is 4.21 to 4.76 oz.
- The relative SE is 0.1248 ÷ 4.4833 = 2.8%, a precise estimate.
From summary numbers. A sample of 36 has s = 12, so SE = 12 ÷ 6 = 2. Quadrupling the sample to 144 would cut the SE to 1.
A proportion. If 40% of 36 respondents agree, SE = √(0.4 × 0.6 ÷ 36) = 0.0816, about 8.2 percentage points, so a rough 95% margin is ±16 points. Small samples give loose percentages.
A difference. Group 1 has s = 8 with 25 people and group 2 has s = 10 with 30. The SEs are 1.6 and 1.826, and the SE of the difference is √(1.6² + 1.826²) = 2.428, larger than either.
Standard error versus standard deviation
| Standard deviation | Standard error | |
|---|---|---|
| Describes | spread of individual values | precision of an estimate |
| Effect of larger n | settles near the population σ | keeps shrinking (÷ √n) |
| Typical use | describing data | confidence intervals, tests |
A common misuse is to plot mean ± SE as if it showed the range of the data. With 100 observations, ±1 SE covers a band one-tenth as wide as ±1 SD; it says nothing about where most individual values lie.
Reading error bars
Two means whose ±1 SE bars just touch are not significantly different at the 5% level; the bars need a gap of roughly one SE for that. Confidence-interval bars behave differently again. When comparing groups, compute the SE of the difference directly, or run the t-test calculator.
For the spread of the data itself, use the standard deviation calculator. To plan a sample that achieves a target precision, use the sample size calculator.
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation describes how spread out individual values are. Standard error describes how precisely a sample statistic, usually the mean, estimates the population value. SD stays roughly the same as you collect more data; SE keeps shrinking, in proportion to 1 ÷ √n.
Should I report SD or SE in a paper?
Report SD when describing the variability of your subjects or measurements, and SE (or better, a confidence interval) when describing the precision of an estimate. Error bars labeled only as ± are ambiguous; always say which one you used.
How do I get a confidence interval from the standard error?
Multiply the SE by a critical value and add and subtract it from the estimate. For a 95% interval that is about 1.96 for large samples, or the t value with n − 1 degrees of freedom for a mean from a small sample. The calculator does this when you supply a mean.
Why does doubling the sample size not halve the standard error?
Because SE depends on the square root of n. Doubling n divides the SE by √2 ≈ 1.41. To halve it, you need four times as many observations, which is why extra precision gets expensive.
What is the standard error of a difference?
For two independent samples, the variances of the means add: SE(x̄₁ − x̄₂) = √(s₁²/n₁ + s₂²/n₂). The SE of a difference is always larger than either group's own SE, a point often missed when judging overlapping error bars.