Case 1
- Standard deviation (SD)
- 15
- Sample size (n)
- 30
2.7386
Open with these valuesStandard Error Calculator
2.7386
Result: 2.7386The standard error says how much a sample mean would wobble if you drew the sample again: SD divided by the square root of n. It is always smaller than the standard deviation, and equal to it only for a sample of one. Quadrupling the sample halves it — precision is bought by the square root.
Worked examples
Case 2
- Standard deviation (SD)
- 10
- Sample size (n)
- 25
2.0000
Open with these valuesCase 3
- Standard deviation (SD)
- 12.5
- Sample size (n)
- 64
1.5625
Open with these valuesHow it's calculated
SE = SD ÷ √n
- StepEnter the standard deviation of your data.
- StepEnter how many observations the sample holds.
- ResultRead the standard error, or solve back for SD or n.
What this number means
The standard error answers a question a single sample cannot answer directly: if you drew a fresh sample of the same size, how far from this one would its mean land? Carry that thought far enough and the means themselves form a distribution — and the standard error is that distribution's standard deviation. It describes the spread of a quantity you only ever get to see once. The formula follows from how averaging works. Adding n independent observations adds their variances, and the sum is then divided by n, so the variance of the mean comes out n times smaller than the variance of one observation. Taking the square root turns that back into a standard deviation, which is why the denominator holds √n and not n. At the values set here, a standard deviation of 15 across 30 observations gives 15 ÷ √30 = 2.7386: individual readings scatter by 15, the mean of thirty of them by under 3. What the number does not say is where the true mean lies, or in which direction this sample missed it. It measures the spread of an estimate, not its accuracy. A sample drawn from the wrong group has a small standard error and a wrong mean all the same — bias appears nowhere in the formula.
It describes the mean, not the data
The standard error says how far a sample mean would wobble across repeated samples. How widely the individual observations scatter is the standard deviation — a different number.
Precision is bought by the square root
Halving the standard error costs four times the sample. With SD = 10 it is 2 at n = 25, 1 at n = 100 and 0.5 at n = 400.
Never larger than the standard deviation
Because SE = SD ÷ √n, the standard error cannot exceed the standard deviation. The two are equal only at n = 1; from two observations on it is strictly smaller.
Where the 1.96 in a confidence interval comes from
A 95 percent interval runs from the mean minus 1.96 standard errors to the mean plus 1.96. For 90 percent it is 1.645 standard errors and for 99 percent 2.576.
Commonly misread
The standard error shows how spread out the individual values are.
That is the standard deviation. The standard error describes only the sample mean, and how far it would move on the next draw.
Twice the sample, half the standard error.
The square root sets the price. With SD = 10 it falls from 2 to 1 only once n goes from 25 to 100 — four times the sample.
A small standard error means the data sit close together.
Mostly it means n is large. With SD = 50 and n = 2500 the standard error is 1, although the data are widely spread.
Reference table
| SD, n | Reading | SE |
|---|---|---|
| 10, 25 | A small sample of moderately spread data | 2 |
| 10, 30 | Five more observations, slightly tighter | 1.8257 |
| 12.5, 64 | More spread, but four times the sample | 1.5625 |
| 15, 100 | A hundred observations | 1.5 |
| 20, 400 | Four hundred observations | 1 |
| 50, 2500 | Wide data, but a very large sample | 1 |
Questions
What is standard error?
The standard error measures how precisely a sample mean estimates the true population mean. A smaller value means the sample mean is likely close to the population mean; a larger one means more uncertainty. It is calculated as SD divided by the square root of n.
What is the difference between standard error and standard deviation?
The standard deviation measures how spread out the individual data points are within one sample. The standard error measures how much the sample mean itself would vary across repeated samples. The standard error is always the smaller of the two, because it divides by the square root of n.
Can the standard error be larger than the standard deviation?
No. Since SE = SD ÷ √n, the standard error can never exceed the standard deviation. The two are equal only when the sample size is 1; from two observations onwards the standard error is strictly smaller.
Why does the formula divide by the square root of n?
The square root comes from the central limit theorem: when you average n observations, individual variations partly cancel out. The variance of the mean falls in proportion to 1 ÷ n, so its standard deviation falls in proportion to 1 ÷ √n. That is why you have to quadruple the sample to halve the standard error.
How does sample size affect the standard error?
It follows a square root, not a straight line. With SD = 10 the standard error is 2 at n = 25, 1 at n = 100 and 0.5 at n = 400. Every halving costs four times the sample, so early observations buy far more precision than later ones.
How is the standard error used for confidence intervals?
A 95 percent interval runs from the mean minus 1.96 standard errors to the mean plus 1.96 standard errors. With a mean of 100 and a standard error of 5 that is 100 ± 9.8, so 90.2 to 109.8. Common alternatives are 1.645 standard errors for 90 percent and 2.576 for 99 percent.
Sources and last check
Information, not professional advice.