Case 1
- Standard deviation (SD)
- 15
- Sample size (n)
- 30
- Input mode
- SD and sample size
2.7386
- Number of observations
- 30
- Sample mean
- Available from raw observations.
- Sample standard deviation
- 15.0000
Standard Error Calculator
Change any value to update the result.
2.7386
Standard error of the mean: 2.7386Number of observations: 30 ; Sample mean: Available from raw observations.; Sample standard deviation: 15.0000
- Number of observations
- 30
- Sample mean
- Available from raw observations.
- Sample standard deviation
- 15.0000
Examples
How the result moves
Sample size (n) → Standard error of the meanYour current valid inputs, with one value varied across the range.
How the result moves
Held fixed: Standard deviation (SD) 15.0000, Input mode SD and sample size.
| Sample size (n) | Standard error of the mean |
|---|---|
| 20 | 3.3541 |
| 30Your value | 2.7386 |
| 40 | 2.3717 |
SE = SD ÷ √n estimates how much a sample mean varies across repeated samples. With positive SD and n > 1, SE is smaller than SD; with SD = 0, both are zero. Holding SD fixed, four times the sample size halves SE. The formula assumes independent observations with a common variance.
Worked examples
Case 2
- Standard deviation (SD)
- 10
- Sample size (n)
- 25
- Input mode
- SD and sample size
2.0000
- Number of observations
- 25
- Sample mean
- Available from raw observations.
- Sample standard deviation
- 10.0000
Case 3
- Standard deviation (SD)
- 12.5
- Sample size (n)
- 64
- Input mode
- SD and sample size
1.5625
- Number of observations
- 64
- Sample mean
- Available from raw observations.
- Sample standard deviation
- 12.5000
How it's calculated
SE = SD ÷ √n; raw sample SD² = Σ(x − mean)² ÷ (n − 1)
- StepEnter the standard deviation of your data.Standard deviation (SD)=15·Decimal convention for raw data=Decimal point · comma / space / newline
- StepEnter how many observations the sample holds.Raw observations=·Sample size (n)=30
- StepRead the standard error, or solve back for SD or n.Standard deviation (SD)=15
- ResultOr choose raw data and its decimal convention; paste two or more values to derive n, mean and sample SD locally.Decimal convention for raw data=Decimal point · comma / space / newline·Sample size (n)=30→2.7386
Raw mode uses sample SD with denominator n − 1, requiring at least two observations. SEM assumes independent observations with a common variance.
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. For n independent observations with a common variance, their sum has n times that variance. Dividing the sum by n divides its variance by n squared, so the variance of the mean is the individual variance divided by n.
Taking the square root gives SD divided by √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 can have 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
At a fixed SD, 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
For n ≥ 1, SE cannot exceed SD. With SD = 0, both are zero at every allowed n; with positive SD, equality occurs at n = 1. That last case is algebraic: one observation cannot supply an estimated sample SD.
When 1.96 is appropriate
The 1.96 multiplier gives a two-sided 95 percent normal interval, not a universal rule. When SD is estimated, the usual interval for independent normal observations uses Student's t with n − 1 degrees of freedom; a normal approximation needs justification.
Commonly misread
MisconceptionThe standard error shows how spread out the individual values are.
CorrectionThat is the standard deviation. The standard error describes only the sample mean, and how far it would move on the next draw.
MisconceptionTwice the sample, half the standard error.
CorrectionThe 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.
MisconceptionA small standard error means the data sit close together.
CorrectionA large n can produce a small standard error even with widely spread data. With SD = 50 and n = 2500 the standard error is 1.
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 a larger 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 describes how a sample mean varies across repeated samples. SD divided by √n estimates it for independent observations with a common variance. A small value indicates precision under those assumptions, not freedom from sampling bias.
What is the difference between standard error and standard deviation?
SD describes the spread of individual observations; SE describes the spread of the sample mean across repeated samples. For positive SD and n > 1, SE is smaller than SD. If SD = 0, both are zero.
Can the standard error be larger than the standard deviation?
For n ≥ 1, SE = SD ÷ √n cannot exceed a nonnegative SD. Equality holds when SD = 0 or n = 1. The n = 1 calculation needs an externally supplied SD; one observation cannot estimate a sample SD.
Why does the formula divide by the square root of n?
Independent observations with a common variance have a sample-mean variance equal to that variance divided by n. Taking the square root gives SD ÷ √n. This variance identity does not require the central limit theorem, although using a normal approximation for an interval needs additional assumptions.
How does sample size affect the standard error?
At a fixed SD, SE scales with 1 ÷ √n. With SD = 10 it is 2 at n = 25, 1 at n = 100 and 0.5 at n = 400. A new sample can also change the estimated SD, so these comparisons hold that input constant.
How is the standard error used for confidence intervals?
A two-sided 95 percent normal interval uses the mean ± 1.96 × SE when the normal model or approximation is justified. With an estimated SD, the usual interval for independent normal observations instead uses a Student's t quantile with n − 1 degrees of freedom. The multiplier is therefore not always 1.96, especially with small samples.
Sources and last check
- MathWorld — standard error of the meanFormula / method
- NIST StRD — NumAcc1Measured data
- NIST StRD — NumAcc2Measured data
- NIST StRD — NumAcc3Measured data
Information, not professional advice.