Sampling Distribution of the Mean Calculator (Standard Error and Sample-Mean Probabilities)
The standard error of a sample mean, the chance that a sample of size n averages above, below or this far from the true mean, and the range holding 95 % of sample means.
Results
The dashed line marks the value you entered.
What this tool does
Individual values scatter with the population's standard deviation σ, but the average of n of them scatters much less, by σ ÷ √n — the standard error. That is why quadrupling the sample only halves the uncertainty. By the central limit theorem the sample means are close to normal once n is moderately large, whatever the shape of the data, which is what the probabilities here assume. If the sample is a big slice of a finite population, sampling without replacement shrinks the error further by √((N − n) ÷ (N − 1)); enter the population size to apply it, or leave it at 0.
Formula
SE = σ ÷ √n × √((N − n) ÷ (N − 1)) · z = (x̄ − μ) ÷ SE · 95 % of sample means lie within μ ± 1.96 SE
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mu | Population mean (μ) | — |
sd | Population standard deviation (σ) | — |
nn | Sample size (n) | — |
pn | Population size (0 if very large) | — |
xb | Sample mean you want to check | — |
SE | Standard error of the mean (σ ÷ √n) | — |
Z | z-score of that sample mean | — |
PB | Chance a sample mean is at or below it | % |
PA | Chance a sample mean is at or above it | % |
PF | Chance of a sample mean this far from μ in either direction | % |
LO | 95 % of sample means fall above | — |
HI | …and below | — |
Worked example
- Population mean (μ)100
- Population standard deviation (σ)15
- Sample size (n)25
- Population size (0 if very large)0
- Sample mean you want to check104
- Standard error of the mean (σ ÷ √n)3.0000
- z-score of that sample mean1.333
- Chance a sample mean is at or below it90.88 %
- Chance a sample mean is at or above it9.12 %
- Chance of a sample mean this far from μ in either direction18.24 %
- 95 % of sample means fall above94.120
- …and below105.880
Limitations
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.