Quick answer
The sample mean has standard error SE = σ ÷ √n and z = (x̄ − μ) ÷ SE. With μ = 100, σ = 15 and n = 36, SE = 2.5; for x̄ = 103, z = 1.2 and P(x̄ < 103) = 0.8849.
Enter the population mean and SD, sample size, a sample mean and the probability type. The calculator returns the probability, standard error and z-score. Use this sampling distribution of the mean calculator to check your own figures.
Related tools: Binomial Distribution Calculator: Binomial probabilities for n trials. Poisson Distribution Calculator: Poisson probabilities for a given mean rate. Normal Distribution Calculator: Probabilities for a normal distribution.
How to use the Sampling Distribution of the Mean Calculator
- Enter population mean (μ).
- Enter population SD (σ) (above 0).
- Enter sample size (a whole number, 1 or more).
- Enter sample mean x̄.
- Choose probability: P(x̄ < value) and P(x̄ > value).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Population mean (μ) | A number | — |
| Population SD (σ) | A number | > 0 |
| Sample size | A whole number | ≥ 1 |
| Sample mean x̄ | A number | — |
| Probability | Choose one: P(x̄ < value); P(x̄ > value) | Starts at P(x̄ < value) |
Formula and method
Formula
SE = σ ÷ √n; z = (x̄ − μ) ÷ SE; P = Φ(z)
IEEE-754 double precision; shown to at most 6 decimal places.
By the central limit theorem the sample mean is approximately normal for large n (or exactly normal if the population is).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Population mean (μ) = 100; Population SD (σ) = 15; Sample size = 36; Sample mean x̄ = 103; Probability = P(x̄ < value).
| Result | Value |
|---|---|
| P(x̄ < value) | 0.88493 |
| Standard error | 2.5 |
| z-score | 1.2 |
Example 2
Inputs: Population mean (μ) = 100; Population SD (σ) = 15; Sample size = 36; Sample mean x̄ = 103; Probability = P(x̄ > value).
| Result | Value |
|---|---|
| P(x̄ > value) | 0.11507 |
| Standard error | 2.5 |
| z-score | 1.2 |
The first line is the headline result (p(x̄ < value)). Below it the sampling Distribution of the Mean Calculator lists standard error and z-score, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Central limit theorem
Sample means are approximately normal for large n, even if the population isn't. The SE shrinks with √n: with n = 144 it would be 1.25.
Caution
For non-normal populations and small n the approximation can be poor. The result concerns the mean of the sample, not individual values. Use the Normal Distribution calculator for individual values.
How the result changes with population mean
Holding every other input at the first worked example, this table changes only population mean. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Population mean | P(x̄ < value) | Standard error | z-score |
|---|---|---|---|
| 50 | 1 | 2.5 | 21.2 |
| 100 | 0.88493 | 2.5 | 1.2 |
| 200 | 0 | 2.5 | -38.8 |
| 400 | 0 | 2.5 | -118.8 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Population SD (σ): entering "0" shows "Population SD (σ) must be greater than 0." beside that field and no result.
Choosing between the Sampling Distribution of the Mean Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Binomial Distribution Calculator](/statistics/binomial-distribution/) | trials, success probability and successes | P(X = 5) |
| [Poisson Distribution Calculator](/statistics/poisson-distribution-calculator/) | average rate and events | P(X = 3) |
| [Normal Distribution Calculator](/statistics/normal-distribution-calculator/) | mean, standard deviation and probability | P(X < x₁) |
| [IQ Percentile Calculator](/statistics/iq-percentile/) | iQ score, mean and standard deviation | Percentile |
| [Empirical Rule Calculator](/statistics/empirical-rule/) | mean and standard deviation | ≈68% (μ ± σ) |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- Approximation for non-normal populations with small n
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for population SD (σ) to confirm this.
Frequently asked questions
What is the SE with n = 100?
With n = 100 and σ = 15 the standard error is 15 ÷ 10 = 1.5.
What is the central limit theorem?
Sample means approach a normal distribution.
Why is SE smaller than σ?
Averaging reduces variability.
What does the Sampling Distribution of the Mean Calculator calculate?
Probability for a sample mean using the central limit theorem. It returns P(x̄ < value), Standard error and z-score.
What formula does the Sampling Distribution of the Mean Calculator use?
SE = σ ÷ √n; z = (x̄ − μ) ÷ SE; P = Φ(z). By the central limit theorem the sample mean is approximately normal for large n (or exactly normal if the population is).
What do I need to enter in the sampling Distribution of the Mean Calculator?
You enter Population mean (μ), Population SD (σ), Sample size, Sample mean x̄ and Probability.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Approximation for non-normal populations with small n.
What does the “Probability” option change?
It sets how the sampling Distribution of the Mean is worked out. The choices are P(x̄ < value) and P(x̄ > value).
What is an example of a sampling Distribution of the Mean result?
With population mean (μ) 100, population SD (σ) 15, sample size 36, sample mean x̄ 103 and probability P(x̄ < value), the p(x̄ < value) is 0.88493.
Related calculators
- Binomial Distribution Calculator — Binomial probabilities for n trials.
- Poisson Distribution Calculator — Poisson probabilities for a given mean rate.
- Normal Distribution Calculator — Probabilities for a normal distribution.
- IQ Percentile Calculator — Percentile of an IQ score on a normal scale.
- Empirical Rule Calculator — 68–95–99.7 intervals for a normal distribution.
- Geometric Probability Calculator — Probability the first success occurs on trial k.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.