About the Sampling Distribution Calculator
This sampling distribution calculator applies the central limit theorem to answer questions like “if the population mean is 70 with standard deviation 12, what is the probability that the mean of 36 randomly chosen scores is above 72?” Enter the population parameters, the sample size and the cut-off values, and it returns the mean and standard error of the sampling distribution, the z-score of each cut-off and the probability below, above or between them, with a shaded curve.
It handles both sample means (from a population mean μ and standard deviation σ) and sample proportions (from a population proportion p). Statistics students use it for central limit theorem homework, quality engineers to judge how much batch averages should vary, and survey researchers to see how far a sample percentage can plausibly stray from the true value. An optional finite population correction handles samples drawn without replacement from small populations.
The normal approximation is exact when the population itself is normal. Otherwise it relies on the central limit theorem, which works well for sample means once n is about 30 or more (sooner for symmetric data), and for proportions when both np and n(1 − p) are at least 10. The calculator checks these conditions and warns you when they fail.
With the default inputs, the probability is 0.1587. Change any value above to recalculate instantly.
How to use the sampling distribution calculator
- 1Choose whether you are working with a sample mean or a sample proportion.
- 2Enter the population mean and standard deviation (or the population proportion).
- 3Enter the sample size n.
- 4Pick less than, greater than or between, and enter the cut-off value(s).
- 5Read the probability, standard error and z-scores; turn on the finite population correction for large samples from small populations.
Formula and method
The central limit theorem says that the means of repeated random samples of size n are approximately normally distributed around the population mean μ, with a standard deviation — the standard error — of σ/√n. Sample proportions behave the same way around p with standard error √(p(1 − p)/n). Converting a cut-off value to a z-score and reading the standard normal curve gives the probability of observing a sample statistic below, above or between the cut-offs.
When you sample without replacement from a finite population and the sample is more than about 5% of it, multiplying the standard error by the finite population correction √((N − n)/(N − 1)) gives a more accurate result. For proportions, the results are shown in percentage points. No continuity correction is applied, so for small samples of proportions the answer is an approximation.
- μ, σ
- Population mean and standard deviation
- p
- Population proportion
- n
- Sample size
- N
- Population size (finite population correction only)
- SE
- Standard error: the standard deviation of the sampling distribution
Worked examples
Average exam score of 36 students above 72
The standard error is 12 ÷ √36 = 2, so a sample mean of 72 sits exactly one standard error above 70 (z = 1). The probability of a z-score above 1 is 0.1587, so about 16% of random classes of 36 would average more than 72.
Mean fill weight between two limits
Bottles average 500 ml with σ = 8 ml. For samples of 16 the standard error is 8 ÷ 4 = 2 ml, so 497 and 503 ml are 1.5 standard errors either side of the mean. About 86.6% of sample means land between them.
Survey proportion: 45% or more support when true support is 40%
With p = 0.40 and n = 200, the standard error is √(0.4 × 0.6 ÷ 200) ≈ 3.46 percentage points. A sample result of 45% is 1.44 standard errors above 40%, which happens in about 7.4% of samples.
Finite population correction: 100 of 500 employees
Without the correction the standard error is 9,000 ÷ 10 = 900. Sampling 20% of a 500-person workforce multiplies it by √(400 ÷ 499) ≈ 0.895, giving about 806, so a sample mean below $51,000 (z ≈ −1.24) has probability ≈ 0.107.
Frequently asked questions
What is a sampling distribution?+
It is the distribution of a statistic — such as the sample mean or sample proportion — over all possible random samples of the same size from a population. Its spread, the standard error, shows how much the statistic varies from sample to sample.
What does the central limit theorem say?+
For a large enough sample size, the sampling distribution of the mean is approximately normal with mean μ and standard deviation σ/√n, whatever the shape of the population, provided the population has a finite variance.
How large does n need to be for the central limit theorem?+
A common rule of thumb is n ≥ 30 for means, smaller for symmetric populations and larger for strongly skewed ones. For proportions, np and n(1 − p) should both be at least 10.
What is the difference between standard deviation and standard error?+
The standard deviation σ describes how individual values vary. The standard error σ/√n describes how sample means vary, so it shrinks as the sample size grows.
When should I use the finite population correction?+
Use it when sampling without replacement and the sample is more than about 5% of the population. It reduces the standard error because a large sample from a small population leaves less room for variation.