# What is the sampling distribution?

Finds the sampling distribution of a sample proportion, mean or sum: its mean, its standard error, and the probability that a sample statistic falls in a range.

- Page: https://www.acalculator.org/statistics/sampling-distribution-calculator
- JSON spec: https://www.acalculator.org/statistics/sampling-distribution-calculator.json
- Version: f53134650a95

## Default answer

Example with the default inputs (Sampling distribution of Sample proportion (p̂), Population proportion (p) 0.4, Sample size (n) 200, Probability Between, Lower value 0.35, Upper value 0.45): The probability is 0.8511, with a sampling distribution of mean 0.4 and standard error 0.034641.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| of | Sampling distribution of | The statistic whose distribution you want: the sample mean, the sample sum or the sample proportion. |
| mu | Population mean (μ) | The mean of the population you sample from. |
| sigma | Population standard deviation (σ) | The standard deviation of the population you sample from. |
| p | Population proportion (p) | The share of the population with the trait, between 0 and 1. |
| n | Sample size (n) | How many values are in each sample. |
| find | Probability | Whether to find the chance of a value between two numbers, below one, or above one. |
| a | Lower value | The lower end, in the units of the statistic. |
| b | Upper value | The upper end, in the units of the statistic. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| probability | Probability | The chance that the statistic falls in the chosen range, by the normal approximation. |
| center | Mean of the sampling distribution | The mean of the statistic: μ for x̄, nμ for Σx, and p for p̂. |
| se | Standard error | The standard deviation of the statistic: σ ÷ √n for x̄, √n σ for Σx, and √(p(1 − p) ÷ n) for p̂. |
| zLo | z of the lower value | How many standard errors the lower value is from the mean. |
| zHi | z of the upper value | How many standard errors the upper value is from the mean. |
| check | Normal approximation | A note on when the normal shape holds for these inputs. |

## Method

p̂ ~ N(p, √(p(1 − p) ÷ n)); x̄ ~ N(μ, σ ÷ √n); Σx ~ N(nμ, √n σ); z = (value − mean) ÷ standard error; the probability is the normal area over the range.

## Assumptions

- The samples are random and independent, and the normal shape comes from the central limit theorem.
- For p̂, the normal approximation is reasonable when np and n(1 − p) are both more than 5.

## Worked examples

1. of = proportion, p = 0.4, n = 200, find = between, a = 0.35, b = 0.45 gives probability = 0.851085, center = 0.4, se = 0.034641, zLo = -1.443376, zHi = 1.443376. Source: OpenStax, Introductory Business Statistics, §7.3 The Central Limit Theorem for Proportions (mean p, standard deviation √(p(1 − p) ÷ n)), https://openstax.org/books/introductory-business-statistics/pages/7-3-the-central-limit-theorem-for-proportions (retrieved 2026-10-05).
2. of = mean, mu = 100, sigma = 12, n = 36, find = below, b = 103 gives probability = 0.933193, center = 100, se = 2, zHi = 1.5. Source: OpenStax, Introductory Statistics 2e, §7.1 The Central Limit Theorem for Sample Means (Averages) (x̄ ~ N(μ, σ ÷ √n); Example 7.1: μ = 90, σ = 15, n = 25, P(85 < x̄ < 92) = 0.6997), https://openstax.org/books/introductory-statistics-2e/pages/7-1-the-central-limit-theorem-for-sample-means-averages (retrieved 2026-10-05).

## FAQ

### What is a sampling distribution?

It is the distribution of a statistic, such as a sample mean or a sample proportion, over all the random samples of the same size you could take. Each sample gives a slightly different value; the sampling distribution shows how those values spread.

### What is the sampling distribution of the sample proportion?

For samples of size n from a population with proportion p, the sample proportion p̂ has mean p and standard error √(p(1 − p) ÷ n). For large samples it is close to normal. With p = 0.4 and n = 200, the standard error is 0.0346.

### What is the sampling distribution of the sample mean?

The sample mean x̄ has mean μ and standard error σ ÷ √n, and by the central limit theorem it is close to normal for large n. With σ = 12 and n = 36, the standard error is 2.

### How do I find a probability from a sampling distribution?

Subtract the mean of the sampling distribution from your value and divide by the standard error to get z. Then read the normal area. For p = 0.4 and n = 200, P(0.35 < p̂ < 0.45) = 0.8511.

### When is the normal approximation good for a proportion?

A usual check is that np and n(1 − p) are both more than 5, so the sample is likely to hold enough of each kind of outcome. The page shows this check under the result.

### How does sample size change the sampling distribution?

The standard error falls with the square root of n. Four times as many observations halve the spread, so sample statistics sit closer to the population value.

## Sources

- OpenStax, Introductory Business Statistics, §7.3 The Central Limit Theorem for Proportions: mean p and standard deviation √(p(1 − p) ÷ n). https://openstax.org/books/introductory-business-statistics/pages/7-3-the-central-limit-theorem-for-proportions (retrieved 2026-10-05)
- OpenStax, Introductory Statistics 2e, §7.1 The Central Limit Theorem for Sample Means (Averages): x̄ ~ N(μ, σ ÷ √n). https://openstax.org/books/introductory-statistics-2e/pages/7-1-the-central-limit-theorem-for-sample-means-averages (retrieved 2026-10-05)
- OpenStax, Introductory Statistics 2e, §7.2 The Central Limit Theorem for Sums: ΣX ~ N(nμ, √n σ). https://openstax.org/books/introductory-statistics-2e/pages/7-2-the-central-limit-theorem-for-sums (retrieved 2026-10-05)
- OpenStax, Introductory Statistics 2e, §8.3 A Population Proportion: the normal approximation needs np and n(1 − p) both more than 5. https://openstax.org/books/introductory-statistics-2e/pages/8-3-a-population-proportion (retrieved 2026-10-05)