# Central limit theorem probability

Uses the central limit theorem to find the mean and standard error of a sample mean, sum or proportion, and the probability that it falls below, above or between values.

- Page: https://www.acalculator.org/statistics/central-limit-theorem-calculator
- JSON spec: https://www.acalculator.org/statistics/central-limit-theorem-calculator.json
- Version: f9298a5853e1

## Default answer

Example with the default inputs (Sampling distribution of Sample mean (x̄), Population mean (μ) 90, Population standard deviation (σ) 15, Sample size (n) 25, Probability Between, Lower value 85, Upper value 92): The probability is 0.6997, with a sampling distribution of mean 90 and standard error 3.

## 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

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

## Assumptions

- The samples are random and independent.
- For a normal population the distribution of x̄ and Σx is exactly normal; otherwise the normal shape is an approximation that improves as n grows.
- For p̂, the normal approximation is reasonable when np and n(1 − p) are both more than 5.

## Worked examples

1. of = mean, mu = 90, sigma = 15, n = 25, find = between, a = 85, b = 92 gives probability = 0.699717, center = 90, se = 3, zLo = -1.666667, zHi = 0.666667. 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).
2. of = sum, mu = 90, sigma = 15, n = 80, find = above, a = 7,500 gives probability = 0.012674, center = 7,200, se = 134.164079, zLo = 2.236068. Source: OpenStax, Introductory Statistics 2e, §7.2 The Central Limit Theorem for Sums (ΣX ~ N(nμ, √n σ); Example 7.5: μ = 90, σ = 15, n = 80, P(Σx > 7,500) = 0.0127), https://openstax.org/books/introductory-statistics-2e/pages/7-2-the-central-limit-theorem-for-sums (retrieved 2026-10-05).
3. of = proportion, p = 0.6, n = 100, find = above, a = 0.65 gives probability = 0.153717, center = 0.6, se = 0.04899, zLo = 1.020621. 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).
4. of = mean, mu = 50, sigma = 10, n = 16, find = below, b = 48 gives probability = 0.211855, se = 2.5, zHi = -0.8. 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 does the central limit theorem say?

If you take many random samples of size n from a population with mean μ and standard deviation σ, the sample means form a distribution that is close to normal, with mean μ and standard deviation σ ÷ √n. This holds whatever the shape of the population, and it gets closer to normal as n grows.

### How do I use the central limit theorem to find a probability?

Find the standard error σ ÷ √n, turn your value into a z-score with z = (x̄ − μ) ÷ (σ ÷ √n), and read the normal area. For μ = 90, σ = 15 and n = 25, the standard error is 3, and the chance that x̄ is between 85 and 92 is 0.6997.

### What is the standard error of the mean?

It is the standard deviation of the sample means: σ ÷ √n. It shrinks as the sample gets bigger, so larger samples give means that sit closer to μ. Four times the sample size halves the standard error.

### Does the central limit theorem work for sums?

Yes. The sum of n values has mean nμ and standard deviation √n × σ, and its distribution is also close to normal for large n. For μ = 90, σ = 15 and n = 80, the sum has mean 7,200 and standard deviation 134.16.

### How large does the sample need to be?

If the population is normal, any n works and the result is exact. Otherwise the normal shape is an approximation that improves with n, and a skewed population needs a larger n. For a proportion, a usual check is that np and n(1 − p) are both more than 5.

### What is the difference between σ and the standard error?

σ is the spread of single values in the population. The standard error is the spread of a statistic, such as the sample mean, from one sample to the next. For a mean it is σ ÷ √n, always smaller than σ when n is more than 1.

## Sources

- 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)
- OpenStax, Introductory Statistics 2e, §7.2 The Central Limit Theorem for Sums: ΣX ~ N(nμ, √n σ); Example 7.5 (n = 80, P(Σx > 7,500) = 0.0127). https://openstax.org/books/introductory-statistics-2e/pages/7-2-the-central-limit-theorem-for-sums (retrieved 2026-10-05)
- 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, §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)