Central limit theorem probability
Enter the population mean and standard deviation (or a proportion) and the sample size. The central limit theorem gives the sampling distribution, and the page finds the chance of a value below, above or between numbers you choose.
- Probability
- 0.6997
The probability is 0.6997, with a sampling distribution of mean 90 and standard error 3.
- Mean of the sampling distribution
- 90
- Standard error
- 3
- z of the lower value
- −1.6667
- z of the upper value
- 0.6667
- Normal approximation
- Exact when the population is normal. For other populations it is an approximation that improves as n grows.
Probability: 0.6997. The probability is 0.6997, with a sampling distribution of mean 90 and standard error 3.
The sampling distribution
How to calculate
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.
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.
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.
- 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
Each example is checked against the calculator on every build.
- 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 gives Probability 0.699717, Mean of the sampling distribution 90, Standard error 3, z of the lower value -1.666667, z of the upper value 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)
- Sampling distribution of Sample sum (Σx), Population mean (μ) 90, Population standard deviation (σ) 15, Sample size (n) 80, Probability Above, Lower value 7,500 gives Probability 0.012674, Mean of the sampling distribution 7,200, Standard error 134.164079, z of the lower value 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)
- Sampling distribution of Sample proportion (p̂), Population proportion (p) 0.6, Sample size (n) 100, Probability Above, Lower value 0.65 gives Probability 0.153717, Mean of the sampling distribution 0.6, Standard error 0.04899, z of the lower value 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)
- Sampling distribution of Sample mean (x̄), Population mean (μ) 50, Population standard deviation (σ) 10, Sample size (n) 16, Probability Below, Upper value 48 gives Probability 0.211855, Standard error 2.5, z of the upper value -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)
How it works
Pick the statistic, then give the population and the sample size n:
| Statistic | Mean of its distribution | Standard error |
|---|---|---|
| Sample mean x̄ | μ | σ ÷ √n |
| Sample sum Σx | n × μ | √n × σ |
| Sample proportion p̂ | p | √(p(1 − p) ÷ n) |
The statistic is treated as normal with that mean and standard error. For each value you type, z = (value − mean) ÷ standard error, and:
- Below a value b: P = Φ(z_b), the normal area to the left.
- Above a value a: P = 1 − Φ(z_a), worked out as Φ(−z_a) so a small tail keeps its digits.
- Between a and b: P = Φ(z_b) − Φ(z_a).
Rules:
- μ is between −10¹² and 10¹², σ is more than 0 and at most 10¹², p is strictly between 0 and 1, and n is a whole number from 1 to 10⁹.
- The value minus the mean, n × μ and p(1 − p) ÷ n are exact on the decimals you type; only the square root and the normal area use floating point.
- A lower value above the upper value gives no answer. So does a z-score too large to show (a standard error far smaller than the distance), with a message.
- The "Normal approximation" line says, for p̂, whether np and n(1 − p) are both more than 5; for x̄ and Σx, that the result is exact for a normal population and approximate otherwise.
Assumptions
- The samples are random, and the values in a sample are independent.
- The probability is the normal approximation, not an exact count for a discrete population.
Worked examples by hand
Sample mean, μ = 90, σ = 15, n = 25, between 85 and 92 (the default, OpenStax Example 7.1). The standard error is 15 ÷ √25 = 3. z = (85 − 90) ÷ 3 = −1.6667 and z = (92 − 90) ÷ 3 = 0.6667. P = Φ(0.6667) − Φ(−1.6667) = 0.7475 − 0.0478 = 0.6997.
Sample sum, μ = 90, σ = 15, n = 80, above 7,500 (OpenStax Example 7.5). The mean is 80 × 90 = 7,200 and the standard error √80 × 15 = 134.164. z = 300 ÷ 134.164 = 2.2361, and P = Φ(−2.2361) = 0.01267.
Sample proportion, p = 0.6, n = 100, above 0.65. The standard error is √(0.6 × 0.4 ÷ 100) = √0.0024 = 0.04899. z = 0.05 ÷ 0.04899 = 1.0206, and P = Φ(−1.0206) = 0.1537.
Sample mean, μ = 50, σ = 10, n = 16, below 48. The standard error is 10 ÷ 4 = 2.5. z = −2 ÷ 2.5 = −0.8, and P = Φ(−0.8) = 0.2119.
Other questions people ask
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.