Central Limit Theorem Calculator
Use our central limit theorem calculator to compute the sampling distribution of the sample mean: the expected sample mean and the standard error (spread) based on population mean (μ), population standard deviation (σ), and sample size (n). Includes a clear central limit theorem explained, when to use central limit theorem, and how to use central limit theorem for probability between two numbers.
What Is the Central Limit Theorem?
The central limit theorem (CLT) is one of the most important ideas in statistics. It says that when you take enough random samples of size n from a population, the distribution of the sample mean becomes approximately normal (bell-shaped) — even if the original population is not normal.
This matters because it lets you use normal-probability tools (z-scores and the normal CDF) to estimate probabilities about sample means. That’s why you’ll see “central limit theorem calculator probability” and “probability between two numbers” — those questions typically refer to probabilities about the sample mean, not individual values.
In CLT problems, the key outputs are (1) the mean of the sampling distribution (the expected sample mean) and (2) the standard deviation of that sampling distribution, usually called the standard error.
Central Limit Theorem Formulas
For the sampling distribution of the sample mean (x̄), CLT uses the population mean μ and population standard deviation σ to describe the mean and spread of x̄.
The sampling distribution of the sample mean is centered at the population mean.
This is the standard deviation of x̄ (often called the standard error).
Φ is the standard normal CDF. This is the common “central limit theorem calculator probability between two numbers” setup.
Larger sample sizes make the sampling distribution tighter around μ.
This is how TI-84 and most CLT probability workflows work under the hood.
Definitions
These are the most common terms you’ll see in central limit theorem problems.
- Central limit theorem (CLT)
- A theorem stating that the distribution of the sample mean becomes approximately normal as sample size n grows, under common sampling conditions.
- Sample mean (x̄)
- The average of a sample. CLT focuses on the distribution of x̄ across many samples.
- Sampling distribution
- The probability distribution of a statistic (like x̄) over repeated random samples of size n.
- Standard error (σx̄)
- The standard deviation of the sampling distribution of x̄. For CLT it is σ/√n.
- Φ (normal CDF)
- The cumulative distribution function of the standard normal distribution used to compute probabilities from z-scores.
How to Use the Central Limit Theorem Calculator
- 1
Enter the population mean (μ).
- 2
Enter the population standard deviation (σ).
- 3
Enter the sample size (n).
- 4
Read the expected sample mean (μx̄) and the standard error (σx̄ = σ/√n).
Frequently Asked Questions
The central limit theorem says that the sample mean x̄ tends to follow an approximately normal distribution as sample size n increases, even if the population distribution is not normal (under typical random sampling conditions).
Use CLT when you’re working with the sample mean (or sums/averages) and want to approximate probabilities using a normal model — especially when the population isn’t known to be normal and your sample size is large enough.
n ≥ 30 is a common rule of thumb, not a universal law. If the population is close to normal, smaller n can work well. If the population is highly skewed or heavy-tailed, you may need a larger n for the normal approximation to be accurate.
Compute the standard error σ/√n, convert both bounds (a and b) into z-scores using z = (x̄ - μ) / (σ/√n), then subtract CDF values: Φ(zb) - Φ(za).
A TI-84 typically converts the bounds into z-scores using the standard error (σ/√n) and then uses a normal CDF function to return the probability.
No. CLT describes the distribution of the sample mean (x̄), not the original population. The population can be skewed and still produce an approximately normal sampling distribution for x̄ when n is large enough.
Most CLT graphs show the sampling distribution of the sample mean — centered at μ with spread σ/√n — and may shade the area between two bounds to represent a probability.