What does the central limit theorem say in simple terms?

The central limit theorem says that if you take many random samples of size n from the same population and compute the sample mean each time, the distribution of those sample means will be approximately normal, even if the original data are not normal, as long as the sample size is not too small and the observations are independent.

When is it reasonable to use the central limit theorem?

The central limit theorem is typically safe to use when the sample size is moderate to large (commonly n ≥ 30), the observations are independent, and the population does not have extremely heavy tails or infinite variance. For strongly skewed or heavy-tailed populations, you may need larger n for the approximation to be reliable.

What is the sampling distribution of the mean under the central limit theorem?

If the population has mean μ and standard deviation σ, and the sample size is n, then under the central limit theorem the sample mean X̄ is approximately normal with mean μ and standard deviation σ/√n, called the standard error. This approximation improves as n increases.

How is the central limit theorem used to choose a sample size?

For estimating a population mean with a given margin of error at a chosen confidence level, the central limit theorem leads to the formula n = (z · σ / E)², where z is the normal critical value for the confidence level, σ is the population standard deviation or a planning estimate, and E is the desired margin of error.

Central Limit Theorem Calculator & Sampling Distribution Explorer

Use the central limit theorem (CLT) to approximate probabilities for the sample mean, compute the standard error, and design sample sizes for a target margin of error at 90%, 95%, or 99% confidence.

Sampling distributions · CLT · Normal approximation Margin of error · Confidence intervals

1. Probability for the sample mean (central limit theorem)

Population mean (μ)

Population standard deviation (σ)

Use a planning estimate if σ is unknown.

Sample size (n)

CLT approximation improves as n grows.

Probability mode

Within a symmetric margin around μ: P(μ − m ≤ X̄ ≤ μ + m) Custom interval for X̄: P(a ≤ X̄ ≤ b)

Margin m

Same units as μ (e.g., kg, points, dollars).

This computes P(μ − m ≤ X̄ ≤ μ + m) using X̄ ~ Normal(μ, σ/√n).

2. Margin of error & sample size (CLT)

Use the central limit theorem to plan a study: choose the sample size for a desired margin of error, or compute the implied margin of error for a given sample size.

Population σ (or estimate)

Confidence level

Solve for

Sample size (n)

What is the central limit theorem?

The central limit theorem (CLT) is one of the pillars of probability and statistics. In its most common form it states that if \( X_1, X_2, \dots, X_n \) are independent and identically distributed (i.i.d.) random variables with mean \( \mu \) and finite variance \( \sigma^2 \), then the sample mean \[ \bar X_n = \frac{1}{n}\sum_{i=1}^{n} X_i \] is approximately normally distributed when \( n \) is large:

\[ \bar X_n \approx \mathcal{N}\!\left(\mu,\; \frac{\sigma^2}{n}\right) \] Equivalently, the standardized mean \[ Z = \frac{\bar X_n - \mu}{\sigma / \sqrt{n}} \] is approximately standard normal \( \mathcal{N}(0, 1) \).

This is why, even when the raw data are skewed, we can often use normal-based methods (z-intervals, t-intervals, normal approximations to sampling distributions) once the sample size is sufficiently large and the independence conditions are reasonably met.

Sampling distribution of the mean

Under the CLT, the sampling distribution of the sample mean has:

Mean of the sampling distribution: \[ \mathbb{E}[\bar X_n] = \mu \]
Standard deviation (standard error): \[ \sigma_{\bar X} = \frac{\sigma}{\sqrt{n}} \]

Our calculator uses exactly this normal approximation, \(\bar X \sim \mathcal{N}(\mu, \sigma^2/n)\), to translate questions about sample means into z-scores and areas under the standard normal curve.

Conditions and practical rules of thumb

Independence. The sampled observations should be independent. This is often reasonable when sampling less than 10% of a large population at random.
Sample size. For roughly symmetric distributions, even n around 20 can be enough. For skewed distributions, many introductory texts recommend n ≥ 30; for very heavy tails, more may be needed.
Finite variance. The CLT requires the underlying variance σ² to be finite. For “wild” heavy-tailed distributions with infinite variance, the classical CLT does not apply.

The calculator does not check these conditions automatically. It assumes that the CLT is appropriate for your context. When in doubt, combine it with diagnostic plots, domain knowledge, and more advanced methods.

FAQ – Central limit theorem calculator

How accurate is the central limit theorem approximation?

Accuracy depends on the sample size and the shape of the underlying distribution. If the population is exactly normal, then the sample mean is exactly normal for any n. For moderately skewed distributions, the approximation becomes good for n around 30–50. For highly skewed or heavy-tailed distributions you may need substantially larger sample sizes for the sampling distribution of the mean to look approximately normal.

What is the difference between the CLT and the law of large numbers?

The law of large numbers says that the sample mean converges to the population mean as n → ∞, but it does not tell you how fast or give you a distribution. The central limit theorem goes further: it describes the approximate distribution of the sample mean for large yet finite n, including its spread (the standard error).

Why do we use z-scores in the CLT calculator?

Once we know that \(\bar X \approx \mathcal{N}(\mu, \sigma^2/n)\), any probability involving the sample mean can be mapped to the standard normal distribution via \[ Z = \frac{\bar X - \mu}{\sigma/\sqrt{n}}. \] The calculator uses this transformation internally and then evaluates the standard normal cumulative distribution function to obtain probabilities.

Can this calculator replace a full statistical analysis?

No. The tool is designed for education, quick checks and planning calculations. It does not perform diagnostic checks, robust estimation, or handle complex designs (stratified sampling, clustering, time series, etc.). For high-stakes decisions in clinical, industrial or financial contexts, always complement this with a full statistical analysis and expert supervision.