Central Limit Theorem Calculator
Try different values to see how sample size affects the distribution of sample means
Simulation Statistics
The Central Limit Theorem (CLT)
The Central Limit Theorem is one of the most important concepts in statistics. It states that if you take sufficiently large samples from any population, the distribution of the sample means will be approximately normally distributed, regardless of the shape of the original population distribution.
More formally, if X₁, X₂, ..., Xₙ are independent, identically distributed random variables with mean μ and finite variance σ², then the sampling distribution of the sample mean X̄ approaches a normal distribution with mean μ and variance σ²/n as the sample size n increases.
Key Implications of the CLT
- Normal approximation: As sample size increases, the distribution of sample means approaches a normal distribution.
- Same mean: The mean of the sampling distribution equals the population mean.
- Reduced variability: The standard deviation of the sampling distribution (standard error) equals the population standard deviation divided by the square root of the sample size: σ/√n.
- Independence from original distribution: This happens regardless of the shape of the original population distribution.
When Does the CLT Apply?
The CLT applies when:
- Samples are random
- Samples are independent
- Sample size is sufficiently large (typically n ≥ 30, but depends on the skewness of the original distribution)
- The population has a finite variance
Key Formulas
Mean of sampling distribution: μX̄ = μ
Standard error (standard deviation of sampling distribution): σX̄ = σ/√n
Standardized sample mean (z-score): z = (X̄ - μ) / (σ/√n)
Common Population Distributions
| Distribution | Parameters | Mean | Variance |
|---|---|---|---|
| Uniform | Min (a), Max (b) | (a + b) / 2 | (b - a)² / 12 |
| Normal | Mean (μ), SD (σ) | μ | σ² |
| Exponential | Rate (λ) | 1/λ | 1/λ² |
| Chi-Squared | Degrees of freedom (df) | df | 2 × df |
| Binomial | Trials (n), Probability (p) | n × p | n × p × (1-p) |