Central Limit Theorem Calculator
Explore the Central Limit Theorem (CLT) and solve real problems about the sample mean. Enter the population mean μ, population standard deviation σ (known), and sample size n — then compute the standard error, z-scores, and probabilities like P(X̄ ≤ x) with clean steps + a mini visual.
Background
The CLT says that when you take many random samples of size n, the distribution of the sample mean X̄ becomes approximately normal as n grows — even if the original population is skewed. For the sample mean: μX̄ = μ and σX̄ = σ / √n (called the standard error).
How to use this calculator
- Enter μ, σ (known), and n.
- Choose Understand CLT to watch the sampling distribution tighten as n grows.
- Choose Solve a probability to compute left tail, right tail, between, or outside probabilities for X̄.
- Turn on step-by-step if you want the z-score math shown.
- Choose Solve for x (quantile) to find the cutoff value x from a target probability p (inverse normal).
How this calculator works
- Sampling distribution mean: μX̄ = μ
- Standard error: σX̄ = σ/√n
- Convert values to z-scores: z = (x-μ)/(σ/√n)
- Use the standard normal CDF to find probabilities.
Formula & Equation Used
Standard error: SE = σ/√n
z-score for X̄: z = (x-μ)/SE
Example Problem & Step-by-Step Solution
Example 1 — Probability for the sample mean
A population has μ = 50 and σ = 10. For samples of size n = 30, find P(X̄ ≤ 54).
- Compute standard error: SE = 10/√30 ≈ 1.826.
- Compute z: z = (54-50)/1.826 ≈ 2.19.
- Probability: P(X̄ ≤ 54) = Φ(2.19).
Example 2 — Between probability
A population has μ = 100 and σ = 15. For samples of size n = 36, find P(97 ≤ X̄ ≤ 103).
- Standard error: SE = 15/√36 = 2.5.
- z-scores: z(97) = (97-100)/2.5 = -1.2, z(103) = (103-100)/2.5 = 1.2.
- Probability: P = Φ(1.2) - Φ(-1.2).
Example 3 — Solve for x (quantile / inverse normal)
A population has μ = 50, σ = 10, and n = 30. Find x such that P(X̄ ≤ x) = 0.95.
- Standard error: SE = 10/√30 ≈ 1.826.
- Find z for 0.95: z = Φ-1(0.95) ≈ 1.645.
- Convert back to x: x = μ + z·SE ≈ 50 + 1.645(1.826) ≈ 53.00.
Frequently Asked Questions
Q: Does the population have to be normal?
No. CLT says X̄ becomes approximately normal as n increases — even if the population is skewed. If the population is extremely skewed or heavy-tailed, you usually need a larger n.
Q: Why does bigger n make things “tighter”?
Because the standard error is σ/√n. As n grows, you divide by a bigger number, so the spread of X̄ shrinks.
Q: How do I find x when the problem gives a probability like P(X̄ ≤ x) = 0.95?
Use Solve for x (quantile). The calculator finds the z-value with the inverse normal (z = Φ-1(p)) and then converts back using x = μ + z·(σ/√n).
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