Central Limit Theorem Calculator
Explore the Central Limit Theorem (CLT) and solve real problems about the sample mean — watch the sampling distribution tighten as n grows, compute probabilities like P(X̄ ≤ x), or work backward from a target probability to a cutoff value — each with its own visual, full step-by-step math, and a callout explaining what the number actually means.
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
- Choose Understand CLT to watch the sampling distribution tighten around μ as n grows, compared side-by-side with the spread of a single observation.
- Choose Solve a Probability to compute left tail, right tail, between, or outside probabilities for X̄.
- Choose Solve for x to find the cutoff value x from a target probability p (inverse normal) — this mode has its own left/right-tail choice, separate from Solve a Probability's.
- Click Calculate to see the visual, the full step-by-step math, and a callout explaining what the result means.
How the Central Limit Theorem Works
The sample mean is centered at the population mean: μX̄ = μ. Averaging doesn't shift where the distribution is centered.
The spread of the sample mean is the standard error: SE = σ/√n. A single observation (n=1) has spread σ; averaging n of them shrinks that spread by a factor of √n.
As n grows, the distribution of X̄ becomes approximately normal — even if the original population is skewed, uniform, or bimodal. This is the theorem's actual claim, and it's why the same normal-CDF machinery works regardless of the population's shape.
To find a probability, convert X̄ to a z-score with z = (x-μ)/SE, then read it off the standard normal CDF Φ(z).
To go the other direction — from a target probability p back to a cutoff x — use the inverse normal z = Φ-1(p), then convert back with x = μ + z·SE.
Left tail, right tail, between, and outside probabilities always pair up as complements: a tail and its opposite side always add to 1, and "between" plus "outside" (with the same bounds) always add to 1 too.
Formula & Equations Used
Standard error: SE = σ/√n
z-score for X̄: z = (x-μ)/SE
Inverse (quantile): x = μ + Φ-1(p)·SE
Example Problems & Step-by-Step Solutions
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).
Step: SE = 10/√30 ≈ 1.826. z = (54-50)/1.826 ≈ 2.19.
Result: P(X̄ ≤ 54) = Φ(2.19) ≈ 0.9857.
Example 2 — Between probability
A population has μ = 100 and σ = 15. For samples of size n = 36, find P(97 ≤ X̄ ≤ 103).
Step: SE = 15/√36 = 2.5. z(97) = -1.2, z(103) = 1.2.
Result: P = Φ(1.2) - Φ(-1.2) ≈ 0.7699.
Example 3 — Solve for x (quantile)
A population has μ = 50, σ = 10, n = 30. Find x such that P(X̄ ≤ x) = 0.95.
Step: SE ≈ 1.826. z = Φ-1(0.95) ≈ 1.645.
Result: x = 50 + 1.645(1.826) ≈ 53.00.
Example 4 — Quality-control outside spec
A filling line targets μ = 500g with σ = 8g. For batches of n = 25, find the chance a batch average lands outside 496–504g.
Step: SE = 8/5 = 1.6. z(496) = -2.5, z(504) = 2.5.
Result: P = Φ(-2.5) + [1-Φ(2.5)] ≈ 0.0124 — about 1.24% of batches.
Frequently Asked Questions
Does the population have to be normal?
No. The CLT says X̄ becomes approximately normal as n increases, even if the population is skewed. Very skewed or heavy-tailed populations often need larger n for a good approximation.
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. Understand CLT mode plots this directly: a single observation (n=1) has spread σ, while your chosen n has the much narrower spread σ/√n.
How do I find x when the problem gives a probability like P(X̄ ≤ x) = 0.95?
Use Solve for x. The calculator finds the z-value with the inverse normal (z = Φ-1(p)) and converts back using x = μ + z·(σ/√n).
Why don't I see "Left tail / Right tail / Between / Outside" in Solve-for-x mode?
Those four options apply to Solve a Probability, where you already know x and want a probability. In Solve for x, you already know the probability and want x back, so that mode has its own, simpler left/right-tail choice instead.
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