IntroStatsCalc

Central Limit Theorem Calculator (Sample Means)

Probabilities for a sample mean x̄ using σ/√n.

Inputs

I want to find
Area

Results update as you type.

Results

P(x̄ < 68) = 0.9452

About 94.52% of sample means (n = 36) fall below 68.

Sampling distribution of x̄ curve with mean 64 and standard deviation 2.5. About 94.52% of sample means (n = 36) fall below 68. The shaded area is 0.9452.56.55961.56466.56971.5

Results

P(x̄ < 68)0.9452
z for 681.6
σx̄ = σ/√n2.5

On the TI-84

2nd VARS (DISTR) → 2:normalcdf(lower, upper, μ, σ). Use −1E99 or 1E99 for an open end (the E is 2nd ,).

normalcdf(-1E99, 68, 64, 2.5)

Show steps
  1. Standard error. The sample mean x̄ is approximately normal with mean μ = 64 and standard deviation σx̄ = σ ÷ √n = 15 ÷ √36 = 2.5.
  2. Standardize. z = (68 − 64) ÷ 2.5 = 1.6.
  3. Find the area. Area to the left of the z-score = 0.9452.
  4. Interpret. About 94.52% of sample means (n = 36) fall below 68.

About this calculator

The Central Limit Theorem

Take many random samples of size n and compute each sample's mean. Those sample means form their own distribution, which is centered at the population mean μ and has a smaller spread: σx̄ = σ ÷ √n, called the standard error. When n is large (a common guideline is n ≥ 30), that distribution is approximately normal even if the population isn't.

Individual vs. average

"What's the chance one person is above 68?" uses σ (the Normal calculator). "What's the chance the average of 36 people is above 68?" uses σ/√n (this page). Averages vary much less than individuals.

More distributions calculators: Normal · t-Distribution · Chi-Square Distribution · F-Distribution