Statistics

Normal Probability Calculator for Sampling Distributions

Find the probability for a sample mean under the CLT.

Calculated Output

Z-score 1.2

P(X̄ ≤ value) 0.8849

Formula: z = (x̄ − μ) ÷ (σ/√n)

Step-by-Step Breakdown

1. Values used:

2. Population mean = 100

3. Population SD = 15

4. Sample size = 36

5. Sample mean = 103

7. Z-score = 1.2

8. P(X̄ ≤ value) = 0.8849

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The math behind it

z = (x̄ − μ) ÷ (σ/√n), then look up the normal CDF for the probability.

μ 100, σ 15, n 36, x̄ 103 → z 1.2, P(≤) ≈ 0.885.

Why divide by √n?

Sample means vary less than individuals, by a factor of √n.