Central Limit Theorem Calculator
Find the mean, standard error and z-score of a sample mean for independent, identically distributed observations with known population SD.
Formula
E(X̄) = μ; SE = σ/√n; z = (x̄ − μ)/SE.
Mean of sample means ≈ (100) · Standard error (same unit) ≈ (15)÷sqrt((100)) · z-score (unitless) ≈ ((103)-(100))÷(1.5)
How to use
- Enter population mean μ and SD σ in the same unit.
- Enter sample size n and observed mean x̄ in that unit.
- Read the sampling mean, SE and z-score.
Examples
- μ = 100, σ = 15, n = 100, x̄ = 103 give SE = 1.5 and z = 2.
Frequently asked questions
When is the normal model valid?
Exactly for a normal population; otherwise approximately for sufficiently large n with finite variance.
Is n = 30 always enough?
No. Strong skew or heavy tails can require larger samples. Independence is essential.
Does this give a probability?
No. It reports SE and z only; the population SD must be known.
Sources
Updated: