Z-score Calculator

Use this Z-Score Calculator to standardize a value, calculate a z-score for a sample mean, or convert between z-scores and percentiles using the standard normal distribution.

Calculation Method

For a single value:

z = (x − μ) ÷ σ

For a sample mean:

z = (x̄ − μ) ÷ (σ ÷ √n)

To convert a z-score to a percentile:

Percentile = Φ(z) × 100

Where Φ(z) is the cumulative distribution function of the standard normal distribution.

Example

Suppose a value is 85, the mean is 70, and the standard deviation is 10:

z = (85 − 70) ÷ 10 = 1.5

The value is therefore 1.5 standard deviations above the mean.

Important Note

Z-to-percentile conversions assume a standard normal model. For a sample mean, this calculator uses the population standard deviation. If population standard deviation is unknown and estimated from the sample, a t-based method may be more appropriate.

Sources

• Penn State STAT 200 — Standard Normal Distribution and Z-Scores
• OpenStax — Using the Normal Distribution