Z-Score Calculator

Calculate the z-score (standard score) of a raw value relative to a population. Find how many standard deviations a value is from the mean and estimate the corresponding percentile.

About this Calculator

Calculate the z-score (standard score) of a raw value relative to a population. Find how many standard deviations a value is from the mean and estimate the corresponding percentile.

Formula & Calculations

Formula

z = (x − μ) / σ

Where:

x=Raw data value
μ=Population mean
σ=Population standard deviation
z=Z-score (number of standard deviations from the mean)

Assumptions

The underlying data follows a normal (Gaussian) distribution.
Percentile is approximated using a standard normal cumulative distribution function.
A positive z-score means the value is above the mean; negative means below.

Calculation Examples

Example 1

Inputs:x = 85, μ = 75, σ = 5

Result:z = 2.00, Percentile ≈ 97.7% (Above Average)

z = (85 − 75) / 5 = 2.00. A z-score of 2.00 means the value is 2 standard deviations above the mean, placing it in the 97.7th percentile.

Example 2

Inputs:x = 110, μ = 130, σ = 15

Result:z = −1.33, Percentile ≈ 9.2% (Below Average)

z = (110 − 130) / 15 = −1.33. The value is 1.33 standard deviations below the mean.

Frequently Asked Questions

What does a negative z-score mean?

A negative z-score indicates that the raw value is below the population mean. For example, a z-score of −1 means the value is exactly 1 standard deviation below the mean. This is not 'bad' — it simply describes the value's position relative to the average.

What z-score is considered statistically significant?

In hypothesis testing, a z-score with an absolute value greater than 1.96 (for a 95% confidence level) is typically considered statistically significant. Z-scores beyond ±2.58 correspond to significance at the 99% confidence level.