What does a z-score actually tell me?

It tells you how far a value is from the mean in units of standard deviation. A z-score of 1.5 means the value is one and a half standard deviations above average, while a negative z-score means it falls below the mean.

How is the percentile computed from the z-score?

The calculator applies the standard normal cumulative distribution function to the z-score, which gives the proportion of the distribution falling at or below the value. Multiplying that proportion by 100 yields the percentile rank.

Is the probability exact?

It uses a well-known rational approximation of the normal CDF (Abramowitz & Stegun), which is accurate to several decimal places across the usual range of z-scores. For typical scoring and grading purposes the result is effectively exact.

What happens if the standard deviation is zero?

A zero standard deviation means every value equals the mean, so a z-score is undefined. The calculator handles this gracefully by returning a z-score of 0, a probability of 0.5, and the 50th percentile.

Z-Score Calculator

Value

Mean

Standard Deviation

Z-Score-1.0000

Probability0.158655

Percentile15.87%

A z-score expresses how many standard deviations a value sits above or below the mean, turning raw data into a standardized score. This calculator computes the z-score from a value, mean, and standard deviation, then uses the standard normal distribution to report the cumulative probability and the percentile rank of that value.

Formula

z = (x − μ) / σ ; percentile = Φ(z) × 100

x: The raw value being standardized
μ: The mean of the distribution
σ: The standard deviation of the distribution
Φ(z): The standard normal cumulative probability of the z-score

How it works

Enter the raw value (x), the distribution mean (μ), and the standard deviation (σ).
The z-score is (x − μ) / σ — positive when the value is above the mean, negative when below.
The calculator feeds the z-score into a normal cumulative distribution approximation to find the probability of scoring at or below the value, then multiplies by 100 to give the percentile. A standard deviation of zero returns a z-score of 0 and a 50th percentile.

Worked example

A test score of 85 in a class with mean 70 and standard deviation 10.

z = (85 − 70) / 10 = 15 / 10 = 1.5.
The standard normal CDF at z = 1.5 is about 0.9332.
Percentile = 0.9332 × 100 ≈ 93.32.

z = 1.5, probability ≈ 0.9332, about the 93rd percentile

Frequently asked questions

What does a z-score actually tell me?: It tells you how far a value is from the mean in units of standard deviation. A z-score of 1.5 means the value is one and a half standard deviations above average, while a negative z-score means it falls below the mean.
How is the percentile computed from the z-score?: The calculator applies the standard normal cumulative distribution function to the z-score, which gives the proportion of the distribution falling at or below the value. Multiplying that proportion by 100 yields the percentile rank.
Is the probability exact?: It uses a well-known rational approximation of the normal CDF (Abramowitz & Stegun), which is accurate to several decimal places across the usual range of z-scores. For typical scoring and grading purposes the result is effectively exact.
What happens if the standard deviation is zero?: A zero standard deviation means every value equals the mean, so a z-score is undefined. The calculator handles this gracefully by returning a z-score of 0, a probability of 0.5, and the 50th percentile.

Related calculators

Standard Deviation Calculator Confidence Interval Calculator Statistics Calculator Sample Size Calculator