How to Calculate a Z-Score and Find Probabilities

A z-score (or standard score) tells you how many standard deviations a value is from the mean of its distribution. It is the key that unlocks the normal distribution: once you have a z-score, you can find the probability of a value falling above or below any point. This guide explains the calculation and how to turn it into a probability.

Follow along with the free Z-Score & Normal Distribution Calculator, which reports the z-score, probabilities, and percentile with steps.

The Z-Score Formula

📐 Formula

z = (x − μ) / σ x = the value, μ = the mean, σ = the standard deviation

A positive z-score means the value is above the mean; a negative z-score means it is below. A z-score of 0 is exactly at the mean.

Step-by-Step Example

IQ scores are normally distributed with a mean μ = 100 and standard deviation σ = 15. What is the z-score for an IQ of 130?

📐 Formula

z = (130 − 100) / 15 = 30 / 15 = 2.0

An IQ of 130 is 2 standard deviations above the mean.

Converting a Z-Score to a Probability

To find the probability that a value is below x, use the standard normal cumulative distribution function, Φ(z). For z = 2.0:

📐 Formula

P(X < 130) = Φ(2.0) ≈ 0.9772 (the 97.72nd percentile) P(X > 130) = 1 − 0.9772 ≈ 0.0228

So about 97.72% of people have an IQ below 130, and only about 2.28% score above it.

The Empirical Rule (68–95–99.7)

For any normal distribution:

About 68% of values fall within z = ±1 (one standard deviation of the mean).
About 95% fall within z = ±2.
About 99.7% fall within z = ±3.

This is why a |z| greater than 2 or 3 is considered unusual.

Why Z-Scores Matter

Comparing across scales. A z-score lets you compare a test score from one exam to another with a different mean and spread.
Detecting outliers. Large |z| values flag unusual observations.
Hypothesis testing. Z-scores underpin confidence intervals and many significance tests.

Try It Yourself

Enter your value, mean, and standard deviation into the Z-Score & Normal Distribution Calculator to get the z-score, P(X < x), P(X > x), and percentile instantly. To first compute the mean and standard deviation from raw data, use the Standard Deviation & Statistics Calculator.

Key Takeaways

A z-score measures how many standard deviations a value is from the mean: z = (x − μ) / σ.
Convert a z-score to a probability with the normal CDF, Φ(z).
For IQ 130 (μ = 100, σ = 15): z = 2.0, and about 97.72% of scores fall below it.
The 68–95–99.7 rule gives a quick sense of how extreme a z-score is.