Z Score Calculator

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

Use this Z Score Calculator to convert any raw value into its standardized Z score. The Z score tells you how many standard deviations a data point is from the mean of its distribution. Enter the value, population mean, and standard deviation to calculate the Z score along with the corresponding cumulative probability and percentile.

What Is a Z Score?

A Z score (also called a standard score or z-value) measures how far a data point is from the mean of its distribution in units of standard deviations. A positive Z score indicates the value is above the mean, while a negative Z score indicates it is below the mean. A Z score of zero means the value is exactly at the mean.

Z scores are fundamental in statistics because they allow you to compare values from different distributions on a common scale. For example, a student who scored 85 on a test with mean 75 and standard deviation 10 has a Z score of 1.0. Another student who scored 90 on a different test with mean 80 and standard deviation 5 also has a Z score of 2.0. By comparing Z scores, you can see that the second student performed relatively better within their group despite both scoring above their respective means.

Z scores also connect directly to probabilities through the standard normal distribution. Once you know the Z score, you can look up the cumulative probability to determine what percentage of the population falls below that value.

Z Score Formula

The Z score is calculated using the standardization formula:

z = (x − μ) / σ

where:

• x is the raw value (data point)
• μ is the population mean
• σ is the population standard deviation

This formula transforms any normal distribution into the standard normal distribution (mean = 0, standard deviation = 1). The cumulative probability is then found using the standard normal CDF: P(Z ≤ z) = Φ(z).

Worked Example

A factory produces light bulbs with a mean lifespan of 1200 hours and a standard deviation of 100 hours. What is the Z score for a bulb that lasts 1350 hours?

1. Identify values: x = 1350, μ = 1200, σ = 100
2. Apply formula: z = (1350 − 1200) / 100 = 150 / 100 = 1.50
3. Interpret: The bulb lasted 1.50 standard deviations above the mean.
4. Find probability: P(Z ≤ 1.50) = Φ(1.50) ≈ 0.9332, so about 93.32% of bulbs last 1350 hours or less. This bulb is in the 93rd percentile.

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