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?
- Identify values: x = 1350, μ = 1200, σ = 100
- Apply formula: z = (1350 − 1200) / 100 = 150 / 100 = 1.50
- Interpret: The bulb lasted 1.50 standard deviations above the mean.
- 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.
Z Score Calculator FAQs
Enter the raw value, the mean, and the standard deviation into this Z score calculator to get z = (x − μ) / σ instantly. On a TI-84, you can compute the same value manually with the expression (x − μ) / σ, since there is no dedicated “Z score” button.
A negative Z score means the value is below the mean of its distribution. A Z score of −1.5, for example, indicates the value is 1.5 standard deviations below the mean. A positive Z score means the value is above the mean, and a Z score of 0 means the value equals the mean exactly.
There is no universal “good” Z score — it depends on context. In most applications, Z scores between −2 and 2 are considered typical, since about 95% of values in a normal distribution fall in that range. Whether a given Z score is “good” depends on whether higher or lower raw values are desirable for the specific measurement.
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