Z-score: what it is, what z = 1.5 means, and how to read it
A z-score, or standard score, says how many standard deviations a value sits from the mean and in which direction. It is the one number that lets you place a value from any normal distribution on a common scale, which is why a question like "what percentile is 1.5 standard deviations above the mean?" has a single answer regardless of the data.
The value 82 sits 1.5 standard deviations above the mean of 70. The shaded area is 93.3% of a normal distribution.
Show the working, step by step
The formula
A value x with mean μ and standard deviation σ standardises to:
z = (x − μ) ÷ σ
Positive z is above the mean, negative below. Because the value is divided by its own standard deviation, the answer is unit-free, which is what makes z-scores comparable between a set of test scores, a set of heights and a set of lab readings.
What a z-score of 1.5 means
A z-score of 1.5 is a value 1.5 standard deviations above the mean. In a normal distribution that is the 93.3rd percentile, above roughly 93% of values. The common landmarks are:
| z-score | Percentile (normal) | Reading |
|---|---|---|
| −1 | 15.9th | One SD below the mean |
| 0 | 50th | At the mean |
| 1 | 84.1st | One SD above the mean |
| 1.5 | 93.3rd | The value this page works through |
| 2 | 97.7th | Two SD above; the usual "high" cut-off |
| 3 | 99.9th | Three SD above; the usual "extreme" cut-off |
Worked example
A national math test is normally distributed with a mean of 8 and a standard deviation of 1.5. A student scores 10.25. Their z-score is
- Standardise the value: z = (10.25 − 8) ÷ 1.5 = 1.5.
- Look up z = 1.5: the area to the left is 0.9332.
- So 10.25 sits at the 93.3rd percentile, above about 93% of all test-takers.
The z-score calculator above runs all three steps on any value, mean and SD, and returns the percentile and the two-tailed p-value alongside the z itself.
When a z-score is the right tool
Use a z-score when you need to compare a value against its own distribution, or to place two values from different distributions on one scale. It is the building block of the z table, of the 68–95–99.7 bands the empirical rule calculator draws, and of the outlier flags the calculators raise past ±3. It assumes the data are roughly normal; for skewed data the percentile still works but the "93.3rd" number is an approximation, and the skewness calculator tells you how far off normal you are.
Common questions
What is a z-score of 1.5?
A value 1.5 standard deviations above the mean. In any normal distribution that puts it at the 93.3rd percentile, higher than about 93% of the data. The z-score calculator gives the exact percentile for any z.
What does a z-score tell you?
How many standard deviations a value sits from the mean, and in which direction: positive above, negative below. It is a standardised score, so it compares values from datasets with different units. A z-score of −1 means one standard deviation below the mean, the same relative position no matter what the data measure.
How do I find the percentile from a z-score?
Read the area under the standard normal curve to the left of z. For z = 1.5 that area is 0.9332, so the value is at the 93.3rd percentile. The percentile calculator and the z table give the same lookup; the calculator does it for any z without rounding to the table's two-decimal grid.
What z-score is a value of 10.25 when the mean is 8 and the standard deviation is 1.5?
z = (x − μ) ÷ σ = (10.25 − 8) ÷ 1.5 = 1.5. The value is 1.5 standard deviations above the mean, so it sits at the 93.3rd percentile of the normal distribution with that mean and SD.
Is a z-score of 2 high?
In a normal distribution yes: it is the 97.7th percentile, above about 98% of values, and values beyond ±2 cover only the outer 5%. That is why 2 and 3 standard deviations are the usual cut-offs for "unusually high" and "an outlier". The outlier and IQR calculator uses the quartile-based cut-offs, and the z-score calculator flags values beyond ±3.
Related calculators
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Z-score calculator
Turn a value, mean and SD into z, with its percentile.
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Percentile calculator
The position of a value in your own data.
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Z table
The printed standard normal areas behind the lookup.
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Empirical rule calculator
The 68–95–99.7 bands the z-scores sit inside.
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Normal distribution calculator
Areas and cut-offs for any mean and SD.