z-score
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Z-Score Calculator: Single Value, Data Set, and Percentile
Compute a z-score with z = (x - mu)/sigma for one value or a whole data set, and convert between z-scores and percentiles. Population SD by default, sample SD one toggle away, method shown.
Enter values and calculate to see the z-score.
Percentile (left area)
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| Value | z |
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Percentile (left area)
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Result appears here.
z = (x - μ) / σ
How?
How this is calculated
A z-score reports how many standard deviations a value sits from the mean: z = (x - μ) / σ. With x = 85, μ = 70, and σ = 10, the score is 15 points above the mean, which is 1.5 standard deviations, so z = 1.5. The numerator and denominator share the same units, so z is dimensionless and comparable across different scales.
Formula: z = (x - μ) / σ
Population or sample standard deviation
Data-set mode standardizes every point against the data set's own mean and standard deviation. It defaults to the population divisor n, because standardizing within a data set treats that set as the reference distribution, and the resulting z-scores then have a population standard deviation of 1. A named toggle switches to the sample divisor n - 1.
This default differs from the descriptive-statistics calculator, which defaults to the sample divisor n - 1 because it estimates a wider population's spread from a sample. The same paste of 2, 4, 4, 4, 5, 5, 7, 9 yields a population SD of 2 here and a sample SD of 2.1381 there. Each default answers the distinct question its tool asks.
From a z-score to a percentile, and back
A percentile here is the left cumulative area under the standard normal curve: percentile = 100 · Φ(z), the share of a standard normal distribution that falls below z. A z of 1.5 lands at the 93.32nd percentile. Run in reverse, the 97.5th percentile corresponds to z = 1.96, the familiar two-sided 95% critical value. The bridge is defined on the open interval from 0 to 100; the exact endpoints have no finite z, so they are rejected.
When t replaces z
A z-score assumes the standard deviation is a known population value. When σ is instead estimated from a small sample, the t-distribution is the correct reference, and its heavier tails give a wider spread than the normal curve. Single-value mode takes μ and σ as given, so it applies no n - 1 correction.
Common questions
How do I get a z-score for each value in a data set?
Paste the column into data-set mode. Every point is standardized against the list's own mean and standard deviation, and the table returns one z per row in input order.
What does a negative z-score mean?
A negative z-score marks a value below the mean. A z of -2 sits two standard deviations below the mean, at roughly the 2.28th percentile.
Why four decimals instead of the two-decimal z-table value?
Printed z-tables round to two decimals as a lookup convenience. The calculation carries full precision and displays four decimals, so a value near a table cell is not rounded before you use it.
Sources
- Standard score (Wikipedia). Wikipedia. Retrieved .
- NIST/SEMATECH e-Handbook, standardizing. NIST. Retrieved .
- Standard deviation worked example (Wikipedia). Wikipedia. Retrieved .
Method last reviewed