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Z-Score Calculator

Find how many standard deviations a value is from the mean.

How it is calculated

The z-score is z = (x - μ) / σ, where x is the observed value, μ is the mean of the distribution and σ is its standard deviation. All three are in the same unit, which cancels out, so z is a pure number counting standard deviations. A positive z means the value is above the mean, a negative z below it, and 0 means exactly at the mean. The standard deviation must be greater than zero; with zero or negative σ the tool returns no result. The answer is rounded to two decimals. The tool only standardises the value; turning z into a percentile requires a normal-distribution table.

Worked example

An exam has a mean of 70 and a standard deviation of 10, and a student scored 85. Difference from the mean: 85 - 70 = 15. Divide by the standard deviation: 15 / 10 = 1.50. The z-score is 1.50, meaning the score is one and a half standard deviations above average.

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Common mistakes

  • Entering the variance instead of the standard deviation. If variance is 100, σ is 10.
  • Mixing sample and population standard deviation without noticing which one the data source reports.
  • Reversing the subtraction (mean - value), which flips the sign and turns an above-average result into a below-average one.
  • Reading z as a percentage. A z of 1.50 is not 1.5%; it is a distance in standard deviations.

About this calculator

The z-score standardizes a value against a distribution using z = (x − μ) ÷ σ, where μ is the mean and σ is the standard deviation — the result tells you how many standard deviations above or below average that value sits, positive for above and negative for below.

Statistics students use it to interpret test scores or survey data, quality-control engineers use it to flag measurements that fall outside a normal manufacturing tolerance, and researchers use it to compare values drawn from different scales (like exam results and lab measurements) on one common footing.

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Comments (1)

  • Editorial team

    A z-score only says how far a value sits from the mean in standard deviations. Reading a score beyond plus or minus 2 as unusual is reasonable only when the data are roughly bell-shaped. Use the standard deviation that describes your group: the population figure if you have everyone, the sample figure if you have a subset. The tool stops at the z value and does not convert it to a percentile. If you compare scores from two tests, post both sets of numbers.

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