Z-Score Calculator - Standard Score & Percentile

About the Z-Score Calculator

A z-score, or standard score, tells you how many standard deviations a value lies above or below the mean. It turns any measurement into a common scale, so you can compare a mark on a hard test with a mark on an easy one, a child's height with a reference population, or a laboratory result with the normal range. Combined with the normal distribution, a z-score also tells you what percentage of values lie below a given point.

This z-score calculator works both ways. It converts a value into a z-score from the mean and standard deviation, or converts a z-score back into a value. For either, it gives the normal percentile — the share of a normal distribution below the point — the tail areas above it and on both sides, and a plain-language reading of how unusual the value is.

How to Use the Z-Score Calculator

Choose Value to z-score or Z-score to value.

Enter the value (or the z-score), the mean and the standard deviation.

The result appears with the percentile, tail areas and interpretation.

The Formulas

  z = (x − μ) ÷ σ         value → z-score
  x = μ + z × σ           z-score → value

  percentile = Φ(z) × 100, where Φ is the standard normal distribution

Step-by-Step Example: Value to Z-Score

A test score of 85, where the mean is 70 and the standard deviation 10.

  z = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5
  Φ(1.5) = 0.933193

The score is 1.5 standard deviations above the mean. If scores are normally distributed, about 93.32% of students scored lower, 6.68% higher, and 13.36% scored at least 1.5 standard deviations away from the mean in either direction.

Step-by-Step Example: Z-Score to Value

The value one standard deviation below a mean of 100, with a standard deviation of 15.

  x = 100 + (−1) × 15 = 85
  Φ(−1) = 0.158655

On an IQ-style scale, a score of 85 is at about the 15.87th percentile.

Key Z-Scores

These values come up again and again in statistics, from confidence intervals to hypothesis tests.

  z       percentile        z       percentile
  −3      0.13%             1       84.13%
  −2      2.28%             1.5     93.32%
  −1.96   2.50%             1.645   95.00%
  −1      15.87%            1.96    97.50%
   0      50.00%            2       97.72%
                            3       99.87%

About 68 percent of a normal distribution lies within one standard deviation of the mean, 95 percent within 1.96, and 99.7 percent within three — the empirical rule.

Comparing Across Scales

Z-scores make different measurements comparable. Suppose a student scores 85 on a history test with a mean of 70 and SD of 10, and 78 on a maths test with a mean of 60 and SD of 6. The history z-score is 1.5, but the maths z-score is (78 − 60) ÷ 6 = 3. Although the maths mark is lower, it is the more exceptional performance relative to the class.

When the Percentile Can Mislead

Calculating a z-score needs only the mean and standard deviation, and it is always valid as a measure of distance from the mean. Converting it to a percentile, however, assumes the data follows a normal, bell-shaped distribution. Heights, measurement errors and many test scores are close to normal; incomes, house prices and waiting times are strongly skewed, and for those the normal percentile can be far from the real one. For skewed data, use the actual data with a percentile calculator.

Z-Scores and Outliers

Because extreme z-scores are rare in normal data, they are a common way to flag unusual observations. Values with |z| above 2 occur about 5 percent of the time; above 3, only about 0.3 percent. Many quality-control and data-cleaning rules flag values beyond ±3 for investigation. As with any outlier rule, a flagged value may be an error or a genuine extreme — check before discarding it.

Z-Scores in Medicine and Growth Charts

Doctors use z-scores to judge whether a measurement is unusual for a patient's age and sex. A bone density T-score, a child's height-for-age z-score and many laboratory reference ranges all express results as standard deviations from a reference mean. A height z-score of −2, for example, means the child is shorter than about 97.7 percent of children of the same age and sex, a common threshold for further assessment.

Understanding Your Result

The headline is the z-score, or the value for the z-score you entered.

The percentile line gives the share of a normal distribution below that point.

The tail areas line gives the share above it, and the share at least as far from the mean on either side, used in two-tailed tests.

The what it means line describes how typical or unusual the value is.

When Should You Use This Calculator?

Use it to compare scores or measurements from different scales.

Use it to find what percentile a result corresponds to in a normal population.

Use it to find the value at a given number of standard deviations from the mean.

Use it for statistics homework on the normal distribution and standard scores.

Common Mistakes

Dividing by the variance. Divide by the standard deviation.

Forgetting the sign. Negative z-scores lie below the mean.

Applying normal percentiles to skewed data. They assume a bell-shaped distribution.

Mixing sample and population statistics. Use the mean and SD of the right group.

Treating |z| > 2 as proof of an error. It only flags a value as unusual.

Rounding z-scores too early. Percentiles change quickly in the tails, so keep two or three decimal places before looking up the area.