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Normal Distribution Calculator

Z-score and percentile for any value

How it is calculated

You give a value x, the distribution's mean μ and its standard deviation σ, all in the same unit. The tool computes the z-score z = (x − μ) ÷ σ, then the cumulative probability Φ(z) = ½ × (1 + erf(z ÷ √2)). The percentile is that probability times 100, the share of the distribution lying below x.

The standard deviation must be greater than zero, otherwise there is no result. The error function is evaluated with a polynomial approximation, so the percentile is accurate to roughly the third or fourth decimal place, which is plenty for two-decimal display. The calculation assumes the data really follow a bell-shaped normal distribution; for skewed data the percentile will not match reality.

Worked example

An IQ-style scale has mean 100 and standard deviation 15. For a score of 130: the distance from the mean is 130 − 100 = 30, and 30 ÷ 15 gives a z-score of 2.00. The cumulative probability at z = 2 is about 0.9772, so the score sits at the 97.72% percentile. Roughly 2.3% of people would score higher.

Try this example →

Common mistakes

  • Entering the variance instead of the standard deviation. With a variance of 225, the tool would treat σ as 225 and understate z.
  • Reading the percentile as the chance of getting exactly that value. It is the share at or below it.
  • Applying it to clearly skewed data such as incomes, where the normal assumption fails.
  • Using the standard error of a sample mean in place of the population standard deviation when scoring an individual value.
  • Confusing a z of 2.00 with 2% of the distribution. It is about the 97.7th percentile, with about 2.3% above.

Sources

About this calculator

This calculator finds the z-score, z = (x − μ) / σ, which measures how many standard deviations a value x sits from the mean μ of a normal distribution with standard deviation σ, then converts that z-score into a percentile using the cumulative distribution function (CDF) of the standard normal distribution, Φ(z) — the area under the bell curve to the left of z. A z-score of 0 sits exactly at the mean (the 50th percentile), while z-scores of ±1, ±2, and ±3 correspond to the well-known 68-95-99.7 rule.

Statisticians and researchers use z-scores to standardize measurements from different scales for comparison, standardized-test companies use them to convert raw scores into percentile rankings, quality-control engineers use them in six-sigma process control to flag outliers, and students learning inferential statistics use them as the entry point into hypothesis testing and confidence intervals.

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Comments (1)

  • Editorial team

    The percentile is the share at or below your value, so for the upper tail subtract it from 100. A value of 120 with mean 100 and standard deviation 15 gives z = 1.33 and 90.88 percent, so about 9.12 percent of the distribution lies above it. The percentile comes from a numerical approximation of the error function, accurate to roughly seven decimal places. What data do you plug in?

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