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Percentile from Z-Score Calculator

Enter a z-score to find its percentile on the normal curve.

How it is calculated

The percentile is the area under the standard normal curve to the left of your z-score: cdf = 0.5 × (1 + erf(z ÷ √2)), percentile = cdf × 100 and right tail = (1 − cdf) × 100. The z-score is unitless, a count of standard deviations from the mean, and can be negative or positive. The error function is computed with the Abramowitz and Stegun polynomial approximation, whose error is around one part in ten million, far below the 4 decimals shown for the percentage and 6 for the cumulative value. It assumes a normal distribution, so it is not valid for skewed data. The right tail is the chance of a value above your z-score. For a two-tailed test, double the smaller tail.

Worked example

For z = 1.96, the cumulative probability is 0.975002. Multiplying by 100 gives a percentile of 97.5002%. The right tail is 100 − 97.5002 = 2.4998%, which is why 1.96 is the usual cut-off for a 5% two-tailed test: 2.4998% in each tail, about 5% in total.

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Common mistakes

  • Typing a raw score instead of a z-score. A test score of 115 must first be converted with (value − mean) ÷ standard deviation.
  • Reading the percentile as a percentage of correct answers. A percentile is the share of the distribution below the value, not marks scored.
  • Using the percentile as a p-value. For a two-tailed test, the p-value is twice the smaller of the percentile and the right tail, not the percentile itself.
  • Applying it to data that are clearly not normal, such as incomes or waiting times. The conversion is only right if the distribution is approximately bell-shaped.
  • Expecting an exact 100% for large z. From about z = 5 the percentile shows as 100.0000% because of rounding, although the true value is never exactly 100.

About this calculator

A z-score measures how many standard deviations a value sits from the mean of a normal distribution. This calculator converts that z-score into a percentile using the standard normal cumulative distribution function, Φ(z) — the area under the bell curve to the left of z — so a z-score of 0 returns the 50th percentile, +1.0 returns roughly the 84th percentile, and −1.96 returns roughly the 2.5th percentile.

Because so many natural and measured quantities are approximately normally distributed, this z-to-percentile conversion underlies standardized test score reports, growth charts, and IQ scores, as well as statistical hypothesis testing where a z-score beyond ±1.96 corresponds to the common 5% two-tailed significance threshold. Statisticians, teachers scoring standardized assessments, and researchers running hypothesis tests use this conversion constantly to translate a raw z-score into a percentage that's easy to interpret and communicate.

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Comments (1)

  • Editorial team

    Here is a sanity check for the most common conversions. On a scale with mean 100 and standard deviation 15, a score of 130 is z = 2, which this tool places at the 97.7250th percentile. For a one-sided test at the 5% level the cut-off is z of about 1.645, which gives 95.0015%, while 1.96 is the two-sided version. If your z lands far from these anchors, recheck the subtraction. What scale are you converting from?

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