HesapHanem

Z-Score from Percentile Calculator

Enter a percentile from 0 to 100 to get its z-score.

How it is calculated

The tool inverts the standard normal distribution: it finds z such that Φ(z) = p, where p is the share of the distribution below z. It uses a rational polynomial approximation of the inverse normal CDF with separate formulas for the tails and the centre, and shows z and p to four decimals. A value above 1 is read as a percentile and divided by 100; a value of 1 or less is read as a probability already. The result must lie strictly between 0 and 1, so exactly 0, 1 (the number one), and 100 give no result. Because of that rule, the 0.5th percentile cannot be entered as 0.5 (it is read as probability 0.5, z = 0): type it as 0.5 only if you mean 50%.

Worked example

Enter the 85th percentile. Since 85 is above 1, it becomes p = 85 / 100 = 0.8500. The inverse normal gives z = 1.0364, meaning a value about 1.04 standard deviations above the mean has 85% of the distribution below it. Check: Φ(1.0364) is about 0.85. A score of mean 100 and standard deviation 15 at that percentile would be about 100 + 1.0364 × 15 = 115.5.

Try this example →

Common mistakes

  • Entering 1 for the first percentile. A value of 1 is read as probability 1 and returns nothing; use 1.01 or another value above 1, or think of it as 0.01.
  • Entering 0.5 for the half-percent point. Values from 0 to 1 are treated as probabilities, so 0.5 gives z = 0.0000 (the median), not the 0.5th percentile.
  • Confusing upper-tail and lower-tail values. The z returned has p of the distribution below it; for the top 5% enter 95.
  • Entering 0 or 100 and expecting a very large z. The tool returns nothing, because those percentiles correspond to infinity.
  • Using the z directly for a non-normal distribution such as a skewed test. It is exact only for a normal distribution.

About this calculator

This calculator inverts the standard normal cumulative distribution function — computing the probit function, Φ⁻¹(p) — to find the z-score below which a given percentage of a normal distribution falls. Enter the 85th percentile, for example, and it returns z ≈ 1.04, meaning a value 1.04 standard deviations above the mean.

Statistics students use it to check inverse-normal problems without a z-table, test-score analysts use it to convert a percentile rank on a standardized test into a z-score for further statistical work, and Six Sigma and quality engineers use the same inverse-normal lookup when setting control limits or interpreting process capability against percentile targets.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    For a two-sided interval, convert the tails into a single percentile first. A 95% confidence interval leaves 2.5% in each tail, so enter 97.5 and read z of about 1.96; for 99% enter 99.5. Entering 95 would give the one-sided cutoff of about 1.64 instead. Which kind of cutoff does your problem need, one-sided or two-sided? Which confidence level does your field usually report?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators