A z-score, also called a standard score, measures how many standard deviations a value is above or below the mean of a distribution. A z-score of 0 means the value equals the mean exactly. A z-score of 1 means the value is one standard deviation above the mean, and a z-score of minus 1 means it is one standard deviation below the mean.

How do I convert a z-score to a percentile?

In a standard normal distribution, you look up the cumulative distribution function (CDF) at the z-score. A z-score of 0 corresponds to the 50th percentile, a z of 1.00 corresponds to roughly the 84th percentile, and a z of 1.96 corresponds to roughly the 97.5th percentile. This calculator uses a polynomial approximation of the normal CDF to convert z directly to a percentile.

What is a good or bad z-score?

There is no universally good or bad z-score; it depends on context. In hypothesis testing, a z-score beyond plus or minus 1.96 often indicates statistical significance at the 5% level. In standardised testing, a z-score above 2 means you scored in roughly the top 2.5% of participants. In quality control, values with z beyond plus or minus 3 are often flagged as outliers.

Z-Score Calculator

This calculator computes the z-score (standard score) for a value in a normal distribution and converts it to an approximate percentile rank. A z-score tells you how many standard deviations a particular observation sits above or below the mean of its distribution. The formula is z = (X minus mu) divided by sigma, where X is your observed value, mu is the population mean, and sigma is the standard deviation. A positive z-score means the value is above average; a negative z-score means it is below average. Because z-scores are dimensionless, they let you compare values that come from distributions with different means and spreads, such as comparing a test score in mathematics with a score in reading. The percentile rank tells you what proportion of the distribution falls below your value. For a standard normal distribution, a z-score of 1.00 corresponds to approximately the 84th percentile, meaning 84% of observations fall below that point. To use the calculator, enter the observed value X, the mean mu, and the standard deviation sigma. You get back the z-score and the approximate percentile. The percentile is computed using a well-known polynomial approximation of the cumulative normal distribution function. This tool is useful for students, teachers, researchers, and anyone interpreting standardised test scores, medical reference ranges, or quality control data. Results assume a normal distribution and are estimates only.

Calculate.co.nz is proud to be partnered with realtor.co.nz, a trusted resource for navigating the New Zealand property market. Their Helpful Articles section offers clear, well-structured insights across buying, selling, and building, making complex real estate topics more accessible. With a focus on up-to-date guidance and practical knowledge, they empower Kiwis to move forward with clarity and confidence in a constantly evolving property landscape.

Value (X)

Mean (μ)

Standard deviation (σ)

1.00

z-score

Percentile~84th

Deviation from mean+5.00

Above / below meanAbove

How it works

The z-score formula is z = (X − μ) ÷ σ. The calculator subtracts the mean from your value and divides by the standard deviation. It then converts the z-score to a percentile using a rational approximation of the standard normal cumulative distribution function (CDF), which gives accurate results across the common range of z-scores. The deviation from the mean is simply X minus mu shown with its sign, and the direction tells you whether the value is above or below average.

Worked example

A student scores 75 on a test where the class mean is 70 and the standard deviation is 5. The z-score is (75 − 70) ÷ 5 = 1.00. Looking up z = 1.00 on the standard normal CDF gives a cumulative probability of approximately 0.8413, meaning the student is at the 84th percentile and scored better than roughly 84% of the class. The deviation from the mean is +5.00, confirming the score is above average.

Related calculators

Normal Distribution Calculator: find P(X<x) and P(X>x) for any normal distribution directly.
Variance Calculator: compute the standard deviation you need as an input to this calculator.
Confidence Interval Calculator: use z-scores to build intervals around a sample mean.
Percent Error Calculator: measure how far a result deviates from an accepted value.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-07-02 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.