Z Score Calculator

How to use this tool

Choose what you have. With a value, the mean and the standard deviation, the z score appears at the top with the share of values below and above it and the p values. With only a z score, pick the second option; to go the other way, from a percentile or a confidence level to a z score, pick the third; and to find the chance of falling between two values, pick the fourth.

The curve shades the area for your answer and labels the axis in both z scores and your own units. The rule chart shows how far out the value sits, and the z table section highlights the entry you would look up by hand.

How z scores and normal areas are worked out

A z score measures distance from the mean in standard deviations: z = (x – μ) / σ. Converting to z puts any normal distribution onto the same standard curve, with mean 0 and standard deviation 1, so a single table or function covers them all.

The share of values below z is the area under that curve to the left of z. Printed tables give it to four decimal places for z in steps of 0.01; this calculator works it out directly to about 15 significant figures, so tail areas far from the mean stay accurate. For percentiles and confidence levels it runs the same function in reverse.

Common z scores and their areas

A worked example you can check on paper

Scores on a test have a mean of 70 and a standard deviation of 10. A score of 85 has z = (85 – 70) / 10 = 1.5. In a z table, row 1.5 and column .00 give 0.9332, so 93.32% of scores are lower and 6.68% higher; the score is at about the 93rd percentile.

To find the score needed for the top 10%, look for an area of 0.9000 below: z = 1.2816. Then x = 70 + 1.2816 x 10 = 82.8, so a score of about 83 or more is in the top tenth.

Mistakes that catch people out

Using the wrong tail. A table of areas to the left gives the share below z. For the share above, subtract it from 1; for a two tailed p value, double the smaller tail.

Mixing up one and two sided values. 1.645 leaves 5% in one tail; 1.96 leaves 5% split across both.

Dividing by the variance. Divide by the standard deviation, the square root of the variance. With a variance of 225, the standard deviation is 15.

Assuming everything is normal. Percentiles from z are only right for bell shaped data. Skewed data, such as incomes, need a different model.

Popular calculations

Each opens the calculator with that result worked out, and every value can be changed once it is open.

From a value, mean and standard deviation

Z score to percentile

Percentile to z score

Confidence level critical values

Probability between two values

How this was checked

The normal area function was tested in Node against reference values to 16 digits, including Φ(1) = 0.8413447460685429, Φ(1.96) = 0.9750021048517795 and Φ(-8) = 6.22 x 10^-16, with errors below 1 in 10^13. The inverse was checked by feeding each z score back in, and the critical values 1.645, 1.960, 2.326 and 2.576 and the 68.27%, 95.45% and 99.73% rule all match published figures.

Every address in the lists above was then opened and checked for a distinct title, the right answer, charts that render, and clean CSV, image and print output containing only that calculation, at desktop, tablet and phone widths.

Common questions

How do you calculate a z score?

Subtract the mean from the value and divide by the standard deviation: z = (x – μ) / σ. A test score of 130 where the mean is 100 and the standard deviation 15 gives (130 – 100) / 15 = 2.

What does a z score of 2 mean?

The value is two standard deviations above the mean. In a normal distribution 97.72% of values lie below it and 2.28% above, so it sits at about the 98th percentile.

What z score is 95% confidence?

1.96 for a two sided interval, which leaves 2.5% in each tail. For a one sided test at 5%, the critical value is 1.645. The 99% two sided value is 2.576.

How do you turn a z score into a percentile?

Find the area under the standard normal curve to the left of z and multiply by 100. z = 1 has an area of 0.8413, the 84th percentile; z = -1 has 0.1587, the 16th percentile.

Can a z score be negative?

Yes. A negative z score is below the mean: z = -1.5 means one and a half standard deviations below, with 6.68% of values lower. The mean itself has a z score of 0.

What counts as an unusual z score?

Beyond ±2 is often treated as unusual, since only 4.55% of a normal distribution lies that far out, and beyond ±3 as rare, 0.27%. Whether that matters depends on the question you are asking of the data.

Related Calculators and Reference

• Standard Deviation Calculator: the mean and standard deviation of your data.
• Mean, Median and Mode Calculator: the centre of a data set.
• Random Number Generator: fair random samples and draws.
• Percentage Calculator: percentages, changes and differences.
• Permutation and Combination Calculator: count outcomes for exact probabilities.
• Sample Size Calculator: the sample needed for a given margin of error.
• Confidence Interval Calculator: intervals for a mean or a proportion.
• Binomial Distribution Calculator: exact binomial probabilities against the normal approximation.
• T Test Calculator: the t test when the standard deviation comes from the sample.
• Test Score Calculator: a score as a percent and letter grade, and how a curve moves it.
• NIST Engineering Statistics Handbook: Normal Distribution: definitions and formulas.