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Z-Score Calculator

Convert a value to a z-score and its normal-distribution percentile, or turn a z-score back into a raw value.

Level: Grades 11–12

  • Common Core standard HSS.ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Printable worksheet

Z-Score Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Find

Result

Z-score

1.5

85 is 1.5 standard deviations above the mean.

Percentile (share of a normal distribution below x)
93.3193%
Share above x
6.6807%

See the step-by-step solution ↓

Step-by-step solution

  1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)

    z=x−μσ

  2. Put in your value, mean and standard deviation

    z=85−7010=1510=1.5

  3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile

    Φ(1.5)=93.3193%

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how far a score or measurement is from average, and what share of results likely fall below it. A z-score, also called a standard score, counts how many standard deviations a value sits above or below the mean. The mean is the average. The standard deviation is the typical distance of values from that average.

A z-score of 0 is exactly average. +1 is one standard deviation above, and −2 is two standard deviations below. Because a z-score drops the original units, you can compare results on different scales, such as a test score and a race time.

The calculator works in both directions. Give it a value, a mean and a standard deviation. You get the z-score and the share of a normal distribution (the familiar bell curve) that falls below your value. Or give it a z-score to get back the raw value it stands for.

Where it's used

Why this math matters outside the classroom:

  • Comparing tests. A 650 on a test with mean 500 and SD 100 (z = 1.5) beats a 28 on a test with mean 21 and SD 5 (z = 1.4).
  • Quality control. An inspector flags a part with z = 3.2 because its length sits more than 3 standard deviations from the mean.
  • Class standing. A student with z = 1 on a normally distributed exam scored higher than about 84% of the class.
  • Market moves. Analysts turn a 4% daily drop into z ≈ −2.67 when daily moves have mean 0 and SD 1.5%.
  • Setting cutoffs. A teacher who wants an honors cutoff at z = 1.5 on a test with mean 72 and SD 8 sets it at 72 + 1.5 × 8 = 84.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSS.ID.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose whether you want the z-score of a value, or the value for a z-score.
  2. Enter the mean and standard deviation of the group you're comparing against. The standard deviation must be more than zero.
  3. Enter either the value x or the z-score z.
  4. Read your answer. You'll also see the percentile (the share below) and the share above. Both assume the data follows a bell curve.

What the inputs mean

Z-score (z)
Positive z-scores are above the mean, and negative ones are below it. On a bell curve, most values have z between −3 and 3.
Mean (μ)
The average of the group you're comparing with, for example the average score of everyone who took a test.
Standard deviation (σ)
The standard deviation of that group, in the same units as the value. Use the figure for the whole population if you have it.

Formula and method

z = (x − μ) ÷ σ
x = μ + z × σ
percentile = Φ(z) × 100

In the formulas, x is your value, μ (the Greek letter mu) is the mean and σ (sigma) is the standard deviation. Subtracting the mean shows how far your value is from the middle. Dividing by the standard deviation turns that gap into a count of standard deviations. Solving the same equation for x takes you the other way.

Φ(z) (phi of z) is the share of a standard bell curve that lies to the left of z. Multiplying by 100 turns it into a percentile. The calculator works it out to about 12 significant digits, so you don't need a printed z-table.

Worked example

A student scores 85 on an exam where the mean is 70 and the standard deviation is 10.

  1. Distance from the mean: 85 − 70 = 15.
  2. Divide by the standard deviation: z = 15 ÷ 10 = 1.5.
  3. Find the share of the bell curve below 1.5: Φ(1.5) ≈ 0.933193, or 93.3193%.
  4. Check it the other way: 70 + 1.5 × 10 = 85.

Result: The score has a z-score of 1.5. If scores follow a bell curve, that puts it at about the 93.3rd percentile. About 6.68% of scores would be higher.

Understanding the result

The sign tells you the direction, and the size tells you how unusual the value is. On a bell curve, about 68% of values have z between −1 and 1. About 95% fall between −2 and 2, and about 99.7% between −3 and 3.

A z-score beyond about ±3 is rare. It's often worth checking whether the value is a mistake or comes from a different group.

You can compare z-scores across different measurements. Say you get a z of 1.2 in one subject and 0.8 in another. The first result stands out more compared with others, whatever the raw scores were.

Assumptions and limitations

  • The percentile and the share above assume the data follows a bell curve, with the mean and standard deviation you entered.
  • The z-score itself makes no assumption about the shape of the data. It just changes the scale.
  • If the data is lopsided (skewed) or has many extreme values (heavy tails), the bell-curve percentile can be well off. In that case, use the percentile calculator on your actual data instead.
  • When the mean and standard deviation come from a small sample, they're only estimates. The z-score then carries the same uncertainty.

Frequently asked questions

What does a negative z-score mean?

The value is below the mean. A z-score of −1.5 is one and a half standard deviations below average. On a bell curve, that's around the 6.7th percentile.

Is the percentile here the same as a percentile from my data?

Not always. This percentile comes from the bell curve, so it's exact only if your data truly follows one. A percentile worked out from your own data counts the real values below a point. It makes no guess about the shape.

Should I use the sample or population standard deviation?

Use whichever describes the group you're comparing with. If you have a published mean and standard deviation for a whole population, use those. If you only have a sample, its sample standard deviation is the usual estimate.

How do I find the score needed to reach a certain percentile?

First find the z-score for that percentile with the normal distribution calculator. For example, it's about 1.2816 for the 90th percentile. Then switch this calculator to 'Value from a z-score' and enter it with your mean and standard deviation.

How this calculator is checked

Method: Standard score definition z = (x − μ)/σ; standard normal CDF from the shared, tested distributions module (Hart/West rational approximation). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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