How do you calculate a z-score by hand?

Subtract the mean from your value, then divide by the standard deviation: z = (x - mu) / sigma. For example, with x = 85, mu = 70, and sigma = 10, you get (85 - 70) / 10 = 1.5. That result means the value is 1.5 standard deviations above the mean.

Can a z-score be negative?

Yes. A negative z-score simply means the value is below the mean. A z-score of -2, for instance, sits two standard deviations below average. The size shows the distance and the sign shows the direction, so negative z-scores are perfectly normal and just as valid as positive ones.

What does a z-score of 0, 1.5, or 2 mean?

A z-score of 0 means the value equals the mean exactly. A z-score of 1.5 means it is 1.5 standard deviations above the mean, about the 93rd percentile. A z-score of 2 means it is two standard deviations above the mean, at roughly the 98th percentile, which the empirical rule treats as fairly uncommon.

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

Look up the z-score in a standard normal table (or let the calculator do it) to find the cumulative area to its left, then multiply by 100. A z-score of 1.0 gives an area of 0.8413, which is the 84th percentile. The calculator uses the standard normal CDF, so its percentiles are exact.

What is the difference between a z-score and standard deviation?

The standard deviation is a fixed property of the whole dataset that measures spread, while a z-score describes one specific value: how many of those standard deviations it lies from the mean. In short, the standard deviation is the ruler and the z-score is the measurement read off that ruler.

How do you find the raw value from a z-score?

Rearrange the formula to x = mu + z * sigma. Multiply the z-score by the standard deviation, then add the mean. For example, with mu = 500, sigma = 100, and z = 1.28, the value is 500 + 128 = 628. The calculator's find-a-value-from-a-z-score mode does this for you.

What z-score corresponds to the 90th or 95th percentile?

The 90th percentile corresponds to a z-score of about 1.28, and the 95th percentile to about 1.645. These come from reading the standard normal table backwards, finding the z-score whose left-tail area equals 0.90 or 0.95, and they appear often in confidence intervals and hypothesis tests.

Z-Score Calculator

Formula

Enter your values

Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

Reviewed by the MathInSite math team — instructors with classroom experience teaching introductory statistics. Last updated September 2026.

What is a z-score?

A z-score tells you how many standard deviations a value sits above or below the mean of its distribution. It is also called a standard score, because it puts any value onto a single common scale where the mean is 0 and one standard deviation is 1 unit. The z-score calculator above turns a raw value, a mean, and a standard deviation into this standardized number instantly, and shows the matching percentile on a normal-distribution curve.

Definition: A z-score is the signed distance between a value and the mean, measured in standard deviations. A z-score of +2 means the value is two standard deviations above the mean; a z-score of −1 means it is one standard deviation below.

The sign tells you the direction. A positive z-score means the value is above the mean, a negative z-score means it is below the mean, and a z-score of 0 means the value equals the mean exactly. Because z-scores share one scale, you can compare results that originally used very different units — a test scored out of 100 and a height measured in centimetres can both be expressed as z-scores and compared directly.

Z-score formula

To calculate a z-score, subtract the mean from the value, then divide by the standard deviation:

z = (x − μ) / σ

Each symbol stands for one piece of information:

x — the raw value (the individual data point you are standardizing).
μ (mu) — the population mean, the average of all values.
σ (sigma) — the population standard deviation, which measures how spread out the values are.

When you are working with the mean of a sample rather than a single value, use the standard-error form, which divides σ by the square root of the sample size n:

z = (x̄ − μ) / (σ / √n)

Which form to use: Use z = (x − μ) / σ when standardizing one data point. Use the (σ / √n) version when testing whether a sample mean differs from a known population mean — for example, in a z-test or a hypothesis test about averages.

How to calculate a z-score (step-by-step example)

Suppose a student scores x = 85 on an exam where the class mean is μ = 70 and the standard deviation is σ = 10. Here is how to find the z-score and what it means:

Write down the three values. Raw value x = 85, mean μ = 70, standard deviation σ = 10.
Subtract the mean from the value. x − μ = 85 − 70 = 15. The score is 15 points above the average.
Divide by the standard deviation. 15 ÷ 10 = 1.5. So z = 1.5.
Interpret the z-score. The score sits 1.5 standard deviations above the mean — clearly above average but not extreme.
Convert to a percentile. Looking up z = 1.5 in the standard normal table gives a cumulative area of 0.9332, so the score is at about the 93rd percentile — higher than roughly 93% of the class.

To check this or try your own numbers, use the z-score calculator above: enter the value, mean, and standard deviation, and it returns z, the percentile, and a shaded curve.

How to read your result — percentile, probability, and tails

A z-score becomes most useful once you turn it into a percentile or probability using the standard normal distribution. Which area you want depends on the question you are asking, and the shaded curve in the result panel shows exactly which region is being measured.

Tail type	Question it answers	What the curve shades
Left tail	What fraction is at or below this value?	Everything to the left of the z-score (the cumulative area / percentile).
Right tail	What fraction is at or above this value?	Everything to the right of the z-score (1 − percentile).
Between two z	What fraction falls between two values?	The middle band between two z-scores.

Negative z-scores work the same way, and the bell curve is symmetric about the mean. Because of that symmetry, the left tail below a negative z-score equals the right tail above the matching positive z-score: P(Z < −1.5) = P(Z > 1.5) = 0.0668, or about 6.7%. So you never need a separate table for negative values — just use symmetry.

Z-score and the 68-95-99.7 rule

For a normal distribution, the empirical rule (the 68-95-99.7 rule) ties z-scores directly to how common a value is, which answers the practical question "is my z-score unusual?"

z between −1 and +1 — about 68% of values fall here. These are typical, ordinary results.
z between −2 and +2 — about 95% of values fall here. Beyond ±2 starts to look uncommon.
z between −3 and +3 — about 99.7% of values fall here. A z-score past ±3 is rare and often flagged as an outlier.

Each band is just a number of standard deviations from the mean, which is exactly what a z-score counts. So a z-score of 0.8 is firmly normal, while a z-score of 2.7 lands far out in the tail among the most extreme few percent.

Z-table (standard normal table) reference

A z-table, or standard normal table, lists the cumulative area (the left-tail probability) for each z-score in the standard normal distribution where μ = 0 and σ = 1. To read it, find your z-score and look up the area to its left — that area is the percentile as a decimal. For example, z = 1.00 gives 0.8413, so a z-score of 1 is at the 84th percentile.

z-score	Cumulative area (left tail)	Percentile
−3.0	0.0013	0.13th
−2.0	0.0228	2nd
−1.0	0.1587	16th
−0.5	0.3085	31st
0.0	0.5000	50th
0.5	0.6915	69th
1.0	0.8413	84th
1.28	0.9000	90th
1.5	0.9332	93rd
1.645	0.9500	95th
1.96	0.9750	97.5th
2.0	0.9772	98th
3.0	0.9987	99.87th

For a right-tail probability, subtract the table value from 1; for a between-values probability, subtract the smaller area from the larger. The calculator above uses the exact standard normal cumulative distribution function (CDF), so its percentile and probability figures match a full z-table and stay accurate between the rows shown here.

How to find a value from a z-score (reverse solve)

Sometimes you know the percentile you want and need the raw value behind it — for instance, "what score is the 90th percentile?" Rearranging the formula gives:

x = μ + z · σ

Say a standardized test has a mean of μ = 500 and a standard deviation of σ = 100, and you want the cut-off for the 90th percentile.

Find the z-score for the percentile. The 90th percentile corresponds to z ≈ 1.28 (from the z-table above).
Multiply z by the standard deviation. 1.28 × 100 = 128.
Add the mean. 500 + 128 = 628. A score of about 628 marks the 90th percentile.

Switch the calculator above to its "find a value from a z-score" mode to do this automatically for any mean, standard deviation, and z-score.

When do you use a z-score?

Z-scores appear across statistics whenever values need to be compared on a common, standardized scale:

Comparing different scales — judge a score on the SAT against one on the ACT by converting both to z-scores.
Hypothesis testing — z-tests use a z-score to decide whether a sample mean differs significantly from a population mean.
Outlier detection — values with |z| greater than about 3 are often flagged as outliers worth a second look.
Quality control — six-sigma processes describe how many standard deviations a measurement is from target, which is a z-score by another name.
Probability and percentiles — converting a value to a z-score lets you read off probabilities from the standard normal distribution, which underpins related tools like the combinations calculator.

Common mistakes

Mistake — using the variance instead of the standard deviationThe formula divides by σ, the standard deviation, not σ². If you were given the variance, take its square root first, then divide.

Mistake — dropping the negative signWhen the value is below the mean, x − μ is negative, so the z-score is negative. The sign is part of the answer — it tells you the value is below average — so keep it.

Mistake — confusing the percentile with the z-scoreA z-score (like 1.5) and its percentile (about 93%) are different numbers. The z-score counts standard deviations; the percentile is the area under the curve to its left. Report whichever the question asks for.

Mistake — reading the wrong tailA z-table gives the left-tail (cumulative) area. For an "at or above" question you must subtract that value from 1, otherwise you report the complement of the answer you wanted.

Mistake — mixing up σ and σ/√nUse plain σ for a single value. Only divide by √n when you are standardizing a sample mean in a z-test; using the wrong denominator changes the z-score and the conclusion.

Studying for AP Statistics? Z-scores anchor the normal distribution work in Unit 2, and the sampling-distribution logic that follows feeds directly into the confidence interval & hypothesis test calculator.

Frequently asked questions

Quick answers to the most common questions about using the z-score calculator and interpreting z-scores.

How do you calculate a z-score by hand?

Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. For example, with x = 85, μ = 70, and σ = 10, you get (85 − 70) / 10 = 1.5. That result means the value is 1.5 standard deviations above the mean.
Can a z-score be negative?

Yes. A negative z-score simply means the value is below the mean. A z-score of −2, for instance, sits two standard deviations below average. The size shows the distance and the sign shows the direction, so negative z-scores are perfectly normal and just as valid as positive ones.
What does a z-score of 0, 1.5, or 2 mean?

A z-score of 0 means the value equals the mean exactly. A z-score of 1.5 means it is 1.5 standard deviations above the mean — about the 93rd percentile. A z-score of 2 means it is two standard deviations above the mean, at roughly the 98th percentile, which the empirical rule treats as fairly uncommon.
How do you convert a z-score to a percentile?

Look up the z-score in a standard normal table (or let the calculator do it) to find the cumulative area to its left, then multiply by 100. A z-score of 1.0 gives an area of 0.8413, which is the 84th percentile. The calculator uses the standard normal CDF, so its percentiles are exact.
What is the difference between a z-score and standard deviation?

The standard deviation is a fixed property of the whole dataset that measures spread, while a z-score describes one specific value — how many of those standard deviations it lies from the mean. In short, the standard deviation is the ruler and the z-score is the measurement read off that ruler.
How do you find the raw value from a z-score?

Rearrange the formula to x = μ + z · σ. Multiply the z-score by the standard deviation, then add the mean. For example, with μ = 500, σ = 100, and z = 1.28, the value is 500 + 128 = 628. The calculator's "find a value from a z-score" mode does this for you.
What z-score corresponds to the 90th or 95th percentile?

The 90th percentile corresponds to a z-score of about 1.28, and the 95th percentile to about 1.645. These come from reading the standard normal table backwards — finding the z-score whose left-tail area equals 0.90 or 0.95 — and they appear often in confidence intervals and hypothesis tests.

Need the probability that goes with your z-score? See how to find a p-value on Desmos and the TI-84.