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Formula
z = (x − μ) ÷ σ; percentile = Φ(z) × 100
- z
- the z-score, the number of standard deviations x lies from the mean
- x
- the value being standardized
- μ
- the mean of the distribution
- σ
- the standard deviation (greater than 0)
- Φ(z)
- the standard normal cumulative distribution, the share of values below z
What a z-score is
A z-score, also called a standard score, re-expresses a value as a distance from the mean measured in standard deviations. The formula is short: subtract the mean and divide by the standard deviation. The result has no units, so a z-score of 1.2 means the same thing whether the original data were test points, inches or milliseconds.
That’s what makes z-scores useful for comparisons. If you scored 82 on a chemistry exam (mean 74, standard deviation 6) and 88 on a history exam (mean 85, standard deviation 5), the chemistry result is the stronger one: z = 1.3333 against z = 0.6.
How to use the z score calculator
- Enter the value you want to standardize (x).
- Enter the mean (μ) and standard deviation (σ) of the group. If you only have raw data, get both from the standard deviation calculator first.
- Read the z-score at the top. The note says how many standard deviations above or below the mean the value is.
- The rows below give the percentile and the probabilities to the left, to the right, between −|z| and |z|, and in both tails.
How to calculate a z-score by hand
Using the chemistry exam above:
- Subtract the mean: 82 − 74 = 8.
- Divide by the standard deviation: 8 ÷ 6 = 1.3333.
- Find the percentile: look up 1.33 in a standard normal table, or use the z score calculator, to get Φ(1.3333) ≈ 0.9088. About 90.9% of a normal distribution lies below this score, so it’s roughly the 91st percentile.
A printed z-table usually stops at two decimal places, so hand lookups are slightly less precise than the calculator, which evaluates the normal curve directly.
Z-score to percentile reference table
Values computed with the same normal distribution routine the calculator uses.
| z-score | Share below (Φ) | Percentile |
|---|---|---|
| −3 | 0.00135 | 0.13th |
| −2 | 0.02275 | 2.28th |
| −1.5 | 0.06681 | 6.68th |
| −1 | 0.15866 | 15.87th |
| −0.5 | 0.30854 | 30.85th |
| 0 | 0.5 | 50th |
| 0.5 | 0.69146 | 69.15th |
| 1 | 0.84134 | 84.13th |
| 1.282 | 0.90008 | 90th |
| 1.645 | 0.95002 | 95th |
| 1.96 | 0.975 | 97.5th |
| 2.326 | 0.98999 | 99th |
| 3 | 0.99865 | 99.87th |
Negative z-scores mirror positive ones: Φ(−z) = 1 − Φ(z).
Left tail, right tail and two tails
- Left tail, P(Z < z): the share of values below x. This is the percentile divided by 100.
- Right tail, P(Z > z): the share above x, equal to 1 − P(Z < z).
- Between −|z| and |z|: the share within that many standard deviations of the mean. For z = 1, 2 and 3 this gives the familiar 68%, 95% and 99.7%.
- Two-tailed: the share at least as far from the mean in either direction, used as the p-value for two-sided z-tests.
Going backward: from a percentile to a value
Sometimes you know the percentile you need and want the matching raw value, for example the score that puts a student in the top 10%. Rearrange the formula to x = μ + z × σ. The top 10% starts at the 90th percentile, which the table shows is z ≈ 1.282. With a mean of 500 and a standard deviation of 100, the cutoff is 500 + 1.282 × 100 ≈ 628. You can check the answer by entering 628 into the z score calculator and confirming that the percentile row reads about 90%.
How accurate is the percentile?
The calculator doesn’t round z to two decimals and look it up in a table. It evaluates the normal distribution function directly, using a rational approximation published by John Hart in 1968 that is accurate to about 14 significant digits. The results agree with Excel’s NORM.S.DIST(z, TRUE) and with statistical software to every digit shown, including far out in the tails, where a z of 7 still gives a right-tail probability of about 1.28 × 10⁻¹² instead of a rounded 0.
Real-life uses
- Comparing test scores across exams with different scales, as in the chemistry and history example.
- Child growth charts. Pediatric height and weight are often reported as z-scores against reference data.
- Finance. Analysts flag returns more than 2 or 3 standard deviations from the average as unusual.
- Quality control. Process measurements beyond ±3 standard deviations trigger an investigation.
- Hypothesis testing. A z-statistic is turned into a p-value with exactly the tail areas shown here.
Common mistakes
- Mixing up the order. It’s (x − μ) ÷ σ, not (μ − x) ÷ σ. Reversing it flips the sign and the percentile.
- Using the variance. Divide by the standard deviation, not the variance. If you have σ², take its square root first.
- Assuming normality. The z-score is always valid, but the percentile only matches reality if the data is close to bell-shaped. For skewed data, rank the value directly with the percentile calculator.
- Reading the wrong tail. “Top 5%” means a right-tail probability of 0.05, which is z ≈ 1.645, not −1.645.
To combine probabilities of separate events rather than areas under a curve, use the probability calculator.
Worked examples
Test score above the mean
A test score of 1,210 where scores average 1,050 with a standard deviation of 200.
Inputs: Value (x) = 1210; Mean (μ) = 1050; Standard deviation (σ) = 200
z = (x − μ) ÷ σ = (1,210 − 1,050) ÷ 200 = 160 ÷ 200 = 0.8 P(Z < 0.8) = Φ(0.8) = 0.788145, so 78.8145% of values lie below x
Answer: z = 0.8: 1,210 is 0.8 standard deviations above the mean, at about the 78.8145% percentile of a normal distribution.
Height below the mean
A 64-inch adult in a group whose heights average 69 inches with a standard deviation of 3 inches.
Inputs: Value (x) = 64; Mean (μ) = 69; Standard deviation (σ) = 3
z = (x − μ) ÷ σ = (64 − 69) ÷ 3 = −5 ÷ 3 = −1.6667 P(Z < −1.6667) = Φ(−1.6667) = 0.04779, so 4.779% of values lie below x
Answer: z = −1.6667: 64 is 1.6667 standard deviations below the mean, at about the 4.779% percentile of a normal distribution.
Value equal to the mean
Edge case: a value exactly at the mean has z = 0 and sits at the 50th percentile.
Inputs: Value (x) = 250; Mean (μ) = 250; Standard deviation (σ) = 12
z = (x − μ) ÷ σ = (250 − 250) ÷ 12 = 0 ÷ 12 = 0 P(Z < 0) = Φ(0) = 0.5, so 50% of values lie below x
Answer: z = 0: 250 is at the mean, at about the 50% percentile of a normal distribution.
Assumptions and limitations
- The z-score itself needs no assumption about the shape of the data. The percentile and probabilities assume the values follow a normal (bell-shaped) distribution.
- Φ(z) is computed with Hart's double-precision rational approximation of the normal distribution, accurate to better than 1 × 10⁻¹⁴, so it matches printed z-tables and Excel NORM.S.DIST to every digit shown.
- The z-score is shown to 4 decimal places and probabilities to 6. Probabilities smaller than 0.0000005 are shown in scientific notation rather than as 0.
- The standard deviation must be greater than 0. Enter the population standard deviation if you know it; with a sample estimate the result is an approximation.
- Two-tailed probability is P(|Z| > |z|), the chance of a value at least as far from the mean in either direction.
Frequently asked questions
What does a z-score tell you?
It tells you how many standard deviations a value is above or below the mean. A z-score of 2 is two standard deviations above, −0.5 is half a standard deviation below, and 0 is exactly at the mean. Because it's unit-free, it lets you compare values from different scales, such as an SAT score and an ACT score.
How do I convert a z-score to a percentile?
Look up Φ(z), the area under the standard normal curve to the left of z, and multiply by 100. A z-score of 1 gives Φ(1) ≈ 0.8413, the 84th percentile. The calculator does this lookup for you and shows the result in the first row.
Is a negative z-score bad?
Not necessarily. A negative z-score only means the value is below the mean. For race times or cholesterol, below the mean may be the outcome you want.
What is a good z-score?
There's no universal cutoff. Values beyond ±2 are unusual (about 5% of a normal distribution), and values beyond ±3 are rare (about 0.3%). Many quality-control and outlier rules use those thresholds.
How do I get a p-value from a z-score?
For a one-sided test use the right tail P(Z > z) or the left tail P(Z < z), depending on the direction of the hypothesis. For a two-sided test use the two-tailed probability. A z of 1.96 gives a two-tailed p-value of about 0.05.
Can I use a sample standard deviation?
Yes, and that's common in practice, but the result is then an estimate. For small samples used in hypothesis tests, statisticians switch from the normal distribution to the t-distribution.
Sources
- 6.1 The Standard Normal Distribution (Introductory Statistics 2e) — OpenStax, Rice University. Checked October 12, 2026.
- Normal Distribution (NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1) — National Institute of Standards and Technology. Checked October 12, 2026.
- §7.2 Error Functions: Definitions (Digital Library of Mathematical Functions) — National Institute of Standards and Technology. Checked October 12, 2026.