Z-score calculator (standard score)
A z-score, also called a standard score, converts a raw value into "how many standard deviations from the mean", which makes numbers from completely different scales comparable. Enter the value, the mean and the standard deviation below; the result also gives the percentile, the percentage of a normal distribution that lies below the value.
The value 82 sits 1.5 standard deviations above the mean of 70. The shaded area is 93.3% of a normal distribution.
Show the working, step by step
Z-score formula
z = (x − μ) / σ
Subtracting the mean re-centres the data on zero; dividing by the standard deviation re-scales it so that one unit of z is one standard deviation. Together those two steps are called standardising, and they are why a test score, a height and a share price can all be placed on the same axis and compared.
Run backwards, the same formula gives x = μ + zσ: the raw score calculator turns a z-score or a percentile back into a value on the original scale.
What the sign and size mean
| z | Percentile | Interpretation |
|---|---|---|
| −3.0 | 0.1% | Far below the mean; rare |
| −2.0 | 2.3% | Well below; unusual |
| −1.0 | 15.9% | Below average |
| 0.0 | 50% | Exactly at the mean |
| +1.0 | 84.1% | Above average |
| +1.96 | 97.5% | The 95% two-tailed cutoff |
| +2.0 | 97.7% | Well above; unusual |
| +3.0 | 99.9% | Far above; rare |
Those percentiles come from the standard normal distribution. They are only meaningful if your data is roughly normal; the z-score itself is valid for any data, but its translation into a percentile is not. To go the other way, from an area or percentile to the cut-off value, use the inverse normal calculator.
Standard deviations to percentiles
For normally distributed data, this is the share of values that fall below a point a given number of standard deviations from the mean. A value 1.5 SD above the mean sits at about the 93rd percentile; 1.5 SD below, at about the 7th.
| SDs from the mean | Percentile if below (−z) | Percentile if above (+z) |
|---|---|---|
| 0 | 50% | 50% |
| 0.5 | 30.85% | 69.15% |
| 1 | 15.87% | 84.13% |
| 1.5 | 6.68% | 93.32% |
| 2 | 2.28% | 97.72% |
| 2.5 | 0.62% | 99.38% |
| 3 | 0.13% | 99.87% |
For a z between these rows, the calculator above gives the exact percentile. To go the other way, from your own data rather than the normal curve, use the percentile calculator, which reads percentiles off the sorted values and needs no assumption about the shape.
Z-score: a worked example
A student scores 82 on a test with a mean of 70 and a standard deviation of 8.
z = (82 − 70) / 8 = 12 / 8 = 1.5
The score is 1.5 standard deviations above the mean, which places it at roughly the 93rd percentile: better than about 93% of the group, assuming the scores are approximately normally distributed. Note that this is a stronger statement than "12 points above average": whether 12 points is a lot depends entirely on σ. On a test with σ = 20, the same 12 points would give z = 0.6 and a percentile of only about 73.
This calculator takes one value at a time. To standardise a whole list, with the mean and SD worked out from the list itself, use the z-score for every value page.
Where z-scores get used
- Outlier screening: flag any observation with |z| above 3. Useful on roughly normal data; unreliable on skewed data, where the IQR rule is safer.
- Comparing across scales: a z of 1.2 on the SAT and a z of 1.2 on the ACT represent the same relative standing, even though the raw scores are nothing alike. IQ and other standardised test scales are built the same way.
- Standardising model inputs: regression and machine-learning features are routinely converted to z-scores so that variables measured in different units contribute on equal footing.
- Process control: control limits are conventionally drawn at ±3σ, which is a z-score threshold in different clothing.
A caution on the percentile
The z-score is a pure rescaling and is valid whatever shape your data has. The percentile this calculator reports is not; it reads the z against a normal distribution. For strongly skewed data such as incomes or waiting times, a z of 2 might correspond to the 90th percentile rather than the 97.7th. Check the shape of your distribution before quoting the percentile.
The z-score sits alongside percentiles, hypothesis tests and regression in the full set of statistics calculators.
Related calculators
-
Standard deviation calculator
Get the σ that a z-score needs, from raw data.
-
Confidence interval
The interval built from the same standard error machinery.
-
Standard error
For z-scores of a mean rather than of a single value.
-
Symbols and notation
What μ, σ and z stand for.
Common questions
What is a z-score?
A z-score says how many standard deviations a value sits from the mean:
z = (x − μ) / σ. A z of +1.5 means the value is one and a half standard
deviations above the mean; −0.4 means it is slightly below.
What counts as a high or unusual z-score?
Common rules of thumb put |z| > 2 at "unusual" (outside roughly 95% of a normal distribution) and |z| > 3 at "rare" (outside roughly 99.7%). Many outlier screens use |z| > 3 as a flag.
Those thresholds assume normality. On skewed data they mislead badly, and the interquartile-range rule is a more robust screen.
Can a z-score be negative?
Yes, and it simply means the value is below the mean. The sign carries direction; the magnitude carries distance. A z of −2.0 and a z of +2.0 are equally unusual.
Should I use the sample or population standard deviation for z?
Strictly, the z-score is defined with the population parameters μ and σ. When you only have sample estimates x̄ and s, the resulting quantity follows a t distribution rather than a normal one, which matters for small n. Above roughly n = 30 the difference is minor and the two are used interchangeably in practice.
What percentile is 1.5 standard deviations above the mean?
About the 93rd. For normally distributed data, 93.3% of values fall below z = +1.5, and 6.7% fall below z = −1.5. The table above lists the other half-step values.
How do I convert a z-score back to a raw value?
Rearrange: x = μ + zσ. So with a mean of 100 and an SD of 15, a z of 1.5
corresponds to 100 + 1.5 × 15 = 122.5.
How do I find a z-score from a percentile?
Use the inverse of the normal distribution: the 90th percentile is z = 1.2816 and the 97.5th is z = 1.96. The inverse normal calculator and the z table do this lookup.
How do I calculate a z-score in Excel?
=STANDARDIZE(x, mean, sd) or =(x - mean) / sd. To turn the z-score into a percentile use =NORM.S.DIST(z, TRUE), and to go back from a percentile use =NORM.S.INV(p).
What does the z in z-score stand for?
Nothing beyond convention: z is the letter traditionally used for a variable on the standard normal distribution, with mean 0 and standard deviation 1. A z-score is also called a standard score.
Frequently asked questions about the z-score calculator
Common questions
How do I calculate a z-score using standard deviation?
A z-score tells you how many standard deviations a value is from the mean: z = (x − μ) / σ. Z-scores standardise different distributions. Use the z-score calculator for computation with percentile info.
How do I convert a z-score back to an original value?
A z-score tells you how many standard deviations a value is from the mean: z = (x − μ) / σ. Z-scores standardise different distributions. Use the z-score calculator for computation with percentile info.