Z-Score Calculator
Enter a value, the mean and the standard deviation to get the z-score and its percentile. You can also turn a z-score back into a raw score, find left, right, two-tailed or between probabilities, or find z for a sample mean.
Last updated November 3, 2026
Any number, decimals allowed.
Any number, decimals allowed.
Must be greater than 0.
Result
z = 1.25: the value 1,300 is 1.25 standard deviations above the mean.
| Measure | Value |
|---|---|
| z-score | 1.25 |
| Percentile (below this value) | 89.435023% |
| Area above this value | 10.564977% |
| Distance from the mean | 250 |
The percentile assumes the data follows a normal (bell-shaped) distribution.
Show the work
Formula: z = (x − mean) ÷ standard deviation; percentile = Φ(z) × 100
- Entered: x = 1,300, mean = 1,050, standard deviation = 200.
- Distance from the mean: 1,300 − 1,050 = 250.
- z = 250 ÷ 200 = 1.25.
- Φ(1.25) = 0.89435, so the percentile is 89.435023%.
Result: z = 1.25: the value 1,300 is 1.25 standard deviations above the mean.
How a z-score is calculated
z = (x − μ) ÷ σ x = μ + z × σ percentile = Φ(z) × 100
A z-score says how many standard deviations a value sits above or below the mean. Subtract the mean from the value, then divide by the standard deviation. A positive z means above average, a negative z means below, and 0 means exactly at the mean.
To go the other way, multiply z by the standard deviation and add the mean. The symbol Φ (phi) stands for the standard normal cumulative distribution: Φ(z) is the share of a normal distribution that lies below z. Multiplying it by 100 gives the percentile.
The standard deviation must be greater than 0, because a spread of zero would mean dividing by zero. The calculator also stops with a message if a number is missing.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Statistics Calculator, Combination Calculator and Sample Size Calculator.
Turning z into a percentile
Percentiles and tail probabilities are only meaningful when the data is roughly normal, which means bell-shaped and symmetric. A z-score itself is always defined, but the area under the curve is a model, not a count of real observations.
Common reference points are Φ(1) = 0.8413, Φ(2) = 0.9772 and Φ(3) = 0.9987. Negative values mirror these: Φ(−1) = 0.1587, because the curve is symmetric around 0.
Left, right, two-tailed and between
The left tail is the area below z and equals Φ(z). The right tail is the area above z and equals 1 − Φ(z). The two-tailed probability is the area beyond both −|z| and +|z|, which is 2 × Φ(−|z|). The area between two z-scores is Φ(upper) − Φ(lower).
A z-score of 1.96 has a two-tailed probability of 0.049996, which is why 1.96 is the usual cutoff for a 5% significance level. A z-score of 1.645 leaves about 5% in a single tail.
The 68-95-99.7 rule
For a normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. These figures are rounded. The exact areas between −1 and 1, −2 and 2, and −3 and 3 are 0.682689, 0.954500 and 0.997300.
Real data is rarely perfectly normal, so treat the rule as an approximation and check it against your own data when it matters.
A z-score for a sample mean
When you compare an average of n observations with a population mean, the spread of that average is smaller than the spread of single values. It is called the standard error and equals the standard deviation divided by the square root of n.
With a population mean of 100, a standard deviation of 15 and 36 observations averaging 105, the standard error is 15 ÷ 6 = 2.5, so z = (105 − 100) ÷ 2.5 = 2.
How to use the Z-Score Calculator
- Choose what to find: the z-score, a raw score, a probability from z, or z for a sample mean.
- Enter the value, mean and standard deviation, or the z-score.
- For probabilities, pick left, right, two-tailed or between.
- Read the result and its percentile, then open Show the work to follow each step.
Worked example
A test has a mean of 1050 and a standard deviation of 200. A score of 1300 is 250 above the mean, so z = 250 ÷ 200 = 1.25. Φ(1.25) = 0.8944, so the score is at about the 89.44th percentile: roughly 89% of scores fall below it, assuming a normal distribution.
| z-score | Percentile | Right tail | Two-tailed |
|---|---|---|---|
| −1 | 15.8655% | 0.841345 | 0.317311 |
| 0 | 50% | 0.5 | 1 |
| 1.25 | 89.435% | 0.105650 | 0.211300 |
| 1.645 | 95.0015% | 0.049985 | 0.099970 |
| 1.96 | 97.5002% | 0.024998 | 0.049996 |
| 2.576 | 99.5002% | 0.004998 | 0.009995 |
Frequently asked questions
What is a good z-score?
It depends on the goal. A higher z-score means further above the mean, which is good for test results, while a z-score near 0 is typical. Scores beyond 2 or below −2 are unusual.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean, and its size shows how many standard deviations below.
What is the difference between a z-score and a percentile?
A z-score counts standard deviations from the mean. A percentile is the share of values below it, which for normal data is Φ(z) × 100.
Why does the standard deviation have to be greater than 0?
The formula divides by the standard deviation, and a value of 0 means all numbers are identical, so no z-score exists.
Is my data stored anywhere?
No. Everything is calculated in your browser, and nothing you enter is saved or sent anywhere.