Z-Score Calculator
Enter a value with the mean and standard deviation — or paste a list of numbers and let the calculator find them — and get the z-score: how many standard deviations the value sits above or below the mean. You also get an approximate percentile and the working with your numbers. Free, no sign-up.
Input mode
Enter a value, mean, and a standard deviation greater than zero to see the z-score.
How to Use the Z-Score Calculator
- 1. Choose the input mode. From value, mean, and SD takes the three numbers directly. From a dataset takes a list of numbers and the value you want to standardise.
- 2. Enter the numbers. In the first mode type x, μ and σ (σ must be greater than zero). In dataset mode paste at least two numbers, separated by commas, spaces or new lines, and enter x.
- 3. Read the result. You get the z-score to four decimals, the approximate percentile, and in dataset mode the computed mean, standard deviation and count. A sentence states the position in words.
- 4. Check the working. The how-calculated line shows z = (x − μ) ÷ σ with your numbers, so you can copy it into a report.
The Formula and What It Means
z = (x − μ) ÷ σ, where x is the value, μ the mean and σ the standard deviation. The z-score is unitless: it measures distance from the mean in standard deviations, so scores from different scales can be compared. A z of 0 is exactly average, +1 is one standard deviation above and −2 is two below.
Two exams, one comparison
Exam A: you scored 82; the class mean was 75 and the SD was 6
z = (82 − 75) ÷ 6 = 1.1667, percentile ≈ 87.83%
Exam B: you scored 78; the class mean was 70 and the SD was 5
z = (78 − 70) ÷ 5 = 1.6, percentile ≈ 94.52%
The 78 is the stronger result, even though the 82 is a higher raw score
In dataset mode, paste 62, 68, 71, 74, 70, 65, 80 and 77 and enter 85. The calculator finds μ = 70.875 and the population σ = 5.6222, so z = (85 − 70.875) ÷ 5.6222 = 2.5124, about the 99.40th percentile.
Reading a Z-Score
| Z-score | Percentile (above / below the mean) | Reading |
|---|---|---|
| 0 | 50.00th | Exactly at the mean |
| +1 / −1 | 84.13th / 15.87th | One SD from the mean; common |
| +1.96 / −1.96 | 97.50th / 2.50th | The edges of the middle 95% |
| +2 / −2 | 97.72nd / 2.28th | Unusual: about 4.6% of values lie beyond ±2 |
| +3 / −3 | 99.87th / 0.13th | Rare: about 0.27% of values lie beyond ±3 |
The percentile applies only when the data are roughly bell-shaped. For skewed data a z-score still measures distance in SDs, but the percentile no longer matches.
The percentile comes from the standard normal distribution function, evaluated with the Abramowitz and Stegun approximation (formula 7.1.26 for the error function, accurate to about 1.5 × 10⁻⁷). Treat it as approximate and use the Z-Table when you need the standard four-decimal lookup values.
Dataset Mode Uses the Population SD
When you paste data, σ is the population standard deviation (dividing by n), which is the usual choice when you are standardising values inside the group you have. If your textbook uses the sample standard deviation s (dividing by n − 1), get it from the Statistics Calculator and type it into the value, mean and SD mode instead. For large samples the difference is small. A z-score for a sample mean uses the standard error, not the SD — that is a Z-Test.
Which Tool Do You Need?
| You want to | Use |
|---|---|
| Standardise one value against a mean and SD | This Z-Score Calculator |
| Look up an area or critical value from a z | Z-Table |
| Convert to T-scores, IQ, stanines or NCE | Standard Score Converter |
| Get the 68-95-99.7 intervals for a mean and SD | Empirical Rule Calculator |
| Turn a z-score into a p-value | P-Value Calculator |
Frequently Asked Questions
- What is a z-score?
- A z-score, or standard score, is the number of standard deviations a value lies above or below the mean: z = (x − μ) ÷ σ. A z of 0 means the value equals the mean, positive values are above it and negative values below.
- How do I calculate a z-score from raw data?
- Find the mean and standard deviation of the data, then subtract the mean from your value and divide by the standard deviation. The calculator's dataset mode does all of it: paste the numbers, enter the value, and it shows the working.
- What does a negative z-score mean?
- The value is below the mean. A z of −1.5 means 1.5 standard deviations below the average, which is the 6.68th percentile if the data are normal.
- What is a good z-score?
- It depends on what you are measuring. Higher is better for a test score and lower for a race time. As a rule of thumb, values beyond ±2 are unusual and beyond ±3 are rare in roughly normal data.
- How accurate is the percentile shown?
- It comes from the standard normal distribution, evaluated with an approximation accurate to about seven decimal places in probability, then shown to two decimals as a percentage. It is only meaningful if the data are roughly normally distributed.
- Does dataset mode use the sample or population standard deviation?
- The population standard deviation, dividing by n. If you need the sample version, calculate s separately and use the value, mean and SD mode.