📊 Z-Score Calculator
By ToolNimba Editorial Team · Updated 2026-06-19
The z-score tells you how many standard deviations a value sits above or below the mean.
A z-score (also called a standard score) tells you how far a single value sits from the mean of its data set, measured in standard deviations. Enter the raw value, the mean and the standard deviation, and this calculator returns the z-score along with a plain reading of whether the value is above or below average and by how much. It is the quickest way to compare a number against the rest of the distribution.
What is the Z-Score Calculator?
A z-score answers a simple question: how unusual is this value compared with the rest of the data? It does this by re-expressing the value as a number of standard deviations away from the mean. The formula is z = (x - mean) / standard deviation. A z-score of 0 means the value equals the mean, a positive z-score means it is above the mean, and a negative z-score means it is below. The size of the number tells you the distance: z = 2 is two standard deviations above the mean, z = -1.5 is one and a half below.
The real power of the z-score is that it strips away the original units. A test score measured out of 100 and a height measured in centimetres live on completely different scales, but once both are converted to z-scores they can be compared directly, because both are now expressed in the universal language of standard deviations. This is called standardizing, and it is why z-scores sit at the heart of statistics: they turn raw values from any distribution into a common, unit-free scale.
When the data follows a normal (bell-shaped) distribution, z-scores map neatly onto probabilities. About 68% of values fall within one standard deviation of the mean (z between -1 and +1), about 95% within two, and about 99.7% within three. This is the empirical rule. A z-score beyond 2 or 3 is therefore quite rare, which is why such values are often flagged as outliers or treated as statistically significant.
When to use it
- Comparing two scores measured on different scales, such as a maths exam out of 50 against an English exam out of 100.
- Spotting outliers in a data set by flagging any value with a z-score beyond 2 or 3.
- Standardizing data before feeding it into a statistical model or machine-learning algorithm.
- Reading where a single measurement (a height, a blood-test result, a delivery time) sits within its overall distribution.
How to use the Z-Score Calculator
- Enter the raw value x you want to evaluate.
- Enter the mean of the data set.
- Enter the standard deviation of the data set (use 0 only if there is no spread, which makes the z-score undefined).
- Read off the z-score and the interpretation showing how many standard deviations the value is above or below the mean.
Formula & method
Worked examples
A student scores 85 on a test where the class mean is 70 and the standard deviation is 10.
- Subtract the mean: x - mean = 85 - 70 = 15
- Divide by the standard deviation: 15 / 10 = 1.5
- The z-score is +1.5
Result: z = +1.5, so the score is 1.5 standard deviations above the mean.
A reading of 68 comes from a process with a mean of 75 and a standard deviation of 4.
- Subtract the mean: x - mean = 68 - 75 = -7
- Divide by the standard deviation: -7 / 4 = -1.75
- The z-score is -1.75
Result: z = -1.75, so the value is 1.75 standard deviations below the mean.
Z-scores under a normal distribution: how common each value is (empirical rule)
| Z-score range | Position | Approx. share of values |
|---|---|---|
| -1 to +1 | Within 1 SD of the mean | About 68% |
| -2 to +2 | Within 2 SD of the mean | About 95% |
| -3 to +3 | Within 3 SD of the mean | About 99.7% |
| Beyond +/- 2 | Unusual | About 5% |
| Beyond +/- 3 | Extreme, often an outlier | About 0.3% |
Reading the sign and size of a z-score
| Z-score | Meaning |
|---|---|
| z = 0 | Value equals the mean |
| z = +1 | One standard deviation above the mean |
| z = -1 | One standard deviation below the mean |
| z = +2.5 | Two and a half standard deviations above the mean |
Common mistakes to avoid
- Dividing by the variance instead of the standard deviation. The z-score divides by the standard deviation, not the variance. The variance is the standard deviation squared, so using it gives a number that is far too small. Take the square root of the variance first if that is what you have.
- Mixing up the sample and population standard deviation. If you computed the spread yourself, be sure you used the right divisor. The population standard deviation divides by n, the sample version divides by n minus 1. They give slightly different z-scores, so use the one that matches your data set.
- Forgetting the sign. A z-score below the mean is negative. Dropping the minus sign loses half the information: a z of -2 and a z of +2 are equally far from the mean but on opposite sides.
- Treating any large z-score as proof of an error. A z-score beyond 3 is rare but not impossible. In a large enough data set some genuine values will land far out. Use it as a flag to investigate, not as automatic grounds to delete the value.
Glossary
- Z-score (standard score)
- The number of standard deviations a value lies above or below the mean, found with z = (x - mean) / standard deviation.
- Mean
- The arithmetic average of the data set, the sum of all values divided by how many there are.
- Standard deviation
- A measure of how spread out the data is around the mean. A larger value means the data is more scattered.
- Standardizing
- Converting raw values to z-scores so data on different scales can be compared on one common, unit-free scale.
- Empirical rule
- For a normal distribution, about 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations of the mean.
- Outlier
- A value far from the rest of the data, often one with a z-score beyond 2 or 3.
Frequently asked questions
What is a z-score?
A z-score, or standard score, measures how many standard deviations a value sits above or below the mean of its data set. It is found with z = (x - mean) / standard deviation. A z-score of 0 equals the mean, a positive z-score is above it, and a negative z-score is below it.
How do you calculate a z-score?
Subtract the mean from your raw value, then divide that difference by the standard deviation: z = (x - mean) / standard deviation. For example, a value of 85 with a mean of 70 and a standard deviation of 10 gives z = (85 - 70) / 10 = 1.5.
What does a negative z-score mean?
A negative z-score means the value is below the mean. For instance, z = -1.75 means the value sits 1.75 standard deviations below the average. The minus sign only tells you the direction, the size tells you the distance.
Is a high z-score good or bad?
It depends on what you are measuring. A high positive z-score on a test score is good because it is well above average, but a high z-score on errors or response times might be bad. The z-score itself is neutral: it only reports distance and direction from the mean.
What is considered a normal or unusual z-score?
Under a normal distribution, about 95% of values fall between z = -2 and z = +2, so anything inside that range is fairly typical. A z-score beyond 2 is unusual and beyond 3 is rare, which is why such values are often investigated as possible outliers.
What happens if the standard deviation is 0?
If the standard deviation is 0, every value in the data set is identical and equal to the mean, so there is no spread to measure against. The z-score formula would divide by zero, which is undefined, so a z-score cannot be calculated in that case.