📉 Z-Score Calculator
A z-score says how many standard deviations a value sits from the mean. It puts any measurement on a common scale, which is how a test score, a height and a blood result can be compared at all.
What a z-score is, and what it lets you compare
A z-score, also called a standard score, says how many standard deviations a value sits from the mean. Positive means above, negative below. It strips away the units, which is what makes it possible to compare a test score with a height, or this year's exam with last year's.
z = (x − μ) / σ
Subtract the mean to centre the data on zero, then divide by the standard deviation to express the distance in standard units. A z of 1.5 means one and a half standard deviations above average, whatever the original scale was measuring.
From z to percentile
| Z-score | Percentile | Share beyond it |
|---|---|---|
| −3.0 | 0.1st | 99.9% above |
| −2.0 | 2.3rd | 97.7% above |
| −1.0 | 15.9th | 84.1% above |
| 0.0 | 50th | 50% above |
| +1.0 | 84.1st | 15.9% above |
| +1.645 | 95th | 5% above |
| +1.96 | 97.5th | 2.5% above |
| +2.0 | 97.7th | 2.3% above |
| +3.0 | 99.9th | 0.1% above |
Two of those rows carry most of the practical weight. A z of ±1.96 marks the central 95% of a normal distribution, which is where the conventional 0.05 significance threshold comes from. A z of ±2.58 marks the central 99%.
Comparing across different scales
This is the thing z-scores are genuinely for. A student scores 82 on a maths paper where the mean was 70 with a standard deviation of 8, and 76 on an English paper where the mean was 60 with a standard deviation of 12. Raw scores say maths was the better result. Z-scores say maths was +1.5 and English +1.33, so maths was still better, but only just, and not by the six points the raw marks suggest.
The same logic underlies standardised test scaling, growth percentiles for children, and quality control limits in manufacturing.
The empirical rule
For data that is roughly normal, 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. This gives a fast sanity check on any z-score: if you calculate a z of 7, either the data is not normal or something went wrong, because a z of 7 should occur about once in a billion.
The percentile conversion assumes a normal distribution. For skewed data, such as income or response times, a z-score is still calculable but its percentile is wrong, sometimes badly. Check the shape of your data before converting.
Z-scores and p-values
A two-tailed p-value is the probability of seeing a result at least this extreme in either direction if nothing were going on. A z of 1.96 gives p = 0.05; a z of 2.58 gives p = 0.01. This is where the z-test comes from, and it is appropriate when the population standard deviation is known and the sample is reasonably large. For small samples with an estimated standard deviation, the t-distribution is the correct tool, with slightly wider tails.
A worked example
A test with a mean of 72 and a standard deviation of 8. A score of 85 gives z = (85 − 72) / 8 = 1.625, which puts it at the 94.8th percentile: better than about 95 students in every hundred. A score of 56 gives z = −2.0, the 2.3rd percentile. The gap between them is 29 raw marks and 3.6 standard deviations.
Two neighbours worth knowing: the standard deviation calculator, for the question this one leads into, and the confidence interval calculator, which answers it from the other direction.
Frequently Asked Questions (FAQs)
What is a good z-score?
It depends entirely on context. For a test result, higher is better and +2 is excellent. For a manufacturing defect rate, you want z near zero. For bone density, a z below −2 is a clinical concern. The score measures distance from average, not quality.
What does a negative z-score mean?
The value is below the mean. A z of −1.5 means one and a half standard deviations below average, at roughly the 7th percentile. The sign carries direction and the magnitude carries distance.
How do I convert a z-score to a percentile?
Through the cumulative normal distribution, which this calculator computes. A z of 0 is the 50th percentile, +1 is the 84th and −1 is the 16th. The conversion assumes the data is normally distributed.
When should I use a t-score instead?
When the sample is small, below about 30, and you are estimating the standard deviation from the sample rather than knowing it for the population. The t-distribution has heavier tails to account for that extra uncertainty, and it converges on the normal as the sample grows.
Is a z-score the same as standard deviation?
No. Standard deviation describes the spread of the whole data set in its original units. A z-score describes where one particular value sits, measured in those standard deviations. One is a property of the data, the other of a point in it.
Can z-scores be used on non-normal data?
They can be calculated on any data, and they still correctly express distance from the mean in standard units. What breaks is the percentile conversion, which assumes normality. For skewed data, report the actual percentile rank instead.