Z score and percentile calculator
How many standard deviations a value sits from the mean and which percentile it occupies.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- How many standard deviations a value sits from the mean and which percentile it occupies.
- Formula used
Z = (X − μ) / σ- Example
- Value 85 in a distribution with mean 70 and deviation 10:
Z score calculator
A z score turns any value into a comparable number: how many standard deviations it lies from the mean.
It underpins the percentile tables used in psychometric testing, quality control and paediatrics.
How it works
The mean is subtracted and the result divided by the standard deviation. The cumulative distribution function of the normal is then used to obtain the percentile.
Formula
Z = (X − μ) / σ
Percentile = Φ(Z) × 100
Φ is the cumulative distribution function of the standard normal
Worked example
Value 85 in a distribution with mean 70 and deviation 10:
Z = (85 − 70) / 10 = 1.5
Percentile: 93.3
Only 6.7% of cases exceed that value.
Explanation
The Z score translates any scale into a common language
A 7 on an exam means nothing without knowing how everyone else did. If the mean was 5 and the standard deviation 1, that 7 is two deviations above and is excellent. If the mean was 6.8, it is unremarkable. The Z score converts a mark into its relative position, which is what allows results from different tests to be compared.
What the sign means
A negative Z indicates the value is below the mean, a positive one that it is above. A Z of zero means it matches the mean exactly. The magnitude says how many standard deviations away it sits.
It only makes sense if the distribution is normal
In a highly skewed distribution such as salaries, the Z is calculated the same way but its translation into percentiles stops being valid. Before reading a Z as a probability, check that the data resemble a bell curve.
Where it is used outside the classroom
In industrial quality control, to detect parts outside specification. In medicine, to place a child's growth relative to their age. In finance, to measure how exceptional a market movement was.
Frequently asked questions
What does a negative z mean?
That the value is below the mean. A z of −1 is exactly one standard deviation below it.
Does it work for any distribution?
The z calculation does, but the percentile is only valid if the data follow an approximately normal distribution.
What does a Z of 2 mean?
That the value sits two standard deviations above the mean. In a normal distribution that places it above roughly ninety-eight per cent of cases.
Can I use Z scores on any data set?
You can calculate them, but reading them as percentiles is only valid if the distribution approximates a normal one.
Need to calculate something else?
These tools are often used alongside this calculator.
Standard deviation
Sample and population deviation, variance and coefficient of variation.
Confidence interval
The range the true mean is likely to fall in at your chosen confidence level.
Margin of error
How precise your survey is, given the sample you actually collected.
Average
Paste your list of numbers and get the mean, median, mode, range and standard deviation.
Sample size
How many responses you need to estimate a percentage or a mean, with finite population, design effect and…
Pearson correlation
Pearson’s r between two variables, with p value, confidence interval, regression line and scatter plot.