This free online z-score calculator tells you how many standard deviations a value is above or below the mean of a data set. It's widely used in statistics to standardize scores and compare data points from different distributions.
How to Use the Z-score Calculator
Enter the data point you want to standardize in the "Value (x)" field.
Enter the data set's average in the "Mean (μ)" field.
Enter the data set's spread in the "Standard Deviation (σ)" field.
Click "Calculate" to see the z-score and a plain-language interpretation.
What a Z-score Tells You
A z-score converts a raw value into a common scale: how many standard deviations it lies from the mean. The calculator subtracts the mean from your value and divides by the standard deviation, z = (x − μ) ÷ σ, then shows the result to 3 decimal places along with a sentence saying whether the value is above or below the mean.
Because z-scores have no units, they let you compare results that were measured on different scales, such as marks from two exams with different difficulty or heights and weights in a growth study.
Worked Example: Which Exam Went Better?
A student scores 70 in Maths, where the class mean is 60 and the standard deviation is 5. The same student scores 80 in Science, where the mean is 72 and the standard deviation is 10.
Maths: z = (70 − 60) ÷ 5 = 2.000
Science: z = (80 − 72) ÷ 10 = 0.800
The Science mark is higher in raw terms, but the Maths result is far more exceptional compared with classmates. A third student with 52 in Maths would get z = −1.600, meaning 1.6 standard deviations below the class average.
Z-score to Percentile (Normal Distribution)
If the data follows a normal (bell-shaped) distribution, each z-score corresponds to the share of values that fall below it. The calculator does not show this percentile, so use the table below to read it off:
Z-score
Share of values below
−3
0.13%
−2
2.28%
−1
15.87%
−0.5
30.85%
0
50%
0.5
69.15%
1
84.13%
1.645
95%
1.96
97.5%
2
97.72%
3
99.87%
So the Maths score above, with z = 2, is better than roughly 97.7% of a normally distributed class. The values 1.645 and 1.96 appear often in statistics because they mark the cut-offs for 90% and 95% two-sided confidence intervals.
Things to Keep in Mind
The percentile table only applies to roughly normal data. For skewed data such as household income or house prices, a z-score still measures distance from the mean, but it cannot be turned into a percentile this way.
A common rule of thumb treats values with a z-score beyond ±3 as possible outliers worth checking for data-entry errors.
The mean and standard deviation must come from the same group as your value. If you only have the raw data, find them first with the Standard Deviation Calculator.
A standard deviation is never negative, so the calculator only accepts values greater than zero. If you get a negative figure from a spreadsheet, check the formula that produced it.
Frequently Asked Questions
It uses z = (x − μ) / σ, where x is your value, μ is the mean, and σ is the standard deviation. For example, if x = 70, μ = 60, and σ = 5, then z = (70 − 60) / 5 = 2.000, meaning the value is 2 standard deviations above the mean.
A negative z-score means your value is below the mean. The calculator reports the direction as "above" or "below" the mean based on the sign of the result, along with how many standard deviations away it is.
Since the standard deviation is the denominator in the z-score formula, a value of zero would make the calculation undefined, so the calculator shows "Standard deviation cannot be zero" instead of a result.
It depends on what is being measured. For exam marks a higher z-score is better, while for something like blood pressure or delivery time a value close to 0 or below the mean may be preferable. As a general guide, about 68% of normally distributed values have a z-score between −1 and 1, so anything beyond ±2 is unusual.
Yes. Rearrange the formula to x = μ + z × σ. With a mean of 60 and a standard deviation of 5, a z-score of 1.5 corresponds to a value of 60 + 1.5 × 5 = 67.5.