Z-score calculator

What is a z-score?

A z-score (also called a standard score) tells you how far a single value sits from the average of its group, measured in standard deviations. A z-score of 0 means the value is exactly average. A positive z-score means the value is above average; a negative one means it is below average. Because the unit is “standard deviations” rather than dollars, centimetres, or test points, z-scores let you compare numbers that come from completely different scales.

This calculator works in four directions. Pick what you want to find, fill in the other three boxes, and it solves for the missing one: the z-score itself, the raw value x, the mean μ, or the standard deviation σ.

How does the calculator work?

The z-score is the gap between a value and the mean, divided by the standard deviation:

Where:

• x is the raw value you are scoring.
• μ (mu) is the mean of the population or sample.
• σ (sigma) is the standard deviation, which must be greater than zero.

The same equation can be rearranged to isolate any of the other quantities:

The calculator simply applies whichever rearrangement matches the value you selected, so you never have to do the algebra by hand.

Worked examples

1. Find the z-score. A student scores x = 85 on a test where the class mean is μ = 70 and the standard deviation is σ = 10. $z = \frac{85 - 70}{10} = 1.5$ The score is 1.5 standard deviations above average.

2. A value equal to the mean. With x = 70, μ = 70, and σ = 10: $z = \frac{70 - 70}{10} = 0$ The value lands exactly on the mean, so its z-score is 0.

3. Solve for the raw value. You know the z-score is z = 2, the mean is μ = 100, and the standard deviation is σ = 15. Switch the calculator to find the raw value: $x = 100 + 2 \times 15 = 130$ A value of 130 sits two standard deviations above the mean.

Practical notes

• The standard deviation must be positive. A standard deviation of zero would mean every value is identical, leaving the z-score undefined.
• Negative z-scores are normal and expected — they simply describe values below the mean.
• Use the population mean and standard deviation when you have data for the whole group, and the sample versions when you are working from a subset.
• Once you have a z-score you can look it up in a standard normal table to estimate percentiles and probabilities.

FAQ

Can a z-score be negative?

Yes. Any value below the mean produces a negative z-score. For instance, a value 1.5 standard deviations below the mean has a z-score of -1.5.

What does a z-score of 0 mean?

It means the value is exactly equal to the mean. There is no deviation from average.

Why must the standard deviation be greater than zero?

The formula divides by σ. If the standard deviation were zero the division would be undefined, and in practice it would mean the data has no spread at all.

Is a z-score the same as a percentile?

Not directly. A z-score measures distance from the mean in standard deviations, while a percentile reports the share of values that fall below a point. You can convert a z-score into a percentile using a standard normal distribution table.