Z-score calculator

Compare a value with its group mean or recover the original value from a known Z-score.

Raw value x
Mean ฮผ
Standard deviation ฯƒ

Z-score

1.5

PositionAbove the mean
Distance from the mean1.5 ฯƒ

Read explanation below

Contents

How the Z-score calculator works

A value marked to the right of the mean on a standard deviation scale

Enter a raw value, its group mean and the standard deviation. The calculator reports how many standard deviations separate the value from the mean, or reverses the calculation when a Z-score is already known.

One number makes different scales comparable

A Z-score replaces the original unit with distance from the mean. Subtract the mean from the observed value and divide by the standard deviation.

z=xโˆ’ฮผฯƒz = \frac{x-\mu}{\sigma}
Read the sign first, then the distance: direction and size answer different questions.

A positive result sits above the mean, a negative result below it, and zero exactly at it. For example, a score of 85 in a group with mean 70 and standard deviation 10 has z = 1.5. A result from another test can be compared by its relative position even when that test uses a different point scale.

The score has no unit. The units carried by x, ฮผ and ฯƒ cancel during division.

Recovering the raw value

Choose โ€œRaw valueโ€ when the mean, standard deviation and Z-score are known. The inverse formula restores the value in the original unit.

x=ฮผ+zฯƒx = \mu + z\sigma

With a mean of 50, a standard deviation of 10 and z = 1.2, the raw value is 62. A negative Z-score moves the answer below the mean automatically.

What the result does not say

A Z-score is not a percentile or a probability. Turning it into either one requires a distribution model, commonly a normal distribution. The calculation itself only needs a valid mean and a positive standard deviation.

Use the mean and deviation from the same population or sample. A large absolute score marks a distant observation, but deciding whether it is an error or an outlier still depends on the data and the subject.

Questions about Z-scores

These answers cover the sign, units and the assumptions that are easy to mix up.

What does a Z-score of zero mean?

The raw value equals the mean. The numerator x minus ฮผ is zero, so the result remains zero for every positive standard deviation.

Can a Z-score be negative?

Yes. A negative score means that the value lies below the mean. Its absolute value gives the distance in standard deviations.

Does Z = 2 mean 2 percent?

No. It means two standard deviations above the mean. A percentage or percentile requires additional information about the distribution.

Why must the standard deviation be positive?

It is the divisor in the formula. Zero would make division undefined, and a standard deviation cannot be negative by definition.

Do I need normally distributed data?

Not to calculate a Z-score. Normality becomes relevant only when the score is used to estimate areas, probabilities or percentiles from a normal model.

Can I compare Z-scores from different tests?

Yes, when each score uses the correct mean and standard deviation for its own reference group. Equal Z-scores describe equal relative positions, not equal raw measurements.

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