How this calculation works
A positive z-score is above the mean; a negative score is below it. A z-score alone does not establish a percentile unless a distribution is specified.
z = (value − mean) ÷ standard deviationA worked example
85 with mean 70 and standard deviation 10 has a z-score of 1.5.
Good to know
Assumes the idealized model described above. Use consistent units and independently verify figures before a consequential decision.
A standardized score is not automatically a percentile
A value two standard deviations above its mean has z = 2. That standardization describes distance on the supplied scale. It does not, by itself, tell you what fraction of a population lies below the value.
Converting z into a normal-distribution percentile assumes an appropriate normal model. A skewed or heavy-tailed population can have a different percentile for the same standardized distance.
The reference population supplies the meaning
Use a mean and standard deviation from the same relevant group and measurement definition as the observed value. Combining a score from one scale with a mean from another creates a meaningless comparison.
The standard deviation must be positive. If every population value is identical, the usual standardized distance divides by zero and is not defined. The sign simply identifies which side of the mean the observation occupies.
A few questions, answered
What does z = 0 mean?
The observed value equals the supplied mean.
Does a negative z-score mean a bad result?
No. It means below the reference mean. Whether that is desirable depends on the measurement.