How to Calculate and Interpret a Z-Score
A z-score rescales a value into standard-deviation units so scores from the same distribution can be interpreted relative to the mean.
Method
- Enter the raw value you want to standardize.
- Enter the relevant mean.
- Enter a positive standard deviation.
- Subtract the mean from the value and divide by the standard deviation.
Worked example
Value 85, mean 75 and standard deviation 5 gives z = 2.
A z-score of −1 means the value is one standard deviation below the entered mean.
Notes
- A z-score does not prove the data are normally distributed.
- Use the mean and standard deviation required by your course or analysis.
- Positive and negative signs indicate direction relative to the mean.
- The magnitude of a z-score describes standardized distance, not importance or abnormality by itself. Whether a value is surprising depends on the distribution and context; skewed data or heavy tails can make normal-style interpretations misleading even when the z calculation is arithmetically correct.
A z-score expresses distance in standard-deviation units
Subtract the mean from the observed value and divide by the standard deviation. A positive z-score lies above the mean, a negative z-score lies below it, and the magnitude shows how many standard deviations separate the value from the mean.
The same raw difference can produce a different z-score when variability changes. A five-point gap is relatively unusual in a tightly clustered distribution and less unusual in a widely spread distribution.
Do not equate a z-score with a percentile unless a model is stated
A z-score is a standardized position. Converting it to a percentile usually assumes a normal distribution. Real data can be skewed, multimodal or heavy-tailed, so the empirical percentile and normal-model percentile may differ.
Use the z-score to compare standardized distances when that interpretation makes sense, and use the actual distribution or a justified probability model when making probability statements.
Compare unusual delivery times without confusing their raw minutes
Consider two fictional delivery teams with different normal operating ranges. Team A has a mean delivery time of 30 minutes and a standard deviation of 5 minutes; one delivery takes 42.5 minutes. Its z-score is (42.5 − 30) ÷ 5 = 2.5. Team B has a mean of 45 minutes and a standard deviation of 10 minutes; one delivery takes 55 minutes. That order has z = (55 − 45) ÷ 10 = 1.0. The 55-minute delivery is longer in absolute time, but the 42.5-minute delivery is farther above its own team's mean in standard-deviation units.
That comparison is useful for deciding which records deserve investigation, not for ranking teams as intrinsically better. Route length, weather, delivery type and the period used to estimate each mean can all explain differences. Check whether the standard deviations come from enough comparable observations and whether the definition of delivery time is the same in both datasets before interpreting the relative distances.
Unit conversion leaves z unchanged; changing the reference group does not
The Team A delivery can be expressed in seconds without altering its standardized position: 42.5 minutes is 2,550 seconds, the mean becomes 1,800 seconds and the standard deviation becomes 300 seconds. The calculation (2,550 − 1,800) ÷ 300 still gives z = 2.5. This is a useful audit of spreadsheet formulas when one data export uses minutes and another uses seconds.
By contrast, recomputing the mean and standard deviation from a different quarter, location or type of order can change the z-score of the same 42.5-minute delivery. A rule such as 'review observations beyond two standard deviations' is an operational flag, not proof that an observation is erroneous or that an assumed normal-model tail probability is valid. Inspect the actual distribution and the underlying record before excluding a value.
Authoritative references
Primary or authoritative references for the external standard, scoring rule or identifier convention used on this page.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- Normal Distribution Calculator — Calculate normal-distribution left-tail, right-tail and between-values probabilities, density, z-scores and inverse percentiles.
- Percentile Rank Calculator — Calculate the percentile rank of a target value within a numeric data set using a stated midrank convention.
- Standard Deviation Calculator — Calculate population and sample variance and standard deviation from a numeric data set.