The calculation, without hidden assumptions
A z-score expresses a raw value in standard-deviation units relative to a mean. This calculator can solve the relationship in either direction and, when z is known, shows standard-normal tail percentages as distribution context. The percentile output assumes a normal distribution; the algebraic z-score itself does not require normality.
How to use this calculator
Choose the quantity to solve for.
Enter the other three quantities; standard deviation must be positive.
Review the substituted formula and, where z is available, the normal CDF below/above percentages.
Where people use it
- •Standardizing test scores or measurements
- •Comparing values from differently scaled distributions
- •Reverse-solving a raw score from a target standard score
Example: x = 84, mean = 70, SD = 10
z = (84 − 70) / 10 = 1.4, meaning the value is 1.4 standard deviations above the mean. Under a normal model, the below-tail percentile is about 91.9%.
What the result does not assume
- •The z-score formula is an algebraic standardization. Percentile/tail interpretations on this page assume a normal distribution.
- •Standard deviation must be greater than zero.
- •A z-score does not by itself establish whether a value is erroneous or practically important.
Frequently asked questions
What does z = 0 mean?+
The raw value equals the mean.
Can z-scores be negative?+
Yes. A negative z-score means the raw value is below the mean.
Does a z-score require normally distributed data?+
No for standardization itself. The normal percentile/tail percentages do require the normal-distribution assumption.
Semantic next steps
Continue the calculation
These links move to a different input, formula or project stage rather than a keyword variation of this page.
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