Z-score calculator
How many standard deviations a value sits away from the mean.
Fill in the fields and the result will appear here automatically.
Converts a single value into standard deviations from the mean so that results measured on different scales become comparable: a score of 80 against a mean of 75 with a spread of 8 is the same 0.625 sigma as a height of 178 against a mean of 172 with a spread of 9.6. The sign is kept — a negative score means the value sits below the mean, and that is an answer, not an input error.
How it works
Formula and logic
z = (x − μ) / σ. The numerator is the ordinary deviation from the mean; the denominator rescales it into units of spread, which is why the z-score does not depend on the original measurement scale. x, μ and σ use the same unit and scale, and σ is positive. Thus z is dimensionless while x−μ retains the original unit. The calculation itself does not require normality and does not produce a p-value or tail probability. Interpreting normal-distribution percentages requires a separate assumption about the data. A nonzero result too small to represent causes a range error instead of a false zero.
Example
A value of 85 with a mean of 70 and a deviation of 10 gives z = 1.5: one and a half standard deviations above the mean.
Fields and units
- Value — data unit
- Mean — data unit
- Standard deviation — data unit
How to use
- — Enter the value you want to place.
- — Give the mean and the standard deviation of the set.
- — Read how many sigmas away it falls.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- NIST: standardization by subtracting the mean and dividing by deviation NIST: normal density and CDF; percentage interpretation requires normality
- Limitation
- The calculation itself does not require normality and does not produce a p-value or tail probability. Interpreting normal-distribution percentages requires a separate assumption about the data.
FAQ
What does a negative z-score mean?
That the value is below the mean. The sign is part of the answer: −1.5 and +1.5 are equally far from the mean, just in opposite directions.
Why is a standard deviation of zero rejected?
Zero spread means every value is identical. There is nothing to divide by, and "infinitely far from the mean" is not a number.
Which z-scores count as large?
For roughly normal data about 68 % of values fall within ±1 and about 95 % within ±2, so a magnitude above 2 already stands out from the rest.
Where do the mean and deviation come from?
They can be computed from the list of values itself — the mean and statistics calculator reports both together.