Understanding your result
Positive z-scores lie above the mean and negative scores below it. The lower-tail percentage is a modeled percentile, while the two-tail result combines both equally distant extremes. These are not empirical ranks from your dataset.
If you have observations but no spread estimate, first use the Standard Deviation Calculator .
The formula
Subtracting the mean centers the observation and dividing by a positive standard deviation expresses its distance in spread units. The normal CDF is evaluated numerically; displayed percentages round to four decimal places.
Worked example
For an observation of 85, mean 70, and standard deviation 10, z = 1.5. Under a normal model, approximately 93.3193% lies below that value, 6.6807% above it, and 13.3614% beyond either −1.5 or +1.5.
How to use this calculator
- Enter the observation and mean using the same measurement scale.
- Supply a strictly positive standard deviation in those units.
- Read the standardized distance, then interpret areas only if a normal model is appropriate.
Standardization does not prove normality
A z-score is a rescaling operation that can be applied to many datasets. Interpreting it as a normal percentile is an additional modeling decision. Skewed, heavy-tailed, or mixed populations can have very different tail frequencies. The density table describes the assumed normal model; it is not fitted from observations entered here.
Tail areas are not automatically test results
A formal hypothesis test needs its own sampling model and assumptions. For example, uncertainty in an estimated mean or standard deviation may require a different distribution. This tool does not choose a significance threshold or establish causation. Very small normal-tail areas can display as zero after rounding even though their mathematical probability is positive.
Assumptions & limitations
What this calculation assumes
- Normal areas use a continuous standard normal distribution.
What to keep in mind
- No normality test, sample-size adjustment, t-distribution, or empirical percentile calculation.
Common questions
Why must standard deviation be positive?
Zero spread makes division undefined. A negative standard deviation is not a valid spread measure.
Does a z-score of zero mean no data?
No. It means the observation equals the supplied mean. The normal reference has half its area on each side.