Z-score calculator
Standardising a value puts it on the one scale every distribution shares. This converts between raw values, z-scores, and percentiles in any direction.
A z-score expresses how far a value sits from the mean in units of the standard deviation, turning a raw number into a position on the standard normal curve. This calculator works in three directions: it converts a raw value into its z-score and percentile, recovers the raw value behind a given z-score, and finds the z-score that marks a chosen percentile. Each result reports the tail probabilities you need for standardising scores or flagging outliers.
z-score
1.500
85.00 sits at the 93.3th percentile.
The standardisation formula
A z-score subtracts the mean and divides by the standard deviation, so a value of x becomes:
z = (x - μ) / σ
To go the other way, the raw value is recovered as x = μ + z × σ. The percentile is the cumulative area of the standard normal distribution to the left of the z-score, computed here from the normal cumulative distribution function rather than a printed z-table, and the percentile-to-z direction inverts that function. The percentile interpretation assumes the underlying distribution is approximately normal; for a heavily skewed variable the z-score still measures distance from the mean, but it no longer maps cleanly onto a percentile.
The guides behind this tool
Frequently asked questions
- How do you calculate a z-score?
A z-score is the raw value minus the mean, divided by the standard deviation, so it expresses how many standard deviations the value sits above or below the mean. A z-score of 1.5 means the value is one and a half standard deviations above average, and a negative z-score means it is below average. Because the units cancel, z-scores let you compare values measured on different scales on a common footing.
- What does a z-score tell you?
A z-score tells you where a single value falls within its distribution relative to the mean, measured in standard deviations. When the distribution is normal, the z-score maps directly to a percentile, so a z of 0 is the 50th percentile and a z of about 1.64 is the 95th. It is the basis for standardising scores, flagging outliers, and converting between raw values and probabilities.
- What z-score corresponds to a 95% confidence level?
For a two-tailed 95% confidence interval the critical z-score is approximately 1.96, because 95% of the standard normal distribution lies between -1.96 and +1.96. For a one-tailed 95% level the value is about 1.645. These critical values are why 1.96 appears so often in confidence interval and hypothesis-testing formulas built on the normal distribution.
- Is a higher z-score better?
It depends entirely on what is being measured. A high positive z-score is good for a test score or a desirable outcome but bad for something like reaction time or error rate, where lower is better. The z-score is a neutral measure of distance from the mean, so its desirability is set by the context, not by the sign itself.