Toolstia / Z-score calculator
Z-score calculator
A z-score tells you how many standard deviations a value is above or below the mean. Enter a raw score, the mean and the standard deviation to get the z-score, the percentile and the probabilities to the left and right. You can also start from a z-score and find the raw score.
Enter the mean and a standard deviation above zero, plus the score.
These areas assume the data follows a normal (bell curve) distribution. The percentile is the share of values expected to fall below the score.
How it works
- Choose whether you know the raw score or the z-score.
- Enter the value, the mean and the standard deviation.
- Read the z-score, percentile and areas.
The formula
z = (x − μ) ÷ σ, where x is the raw score, μ the mean and σ the standard deviation. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below.
Percentile and areas
The area to the left of z under the standard normal curve is the share of values expected to fall below your score. The tool also gives the area to the right, the two-tailed p-value, and the area between −|z| and +|z|.
Precision
The normal distribution is computed directly with a series that is accurate to about 15 digits, so the tail areas are reliable even for large z-scores.
When it applies
The areas assume the data is roughly normal. For skewed data, z-scores still measure distance from the mean, but the percentiles will not be accurate.
Questions
What counts as an unusual z-score?
Beyond ±2 is uncommon (about 5% of values) and beyond ±3 is rare (about 0.3%).
Is the p-value one-tailed or two-tailed?
Both are shown: the left and right areas are one-tailed, and the two-tailed value is shown separately.
Is anything uploaded?
No. Your numbers stays in your browser and is never sent anywhere.