Z-score Calculator
Enter a value, the mean and the standard deviation. The calculator returns the z-score, rounded to at most four decimals, and the percentage of a normal distribution that falls below and above that value. Switch to the second mode to go the other way, from a z-score to the raw score.
How to use the Z-score Calculator
- Enter your numbers in the Z-score Calculator above.
- Press the Calculate Z-Score or Calculate Raw Score button to see the result.
- To try other numbers, change any value and press the button again.
The z-score formulas
z = (x − μ) ÷ σ · · · x = μ + z × σ · · · Below = Φ(z) · · · Above = 1 − Φ(z)
A z-score is the number of standard deviations a value sits from the mean. Φ is the cumulative standard normal distribution. z = 0 is the mean, with 50% below it. The standard deviation must be above 0. The raw score is the same formula solved for x, so a z-score of 0 gives back the mean.
Worked example
With a mean of 70 and a standard deviation of 10, a value of 75 has z = (75 − 70) ÷ 10 = 0.5. The calculator puts 69.15% of the distribution below it and 30.85% above. A value of 62 gives z = −0.8, with 21.19% below. In the other direction, z = 1.5 gives a raw score of 70 + 1.5 × 10 = 85, with 93.32% below.
Frequently asked questions
What is the 68-95-99.7 rule?
For normal distributions: 68% of values fall within ±1 SD (z between −1 and 1); 95% within ±2 SD; 99.7% within ±3 SD. Useful for quick mental checks.
What z-score is "extreme"?
A |z| above 2 is unusual, about 4.6% of values in a normal distribution. Above 3 is rare, about 0.3%. Beyond 4 it almost never happens if the data really is normal.
How do I find a raw score from a z-score?
Multiply the z-score by the standard deviation and add the mean. With a mean of 70 and a standard deviation of 10, z = 1.5 is 70 + 1.5 × 10 = 85, and z = −2 is 70 − 2 × 10 = 50. Pick Z-score to raw score and the calculator does this for you, along with the percentage of the distribution below and above that score.
Does z-score require normal distribution?
The z formula works for any distribution. But interpreting it as a percentile from the standard normal table only works if the data is actually normal.