Z-Score Calculator
Calculate the Z-score of a value relative to its mean and standard deviation, with approximate percentile and interpretation.
Percentile: 73.4th
Z-Score
0.625
Percentile
73.4th
CDF Value
0.734
Interpretation
Within one standard deviation — fairly common.
How the Z-Score Works
A Z-score (standard score) tells you how many standard deviations a data point is from the mean. The formula is z = (X − μ) / σ, where X is the value, μ is the population mean, and σ is the standard deviation.
A positive z-score means the value is above the mean; a negative z-score means it is below. The z-score is then used to find the percentile — the percentage of data points that fall below this value — using the standard normal distribution.
Example
If a student scored 75 on a test where the mean was 70 and the standard deviation was 8: z = (75 − 70) / 8 = 0.625. This corresponds to roughly the 73rd percentile — the student scored higher than about 73% of test-takers.
Common Use Cases
- Comparing scores from different distributions (e.g. SAT vs. ACT).
- Identifying outliers in a dataset.
- Standardizing test scores, financial returns, or measurement data.
- Quality control and statistical process control.
FAQs
What does a z-score of 0 mean?
A z-score of 0 means the value is exactly equal to the mean — it sits right in the center of the distribution, at the 50th percentile.
Can z-scores be negative?
Yes — a negative z-score simply means the value is below the mean. A z-score of −2 means the value is two standard deviations below the mean.
Is the percentile exact?
The percentile shown is an approximation based on the standard normal distribution (using the error function). For very large or very small z-scores, the result approaches 0% or 100% but never exactly reaches them.