About the Z-Score Calculator
What is the Z-Score Calculator?
A z-score (standard score) tells you how many standard deviations a value lies above or below the mean. A z-score of 1.5 means the value is 1.5 standard deviations above average; −2 means two standard deviations below.
Z-scores put different scales on a common footing — you can compare a maths mark with an English mark, or a child's height with growth charts. For normally distributed data, the calculator also converts the z-score into a percentile using the standard normal distribution.
How to use the Z-Score Calculator
- Value (x) (sample: 85)
- Mean (μ) (sample: 70)
- Standard Deviation (σ) (sample: 10)
- Press Calculate Z-Score (results also refresh as you type). The result card shows the Z-Score, along with Percentile (share below x), Share above x and Share between −|z| and +|z|.
Formula used
z = (x − μ) ÷ σ
- x = the value
- μ = the mean
- σ = the standard deviation
Percentile = Φ(z) × 100, where Φ is the cumulative standard normal distribution. The percentile interpretation assumes the data is approximately normal.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Value (x) | 85 |
| Mean (μ) | 70 |
| Standard Deviation (σ) | 10 |
| Result | Value |
|---|---|
| Z-Score | 1.5 |
| Percentile (share below x) | 93.32% |
| Share above x | 6.68% |
| Share between −|z| and +|z| | 86.64% |
Tips for using the Z-Score Calculator
- About 68% of values have |z| < 1, 95% have |z| < 2 and 99.7% have |z| < 3 in a normal distribution.
- Values with |z| > 3 are often treated as outliers.
- A z-score of 0 is exactly the mean, which is the 50th percentile.