A z-score tells you how many standard deviations a value lies from the mean. It lets you compare results from different scales, such as two exams with different averages.
How to use it
Enter the value, the mean and the standard deviation. Type a target z into the result to find the raw value that corresponds to it.
Interpreting z
- About 68% of normal data lies within z = ±1.
- About 95% lies within ±1.96.
- |z| above 3 is unusual.
Percentiles assume the data are approximately normally distributed.
Inputs explained
| Field | What to enter |
|---|---|
| Value (x) | A number |
| Mean (μ) | A number |
| Standard deviation (σ) | A number |
What each result means
- z-score = ([Value (x)] – [Mean (μ)]) ÷ [Standard deviation (σ)]
- Percentile (area below) = ncdf([z-score]) × 100
- Area above = (1 – ncdf([z-score])) × 100
- Two-tailed p-value = 2 × (1 – ncdf(abs([z-score])))
Example values
How the main result changes when Value changes and everything else stays at the example values above:
| Value (x) | z-score |
|---|---|
| 41 | -3.625 |
| 61.5 | -1.063 |
| 82 | 1.5 |
| 102.5 | 4.063 |
| 123 | 6.625 |
| 164 | 11.75 |
Formula
z = (x − μ) ÷ σPercentile = Φ(z)Worked example
Score 82 on a test with mean 70 and SD 8.
z = (82 − 70) ÷ 8 = 1.5. About 93.3% of scores fall below 82.
Key terms
- Standard deviation
- Typical distance of values from the mean.
- Standard normal distribution
- Normal distribution with mean 0 and SD 1.
- p-value
- Probability of a result at least as extreme as observed, assuming the null hypothesis.
Frequently asked questions
Can a z-score be negative?
Yes, it means the value is below the mean.
Do I need normal data?
The z-score itself does not; percentiles and p-values do.
What z is the 95th percentile?
About 1.645.
What formula does the Z-Score Calculator use?
z = (x − μ) ÷ σ.
Can I work backwards from the result?
Yes. Type the result you want into z-score and the calculator finds the matching value (x).