About the Z-Score Calculator
This z-score calculator converts a raw score into a standard score — the number of standard deviations it sits above or below the mean — and then looks up the matching percentile and probabilities from the standard normal distribution, so you do not need a printed z-table. Enter the score, the mean and the standard deviation, or switch modes and enter a z-score directly.
Students use it for statistics homework and to compare scores on different tests (an 85 on one exam and a 1,200 on another can be compared once both are z-scores). Teachers use it to interpret results, researchers to standardise measurements and flag outliers, and analysts to compute tail probabilities for hypothesis tests. The shaded chart shows the area to the left of your z-score, which is the percentile.
Percentiles and probabilities assume the underlying data follow a normal (bell-shaped) distribution. The z-score itself is valid for any data, but converting it to a percentile is only accurate when the distribution is approximately normal.
With the default inputs, the z-score is 1.5. Change any value above to recalculate instantly.
How to use the z-score calculator
- 1Choose whether you are starting from a raw score or from a z-score.
- 2Enter the raw score, the mean and the standard deviation (or the z-score directly).
- 3Read the z-score and the percentile — the share of values below it.
- 4Use the left-, right- or two-tailed probability that matches your question or hypothesis test.
Formula and method
The z-score subtracts the mean from the raw score and divides by the standard deviation, which rescales any normal distribution to the standard normal distribution with mean 0 and standard deviation 1. A positive z is above the mean, a negative z below it, and z = 0 is exactly average.
Φ(z) is the cumulative distribution function of the standard normal distribution — the area under the bell curve to the left of z — which is what a printed z-table lists. The right-tail probability is 1 − Φ(z) and the two-tailed probability is 2 × (1 − Φ(|z|)). Φ is computed numerically here with a maximum error below 0.0000002. To go from a z-score back to a raw score, use x = μ + zσ.
- z
- Standard score (standard deviations from the mean)
- x
- Raw score or observation
- μ
- Mean of the distribution
- σ
- Standard deviation of the distribution
- Φ(z)
- Standard normal cumulative probability (area to the left of z)
Worked examples
Test score of 85 (mean 70, SD 10)
z = (85 − 70) ÷ 10 = 1.5. The area to the left of 1.5 on the standard normal curve is 0.9332, so the score is at about the 93rd percentile; only 6.7% of students would score higher.
SAT-style score of 1,200 (mean 1,050, SD 200)
z = (1200 − 1050) ÷ 200 = 0.75, which corresponds to Φ(0.75) ≈ 0.7734 — roughly the 77th percentile.
Below-average value: z = −1.2
A score of 58 with mean 70 and SD 10 gives z = −1.2. About 11.5% of values fall below it, and 77% of values lie between −1.2 and +1.2 standard deviations.
Critical value z = 1.96
z = 1.96 leaves 2.5% in each tail, so the two-tailed probability is 0.05 — the cut-off for 95% confidence. With mean 70 and SD 10 it corresponds to a raw score of 70 + 1.96 × 10 = 89.6.
Frequently asked questions
What is a z-score?+
A z-score (standard score) tells you how many standard deviations a value is from the mean. z = 2 means two standard deviations above average; z = −1 means one standard deviation below. It lets you compare values from different scales.
How do you convert a z-score to a percentile?+
Find the area under the standard normal curve to the left of the z-score, Φ(z), and multiply by 100. For example Φ(1) ≈ 0.8413, so z = 1 is about the 84th percentile. This calculator does the lookup for you.
What is a good z-score?+
It depends on context. In testing, a positive z-score means above average — z = 1 beats about 84% of people and z = 2 about 98%. In quality control or outlier detection, values beyond ±2 or ±3 are flagged as unusual.
Can a z-score be negative?+
Yes. A negative z-score means the value is below the mean. z = −1.5 is one and a half standard deviations below average, around the 7th percentile of a normal distribution.
What z-score corresponds to 95% confidence?+
For a two-sided 95% confidence interval the critical value is z = 1.96, because 2.5% of the normal distribution lies beyond +1.96 and 2.5% beyond −1.96. For 90% it is 1.645 and for 99% it is 2.576.