TimelyScholarFree Tools › Z-Score Calculator
Convert a value to a z-score, find the percentile and tail probabilities under the normal distribution, or find the z-score for a percentile.
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| z-score | 2.000 |
|---|---|
| Percentile (area below) | 97.72% |
| Area above | 2.28% |
| Two-tailed p-value | 0.046 |
Value to z, z to probability, or percentile to z.
Value, mean and SD, a z-score or a percentile.
A z-score tells you how many standard deviations a value lies from the mean. A z-score of 2 means the value is two standard deviations above the mean.
Under a normal distribution, z-scores convert directly to percentiles: about 95% of values lie between z = −1.96 and z = 1.96.
Worked example: on a test scored with mean 100 and standard deviation 15, a score of 130 has z = (130 − 100) / 15 = 2.00. That is the 97.7th percentile, so about 2.3% of people score higher. Working backwards, the 90th percentile is z = 1.28, a score of about 119.
Because z-scores are in standard-deviation units, they let you compare values measured on different scales, such as results from two exams with different means and spreads. They also help flag unusual values: under a normal distribution only about 0.3% of values lie beyond ±3.
The percentile conversions assume the data are roughly normal. For skewed measures such as income or length of hospital stay, a z-score still tells you how far a value is from the mean, but the percentile it implies will be wrong; use the observed percentile in your data instead.
About 1.645. The 97.5th percentile is about 1.96, which is why 1.96 appears in 95% confidence intervals.
Yes. A negative z-score means the value is below the mean. For example, z = −1 is one standard deviation below the mean, about the 16th percentile.
Values beyond ±2 occur in about 5% of a normal distribution and values beyond ±3 in about 0.3%, so they are often used as flags. Whether a value is a problem depends on the context.
Use the population value when it is known, such as published norms for a test. Otherwise use the sample SD; with small samples, test statistics based on it follow the t distribution rather than the normal.