Understanding your z-score
The table below shows selected z-scores and their standard normal percentiles.
| Z-score | Percentile (share below) | Interpretation |
|---|---|---|
| -2 | 2.28% | Two standard deviations below the mean; unusually low |
| -1 | 15.87% | One standard deviation below the mean |
| 0 | 50% | Exactly at the mean (and the median, for a normal distribution) |
| +1 | 84.13% | One standard deviation above the mean |
| +2 | 97.72% | Two standard deviations above the mean; unusually high |
| +3 | 99.87% | Three standard deviations above the mean; rare under normality |
- Percentiles assume the underlying distribution is approximately normal. For skewed or heavy-tailed data, the z-to-percentile conversion is unreliable even though the z-score itself is well defined.
- By the empirical rule, about 68% of a normal distribution lies within z = -1 to +1, about 95% within -2 to +2, and about 99.7% within -3 to +3.
- A z-score says nothing about whether a value is good or bad — only how far it sits from the mean of its own distribution.
What is a z-score?
A z-score standardizes a value by expressing its distance from the mean in units of standard deviation. A z-score of 0 means the value equals the mean; +1 means one standard deviation above the mean; -2 means two standard deviations below. Standardizing puts measurements from different scales on a common footing, which is why z-scores are used to compare test results, growth measurements and laboratory values across different distributions.
When the underlying distribution is normal (bell-shaped), the z-score maps directly to a percentile through the standard normal cumulative distribution function (CDF). The CDF gives the proportion of the distribution lying below a given z — for example, about 84.13% of a normal distribution lies below z = 1, and about 97.72% below z = 2. This calculator evaluates the standard normal CDF with the Abramowitz and Stegun approximation (formula 26.2.17), accurate to within about 7.5 x 10^-8.
The percentile interpretation depends on normality. Z-scores can be computed for any data, but converting them to percentiles with the normal CDF is only accurate when the distribution is approximately normal; for skewed or heavy-tailed data the stated percentiles can be misleading.
How to use this z-score calculator
- Enter the value (x) you want to standardize — for example, an individual test score.
- Enter the mean of the distribution the value comes from.
- Enter the standard deviation of the distribution, which must be greater than zero.
- Read the z-score, the percentile (share of the distribution below the value, assuming normality), and the share above.
The z-score formula
The z-score is the value minus the mean, divided by the standard deviation: z = (x - mean) / sd. A positive z means the value is above the mean, a negative z below it.
Worked example: x = 130, mean = 100, standard deviation = 15 (the classic IQ scale). z = (130 - 100) / 15 = 30 / 15 = 2. The value sits exactly two standard deviations above the mean. From the standard normal CDF, the proportion below z = 2 is 0.9772, so the value is at the 97.72nd percentile and about 2.28% of the distribution lies above it.
Percentiles come from the standard normal cumulative distribution function: percentile = CDF(z) x 100, and share above = (1 - CDF(z)) x 100. The CDF has no closed-form expression; this calculator uses the Abramowitz and Stegun (1964) approximation 26.2.17.
Common mistakes
- Converting z-scores to percentiles for clearly non-normal data — the normal CDF mapping assumes a bell-shaped distribution.
- Mixing up the direction: a negative z-score means the value is below the mean, not an error or an impossible result.
- Dividing by the variance instead of the standard deviation — the denominator of the z formula is the standard deviation.
- Using the population mean and standard deviation from one group to standardize a value from a different group, which produces a meaningless comparison.
- Confusing percentile with percentage score: being at the 97.7th percentile means scoring above about 97.7% of the distribution, not scoring 97.7% on the test.
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How do I calculate a z-score?
Subtract the mean from the value, then divide by the standard deviation: z = (x - mean) / sd. For example, a score of 130 on a scale with mean 100 and standard deviation 15 gives z = (130 - 100) / 15 = 2, meaning the score is two standard deviations above the mean.
What percentile is a z-score of 2?
About the 97.72nd percentile. The standard normal cumulative distribution function evaluated at z = 2 is 0.9772, meaning 97.72% of a normal distribution lies below that value and 2.28% lies above it.
What is a good z-score?
Z-scores are neutral measurements, not grades — whether a given z is desirable depends entirely on context. A z of +2 is excellent for a test score but alarming for blood pressure. Values beyond about -2 or +2 are conventionally regarded as unusual, because fewer than 5% of a normal distribution lies outside that range.
Can a z-score be negative?
Yes. A negative z-score simply means the value lies below the mean. For example, z = -1.5 indicates a value one and a half standard deviations below the mean, at roughly the 6.68th percentile of a normal distribution.
Do z-scores require a normal distribution?
The z-score itself — distance from the mean in standard deviation units — is defined for any distribution. However, converting a z-score to a percentile using the standard normal CDF, as this calculator does, is only accurate when the distribution is approximately normal. For skewed data, percentiles should be computed from the data directly.
How accurate is the percentile shown?
The standard normal CDF has no exact closed-form expression, so this calculator uses the Abramowitz and Stegun (1964) rational approximation 26.2.17, whose absolute error is below 7.5 x 10^-8 — far more precise than the two decimal places displayed.
Kaynaklar
- Abramowitz M, Stegun IA (eds). Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series 55, 1964 (normal CDF approximation 26.2.17).
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.6.1: Normal distribution. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (standardized values and the normal distribution).