The z-score formula: standardizing a value
A z-score, also called a standard score, converts a raw value into a measure of how far that value sits from its distribution's mean, expressed in units of standard deviation rather than in the original units. The formula is z = (x - mean) / standard deviation, where x is the value being standardized. A z-score of 0 means the value equals the mean exactly; a positive z-score means the value is above the mean; a negative z-score means the value is below the mean.
Standardizing values this way makes it possible to compare measurements that come from different scales or different distributions -- for instance, comparing a student's performance on two different tests with different means and spreads, or comparing an individual laboratory result against a reference range. Because the z-score removes both the original units and the original scale, a z-score of +1 always means the same relative thing (one standard deviation above the mean of whatever distribution it came from), regardless of what is being measured.
What does 'two standard deviations above the mean' mean?
Saying a value is two standard deviations above the mean is simply another way of saying its z-score is +2: the value is further from the mean than a typical observation, by an amount equal to twice the distribution's standard deviation. Because standard deviation measures the typical spread of the data, a z-score of 2 or higher (or -2 or lower) is conventionally considered unusual, since relatively few observations in a roughly normal distribution fall that far from the center.
This phrasing is purely descriptive of position within a distribution -- it does not by itself say whether being two standard deviations above the mean is good, bad, or neutral. A z-score of +2 on a test-score distribution generally indicates a strong result, while a z-score of +2 on a blood-pressure distribution generally indicates a reading well above typical and worth clinical attention. The z-score reports distance from the mean; interpreting what that distance means always depends on what is being measured.
The standard normal distribution and percentiles
When the underlying data are approximately normal (bell-shaped), a z-score can be converted into a percentile using the standard normal cumulative distribution function, which gives the proportion of the distribution that lies below a given z-score. This conversion is what allows a z-score to be restated as 'this value is higher than roughly X% of the distribution.' The table below lists standard, well-established percentiles for whole-number z-scores from -2 to +3.
| Z-score | Percentile (share of distribution below) | Share above |
|---|---|---|
| -2 | 2.28% | 97.72% |
| -1 | 15.87% | 84.13% |
| 0 | 50.00% | 50.00% |
| +1 | 84.13% | 15.87% |
| +2 | 97.72% | 2.28% |
| +3 | 99.87% | 0.13% |
Worked example: converting a raw score to a percentile
Consider a value of x = 130 drawn from a distribution with a mean of 100 and a standard deviation of 15 (the classic scale used for IQ scores). Step 1: compute the z-score, z = (130 - 100) / 15 = 30 / 15 = 2. Step 2: look up z = 2 on the standard normal distribution, which corresponds to the 97.72nd percentile, meaning approximately 97.72% of the distribution lies at or below a value of 130, and approximately 2.28% lies above it.
A second example using the same scale: a value of x = 85 gives z = (85 - 100) / 15 = -15 / 15 = -1, which corresponds to the 15.87th percentile -- meaning this value sits below roughly 84.13% of the distribution and above roughly 15.87% of it. In both cases, the entire calculation reduces to two steps: standardize the value into a z-score, then read the corresponding percentile off the standard normal distribution.
The 68-95-99.7 empirical rule
For approximately normal data, the empirical rule (sometimes called the 68-95-99.7 rule) gives a quick way to judge how unusual a z-score is without a full percentile lookup: about 68% of values fall within one standard deviation of the mean (z between -1 and +1), about 95% fall within two standard deviations (z between -2 and +2), and about 99.7% fall within three standard deviations (z between -3 and +3).
This rule follows directly from the percentile table above: the share between z = -1 and z = +1 is 84.13% - 15.87% = 68.26%, the share between z = -2 and z = +2 is 97.72% - 2.28% = 95.44%, and the share between z = -3 and z = +3 is 99.87% - 0.13% = 99.74% -- all consistent with the commonly cited 68%, 95% and 99.7% approximations. The rule applies only when the underlying distribution is approximately normal; skewed or heavy-tailed data can depart substantially from these proportions even though z-scores can still be calculated for them.
常见问题
How do you calculate a z-score?
Subtract the mean from the value, then divide by the standard deviation: z = (x - mean) / sd. For example, a value of 130 on a scale with mean 100 and standard deviation 15 gives z = (130 - 100) / 15 = 2, meaning the value sits two standard deviations above the mean.
What percentile corresponds to a z-score of 2?
Approximately the 97.72nd percentile. The standard normal distribution places about 97.72% of values at or below a z-score of 2, meaning a value with that z-score is higher than roughly 97.72% of the distribution and lower than about 2.28% of it, assuming the underlying data are approximately normal.
What does it mean to be two standard deviations above the mean?
It means the value has a z-score of +2: it is further above the average than a typical observation, by an amount equal to twice the distribution's standard deviation. For approximately normal data, roughly 97.72% of the distribution lies at or below this value, making it a relatively uncommon result -- though whether that is desirable depends entirely on what is being measured.
Do z-scores require the data to be normally distributed?
The z-score calculation itself, z = (x - mean) / sd, is defined for any distribution and does not require normality. However, converting a z-score into a percentile using the standard normal distribution -- as in the examples above -- is only accurate when the underlying data are approximately normal (bell-shaped); for skewed or heavy-tailed data, percentiles should instead be read directly from the data.
What is the 68-95-99.7 rule?
It is a shorthand for how much of an approximately normal distribution falls within one, two and three standard deviations of the mean: about 68% within one standard deviation (z between -1 and +1), about 95% within two (z between -2 and +2), and about 99.7% within three (z between -3 and +3). It offers a quick way to judge how unusual a value is without doing a full percentile lookup.
参考文献
- 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).