“In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores.
It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This process of converting a raw score into a standard score is called standardizing or normalizing. However, “normalizing” can refer to many types of ratios; see normalization for more.
Standard scores are most-commonly called z-scores. The two terms may be used interchangeably, as they are in this article. Other terms include z-values, normal scores, and standardized variables.
Computing a z-score requires knowing the mean and standard deviation of the complete population to which a data point belongs. If one only has a sample of observations from the population, then the analogous computation with sample mean and sample standard deviation yields the t-statistic.
If the population mean and population standard deviation are known, a raw score x is converted into a standard score by
- {\displaystyle z={x-\mu \over \sigma }}
where:
- μ is the mean of the population.
- σ is the standard deviation of the population.
The absolute value of z represents the distance between that raw score x and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.
Calculating z using this formula requires the population mean and the population standard deviation, not the sample mean or sample deviation. But knowing the true mean and standard deviation of a population is often unrealistic, except in cases such as standardized testing, where the entire population is measured.
When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean and sample standard deviation as estimates of the population values.
In these cases, the z-score is
- {\displaystyle z={x-{\bar {x}} \over S}}
where:
- {\displaystyle {\bar {x}}} is the mean of the sample.
- S is the standard deviation of the sample.
In either case, since the numerator and denominator of the equation must both be expressed in the same units of measure, and since the units cancel out through division, z is left as a dimensionless quantity.
Wikipedia, “Standard score” (https://en.wikipedia.org/wiki/Standard_score)
