Standard score (Redirected from Z-score)
In statistics, a standard score (z) is a dimensionless quantity derived by subtracting the population mean from an individual (raw) score and then dividing the difference by the population standard deviation:
Knowing the true σ of a population is often unrealistic except in cases such as standardized testing in which the entire population is known. In cases where it is impossible to measure every member of a population, the standard deviation may be estimated using a random sample.
The z score calculation requires the following to be known:
The standard score is
The quantity z represents the distance between the raw score and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.
Another name for a standard score is a z-score. The conversion process itself is sometimes called standardizing.
The key point to remember for the z score is that it is calculated using the population mean and the population standard deviation and not the sample mean or sample deviation. Calculation of the z score requires knowledge of the population statistics as opposed to the statistics of a sample drawn from the population of interest.
Population statistics are rarely known in the real world except for circumstances such as standardized testing. The population of people taking a standardized test is known and the population statistics can be calculated because all of the scores of the test takers are available. On the other hand, a population such as people who smoke cigarettes is not fully described so the population statistics are approximated using samples of the population.
When a population is normally distributed, the percentile rank may be determined from the standard score and ubiquitous tables.
Standardizing in mathematical statistics
In mathematical statistics, a random variable X is standardized using the theoretical (population) mean and standard deviation:
where μ = E(X) is the mean and σ² = Var(X) the variance of the probability distribution of X.
If the random variable under consideration is the sample mean:
then the standardized version is
See also
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