Standard scores



In statistics, a standard score (also z-score or normal score) 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:

The standard score, which is also commonly known as the z-score, is not the same as, but is sometimes confused with, the Z-Factor used in the analysis of high-throughput screening data.

Knowing the true &sigma; 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:
 * &sigma; (the standard deviation of the population)
 * &mu; (the mean of the population)
 * X (a raw score)

The standard score is


 * $$ z = {X - \mu \over \sigma}.$$

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:


 * $$Z = {X - \mu \over \sigma}$$

where &mu; = E(X) is the mean and &sigma;&sup2; = Var(X) the variance of the probability distribution of X.

If the random variable under consideration is the sample mean:


 * $$\bar{X}={1 \over n} \sum_{i=1}^n X_i$$

then the standardized version is


 * $$Z={\bar{X}-\mu\over\sigma/\sqrt{n}}$$