Standard score
From Wikipedia, the free encyclopedia
In statistics, a standard score (also called 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 z score reveals how many units of the standard deviation a case is above or below the mean. The z score allows us to compare the results of different normal distributions, something done frequently in research.
≈The standard score is not the same as the z-factor used in the analysis of high-throughput screening data, but is sometimes confused with it.
The conversion process is called standardizing.
The standard score is: 
.
where
- X is a raw score to be standardized
- σ is the standard deviation of the population
- μ is the mean of the population
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.
A key point is that calculating z requires the population mean and the population standard deviation, not the sample mean or sample deviation. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest.
But knowing the true standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured. In cases where it is impossible to measure every member of a population, the standard deviation may be estimated using a random sample. For example, a population of people who smoke cigarettes is not fully measured.
When a population is normally distributed, the percentile rank may be determined from the standard score and ubiquitous tables.
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[edit] 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
[edit] References
- Abdi, H. (2007). Z-scores. In N.J. Salkind (Ed.), Encyclopedia of Measurement and Statistics. Thousand Oaks, CA: Sage.
[edit] External links
Please see Wikipedia:External links for guidelines on appropriate external links.
- Free z-Score (Standardized Score) Calculator Calculate z-score given sample mean, sample standard deviation and the unstandardized value.
- Z-Score to percentile conversion table With a given Z-Score, calculate the value's percentile rank.
