Standard score
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In statistics, a standard score (z) is a dimensionless quantity derived by subtracting the sample mean from an individual (raw) score and then dividing the difference by the sample standard deviation:
- <math> z = {X - \bar{X} \over s}<math>
The quantity z represents the number of standard deviations between the raw score and the mean; it 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.
Conversion to standard scores enables the comparison and combination of measures made on different scales.
For example, the exam scores of two students in different classes may be compared by converting each student's exam score based on the mean and standard deviation of the scores in his or her own class. The student with the higher standard score performed better, even if the raw score was lower. [1] (http://soeweb.syr.edu/faculty/ddgilbri/assess/StandardScores.html)
If data are being combined or scaled, the standardization eliminates accidental weighting due to differences in means or standard deviations. This is particularly important in linear regression when the data are sample means rather than individual observations.
Standard scores are chiefly appropriate for data that are normally distributed, although that is not to say that they can never provide useful information about skewed data. The standard score also provides an estimate of the percentile rank of scores in a normal distribution.
Standardizing in mathematical statistics
In mathematical statistics, a random variable X is standardized using the theoretical (population) mean and standard deviation:
- <math>Z = {X - \mu \over \sigma}<math>
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:
- <math>\bar{X}=\sum_{i=1}^n X_i<math>
then the standardized version is
- <math>Z={\bar{X}-\mu\over\sigma/\sqrt{n}}<math>