Normal score
Encyclopedia
The term normal score is used with two different meanings in statistics
. One of them relates to creating a single value which can be treated as if it had arisen from a standard (zero mean, unit variance) normal distribution. The second relates to assigning alternative values to data points within a dataset, with the broad intention of creating data values than can be interpreted as being approximations for values that might have been observed had the data arisen from a standard normal distribution.
The first meaning is as an alternative name for the standard score
or z score, where values are standardised by subtracting the sample or estimated mean and dividing by the sample or other estimate of the standard deviation. Particularly in applications where the name "normal score" is used, there is usually a presumption that the value can be referred to a table of standard normal probabilities as a means of providing an informal significance test of some hypothesis, such as a difference in means.
The second meaning of normal score is associated with data values derived from the rank
s of the observations within the dataset. A given data point is assigned a value which is either exactly, or an approximation, to the expectation of the order statistic
of the same rank in a sample of standard normal random variables of the same size as the observed data set. Thus the meaning of a normal score of this type is essentially the same as a rankit
, although the term "rankit" is becoming obsolete. In this case the transformation creates a set of values which is matched in a certain way to what would be expected had the original set of data values arisen from a normal distribution.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. One of them relates to creating a single value which can be treated as if it had arisen from a standard (zero mean, unit variance) normal distribution. The second relates to assigning alternative values to data points within a dataset, with the broad intention of creating data values than can be interpreted as being approximations for values that might have been observed had the data arisen from a standard normal distribution.
The first meaning is as an alternative name for the standard score
Standard score
In statistics, a standard score indicates how many standard deviations an observation or datum is above or below the mean. It 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...
or z score, where values are standardised by subtracting the sample or estimated mean and dividing by the sample or other estimate of the standard deviation. Particularly in applications where the name "normal score" is used, there is usually a presumption that the value can be referred to a table of standard normal probabilities as a means of providing an informal significance test of some hypothesis, such as a difference in means.
The second meaning of normal score is associated with data values derived from the rank
Ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either 'ranked higher than', 'ranked lower than' or 'ranked equal to' the second....
s of the observations within the dataset. A given data point is assigned a value which is either exactly, or an approximation, to the expectation of the order statistic
Order statistic
In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference....
of the same rank in a sample of standard normal random variables of the same size as the observed data set. Thus the meaning of a normal score of this type is essentially the same as a rankit
Rankit
In statistics, rankits of a set of data are the expected values of the order statistics of a sample from the standard normal distribution the same size as the data. They are primarily used in the normal probability plot, a graphical technique for normality testing.-Example:This is perhaps most...
, although the term "rankit" is becoming obsolete. In this case the transformation creates a set of values which is matched in a certain way to what would be expected had the original set of data values arisen from a normal distribution.