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In
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the standard score is the number of
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
s by which the value of a raw score (i.e., an observed value or data point) is above or below the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arith ...
value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores. It is calculated by subtracting the
population mean In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hyp ...
from an individual raw score and then dividing the difference by the
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
standard deviation. This process of converting a raw score into a standard score is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
for more). Standard scores are most commonly called ''z''-scores; the two terms may be used interchangeably, as they are in this article. Other equivalent terms in use include z-values, normal scores, standardized variables and pull in high energy physics. Computing a z-score requires knowledge of the mean and standard deviation of the complete population to which a data point belongs; if one only has a sample of observations from the population, then the analogous computation using the sample mean and sample standard deviation yields the ''t''-statistic.


Calculation

If the population mean and population standard deviation are known, a raw score ''x'' is converted into a standard score by :z = where: : ''μ'' is the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arith ...
of the population, : ''σ'' is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
of the population. The absolute value of z represents the distance between that raw score ''x'' and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above. Calculating z using this formula requires use of the population mean and the population standard deviation, not the sample mean or sample deviation. However, knowing the true mean and standard deviation of a population is often an unrealistic expectation, except in cases such as
standardized testing A standardized test is a test that is administered and scored in a consistent, or "standard", manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predet ...
, where the entire population is measured. When the population mean and the population standard deviation are unknown, the standard score may be estimated by using the sample mean and sample standard deviation as estimates of the population values. In these cases, the z-score is given by :z = where: : \bar is the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arith ...
of the sample, : ''S'' is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
of the sample. Though it should always be stated, the distinction between use of the population and sample statistics often is not made. In either case, the numerator and denominator of the equations have the same units of measure so that the units cancel out through division and z is left as a
dimensionless quantity A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
.


Applications


Z-test

The z-score is often used in the z-test in standardized testing – the analog of the
Student's t-test A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
for a population whose parameters are known, rather than estimated. As it is very unusual to know the entire population, the t-test is much more widely used.


Prediction intervals

The standard score can be used in the calculation of
prediction interval In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are ...
s. A prediction interval 'L'',''U'' consisting of a lower endpoint designated ''L'' and an upper endpoint designated ''U'', is an interval such that a future observation ''X'' will lie in the interval with high probability \gamma, i.e. :P(L For the standard score ''Z'' of ''X'' it gives: :P\left( \frac < Z < \frac \right) = \gamma. By determining the quantile z such that :P\left( -z < Z < z \right) = \gamma it follows: :L=\mu-z\sigma,\ U=\mu+z\sigma


Process control

In process control applications, the Z value provides an assessment of the degree to which a process is operating off-target.


Comparison of scores measured on different scales: ACT and SAT

When scores are measured on different scales, they may be converted to z-scores to aid comparison. Dietz et al. give the following example, comparing student scores on the (old)
SAT The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and scoring have changed several times; originally called the Scholastic Aptitude Test, it was later called the Schol ...
and ACT high school tests. The table shows the mean and standard deviation for total scores on the SAT and ACT. Suppose that student A scored 1800 on the SAT, and student B scored 24 on the ACT. Which student performed better relative to other test-takers? The z-score for student A is z = = = 1 The z-score for student B is z = = = 0.6 Because student A has a higher z-score than student B, student A performed better compared to other test-takers than did student B.


Percentage of observations below a z-score

Continuing the example of ACT and SAT scores, if it can be further assumed that both ACT and SAT scores are normally distributed (which is approximately correct), then the z-scores may be used to calculate the percentage of test-takers who received lower scores than students A and B.


Cluster analysis and multidimensional scaling

"For some multivariate techniques such as multidimensional scaling and cluster analysis, the concept of distance between the units in the data is often of considerable interest and importance… When the variables in a multivariate data set are on different scales, it makes more sense to calculate the distances after some form of standardization."


Principal components analysis

In principal components analysis, "Variables measured on different scales or on a common scale with widely differing ranges are often standardized."


Relative importance of variables in multiple regression: Standardized regression coefficients

Standardization of variables prior to
multiple regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
is sometimes used as an aid to interpretation. (page 95) state the following. "The standardized regression slope is the slope in the regression equation if X and Y are standardized… Standardization of X and Y is done by subtracting the respective means from each set of observations and dividing by the respective standard deviations… In multiple regression, where several X variables are used, the standardized regression coefficients quantify the relative contribution of each X variable." However, Kutner et al. (p 278) give the following caveat: "… one must be cautious about interpreting any regression coefficients, whether standardized or not. The reason is that when the predictor variables are correlated among themselves, … the regression coefficients are affected by the other predictor variables in the model … The magnitudes of the standardized regression coefficients are affected not only by the presence of correlations among the predictor variables but also by the spacings of the observations on each of these variables. Sometimes these spacings may be quite arbitrary. Hence, it is ordinarily not wise to interpret the magnitudes of standardized regression coefficients as reflecting the comparative importance of the predictor variables."


Standardizing in mathematical statistics

In mathematical statistics, a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X'' is standardized by subtracting its
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
\operatorname /math> and dividing the difference by its
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
\sigma(X) = \sqrt: :Z = If the random variable under consideration is the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger pop ...
of a random sample \ X_1,\dots, X_n of ''X'': :\bar= \sum_^n X_i then the standardized version is :Z = \frac.


T-score

In educational assessment, T-score is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10. It is also known as '' hensachi'' in Japanese, where the concept is much more widely known and used in the context of university admissions. In bone density measurements, the T-score is the standard score of the measurement compared to the population of healthy 30-year-old adults, and has the usual mean of 0 and standard deviation of 1.


See also

*
Normalization (statistics) In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging ...
* Omega ratio *
Standard normal deviate A standard normal deviate is a normally distributed deviate. It is a realization of a standard normal random variable, defined as a random variable with expected value 0 and variance 1.Dodge, Y. (2003) The Oxford Dictionary of Statisti ...


References


Further reading

* *


External links


Interactive Flash on the z-scores and the probabilities of the normal curve
by Jim Reed {{Statistics Statistical ratios