Deviation (statistics)

# Deviation (statistics)

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and statistics
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....

, deviation is a measure of difference for interval and ratio variables between the observed value and the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

. The sign of deviation (positive or negative), reports the direction of that difference (it is larger when the sign is positive, and smaller if it is negative). The magnitude of the value indicates the size of the difference.
Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and statistics
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....

, deviation is a measure of difference for interval and ratio variables between the observed value and the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

. The sign of deviation (positive or negative), reports the direction of that difference (it is larger when the sign is positive, and smaller if it is negative). The magnitude of the value indicates the size of the difference.

Deviations are known as errors or residuals
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

: deviations from the population mean are errors, while deviations from the sample mean are residuals.

The sum of the deviations across the entire set of all observations from the mean is always zero, and the average deviation is zero.

### Dispersion

Statistics of the distribution of deviations are used as measures of statistical dispersion
Statistical dispersion
In statistics, statistical dispersion is variability or spread in a variable or a probability distribution...

.

Standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

is the frequently used measure of dispersion: it uses squared deviations, and has desirable properties, but is not robust
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

.

Average absolute deviation
Absolute deviation
In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median or sometimes the mean of the data set.D_i = |x_i-m|...

,
sometimes called the "average deviation" is calculated using the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of deviation – it is the sum of absolute values of the deviations divided by the number of observations.

Median absolute deviation
Median absolute deviation
In statistics, the median absolute deviation is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample....

is a robust statistic which uses the median, not the mean, of absolute deviations.

Maximum absolute deviation is a highly non-robust measure, which uses the maximum absolute deviation.

### Dimensional analysis

For more on Studentizing, see Studentization
Studentization
In statistics, Studentization, named after William Sealy Gosset, who wrote under the pseudonym Student, is the adjustment consisting of division of a first-degree statistic derived from a sample, by a sample-based estimate of a population standard deviation...

, Studentized residual
Studentized residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard...

, and Studentized range.

Deviations have units of the measurement scale (for instance, meters if measuring lengths); one can nondimensionalize
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...

them by dividing by a measure of scale (statistical dispersion
Statistical dispersion
In statistics, statistical dispersion is variability or spread in a variable or a probability distribution...

), most often either the population standard deviation, in standardizing, or the sample standard deviation, in studentizing.

One can also scale by location, not dispersion: the formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

for a percent deviation is the accepted value minus observed value divided by the observed value multiplied by 100.

• Errors and residuals in statistics
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

• 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...

• Studentized residual
Studentized residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard...

• Absolute deviation
Absolute deviation
In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median or sometimes the mean of the data set.D_i = |x_i-m|...

• Standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

• Squared deviations
Squared deviations
In probability theory and statistics, the definition of variance is either the expected value , or average value , of squared deviations from the mean. Computations for analysis of variance involve the partitioning of a sum of squared deviations...

• Variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...