mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

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 ...

, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's

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 '' ar ...

. The

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...

of the deviation reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference.

Types

A deviation that is a difference between an observed value and the ''true value'' of a quantity of interest (where ''true value'' denotes the Expected Value, such as the population mean) is an error. A deviation that is the difference between the observed value and an ''estimate'' of the true value (e.g. the sample mean; the Expected Value of a sample can be used as an estimate of the Expected Value of the population) is a residual. These concepts are applicable for data at the interval and

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

levels of measurement.

Unsigned or absolute deviation

, the absolute deviation of an element of a data set is the

absolute difference The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for ...

between that element and a given point. Typically the deviation is reckoned from the central value, being construed as some type of

average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...

, most often the

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...

or sometimes the

of the data set:

D_i = , x_i - m(X), ,

where *''D''_''i'' is the absolute deviation, *''x''_''i'' is the data element, *''m''(''X'') is the chosen measure of central tendency of the data set—sometimes the

(

\overline

), but most often the

Measures

Mean signed deviation

For an unbiased estimator, the average of the signed deviations across the entire set of all observations from the unobserved population parameter value averages zero over an arbitrarily large number of samples. However, by construction the average of signed deviations of values from the sample mean value is always zero, though the average signed deviation from another measure of central tendency, such as the sample median, need not be zero.

Dispersion

Statistics of the distribution of deviations are used as measures of statistical dispersion. *

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 ...

is the frequently used measure of dispersion: it uses

squared A square is a regular quadrilateral with four equal sides and four right angles. Square or Squares may also refer to: Mathematics and science *Square (algebra), multiplying a number or expression by itself *Square (cipher), a cryptographic block ...

deviations, and has desirable properties, but is not

robust Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...

. * Average absolute deviation, is the sum of absolute values of the deviations divided by the number of observations. *

Median absolute deviation In statistics, the median absolute deviation (MAD) is a Robust statistics, robust measure of the statistical dispersion, variability of a univariate sample of quantitative data. It can also refer to the statistical population, population paramete ...

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

Maximum absolute deviation The average absolute deviation (AAD) of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, media ...

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

Normalization

Deviations have units of the measurement scale (for instance, meters if measuring lengths). One can nondimensionalize in two ways. One way is by dividing by a measure of scale ( statistical dispersion), most often either the population standard deviation, in

standardizing In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mea ...

, or the sample standard deviation, in studentizing (e.g., Studentized residual). One can scale instead by ''location,'' not '' dispersion:'' the

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

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

References

{{DEFAULTSORT:Deviation (Statistics) Statistical deviation and dispersion Statistical distance