In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
,
deviation is a
measureMeasure can mean:* Measurement, the process of establishing the magnitude of some attribute of an object relative to some unit of measurement* Measure , a way to assign non-negative real numbers to subsets...
of
differenceDifference is the contrary of equality, in particular of objects. Differences can only be stated on the basis of a comparison or categorization...
for interval and ratio variables between the observed value and the
meanIn 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, either positive or negative, indicates whether the observation is larger than or smaller than the mean. The magnitude of the value reports how different (in the relevant numerical scale) an observation is from the mean. Deviations are known as
errors or residualsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of "deviation of a sample from the mean": the error of a sample is the deviation of the sample from the population mean or actual function, while the residual of a sample is the...
: deviations from the population mean are errors, while deviations from the sample mean are residuals.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
,
deviation is a
measureMeasure can mean:* Measurement, the process of establishing the magnitude of some attribute of an object relative to some unit of measurement* Measure , a way to assign non-negative real numbers to subsets...
of
differenceDifference is the contrary of equality, in particular of objects. Differences can only be stated on the basis of a comparison or categorization...
for interval and ratio variables between the observed value and the
meanIn 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, either positive or negative, indicates whether the observation is larger than or smaller than the mean. The magnitude of the value reports how different (in the relevant numerical scale) an observation is from the mean. Deviations are known as
errors or residualsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of "deviation of a sample from the mean": the error of a sample is the deviation of the sample from the population mean or actual function, while the residual of a sample is the...
: deviations from the population mean are errors, while deviations from the sample mean are residuals.
One of the features of the mean is that the sum of the deviations across the entire set of all observations is always zero, corresponding to the fact that the
average deviation is zero.
Dispersion
Statistics of the distribution of deviations are used as measures of
statistical dispersionIn statistics, statistical dispersion is variability or spread in a variable or a probability distribution...
.
Standard deviationIn probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...
is the frequently used measure of dispersion: it uses
squareIn algebra, the square of a number is that number multiplied by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x²...
d deviations, and has desirable properties, but is not robust.
Average deviation, or more precisely "average
absolute deviation" is calculated using the
absolute valueIn mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....
of deviation – it is the sum of absolute values of the deviations divided by the number of observations.
Median absolute deviationIn statistics, the median absolute deviation is a robust measure of the variability of a univariate sample of quantitative data...
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
Deviations have units of the measurement scale (for instance, meters if measuring lengths); one can
nondimensionalizeNondimensionalization 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 dispersionIn 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
formulaIn mathematics and other sciences, a formula In mathematics and other sciences, a formula In mathematics and other sciences, a formula (plural: formulas or formulaeis a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between...
for a
percent deviation is the accepted value minus observed value divided by the observed value multiplied by 100.
See also
- Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of "deviation of a sample from the mean": the error of a sample is the deviation of the sample from the population mean or actual function, while the residual of a sample is the...
- Standard score
In statistics, a standard score indicates how many standard deviations an observation 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
In statistics, a studentized residual is the quotient resulting from 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 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.whereSeveral...
- Standard deviation
In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...
- 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
In probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...