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Example:Finding Mean Deviation

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In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values.


In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median (in the case of an even sample size), and, if there is such a member, there may be more than one so that the median may not uniquely identify a sample member. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.

At most, half the population has values less than the median, and, at most, half have values greater than the median. If both groups contain less than half the population, then some of the population is exactly equal to the median. For example, if a < b < c, then the median of the list {abc} is b, and, if a < b < c < d, then the median of the list {abcd} is the mean of b and c; i.e., it is (b + c)/2.




Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term "arithmetic mean" is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.


Suppose we have sample space . Then the arithmetic mean A is defined via the equation



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.

Di = | xi − m(X) |


Di is the absolute deviation,

xi is the data element

and m(X) is the chosen measure of central tendency of the data set-sometimes the mean (), but most often the median.

Average absolute deviation

The average absolute deviation, or simply average deviation of a data set is the average of the absolute deviations and is a summary statistic of statistical dispersion or variability. It is also called the mean absolute deviation, but this is easily confused with the median absolute deviation.

The average absolute deviation of a set {x1, x2, ..., xn} is


Mean absolute deviation

The mean absolute deviation (MAD) is the mean absolute deviation from the mean. A related quantity, the mean absolute error (MAE), is a common measure of forecast error in time series analysis, where this measures the average absolute deviation of observations from their forecasts.

Although the term mean deviation is used as a synonym for mean absolute deviation, to be precise it is not the same; in its strict interpretation (namely, omitting the absolute value operation), the mean deviation of any data set from its mean is always zero.

(Our solved example in mathguru.com uses this concept)




Median absolute deviation


In statistics, the median absolute deviation (MAD) 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.

For a univariate data set X1X2, ..., Xn, the MAD is defined as the median of the absolute deviations from the data's median:

that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values.




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.