WordNet

have a specified degree of importance; "My ex-husband means nothing to me"; "Happiness means everything"
an average of n numbers computed by adding some function of the numbers and dividing by some function of n (同)mean_value
mean or intend to express or convey; "You never understand what I mean!"; "what do his words intend?" (同)intend
denote or connote; "`maison means `house in French"; "An example sentence would show what this word means" (同)intend, signify, stand_for
(used of persons or behavior) characterized by or indicative of lack of generosity; "a mean person"; "he left a miserly tip" (同)mingy, miserly, tight
destine or designate for a certain purpose; "These flowers were meant for you"
excellent; "famous for a mean backhand"
rich in significance or implication; "a meaning look" (同)pregnant, significant
the message that is intended or expressed or signified; "what is the meaning of this sentence"; "the significance of a red traffic light"; "the signification of Chinese characters"; "the import of his announcement was ambiguous" (同)significance, signification, import
the idea that is intended; "What is the meaning of this proverb?" (同)substance
considerable capital (wealth or income); "he is a man of means" (同)substance
how a result is obtained or an end is achieved; "a means of control"; "an example is the best agency of instruction"; "the true way to success" (同)agency, way
an instrumentality for accomplishing some end
deviate behavior (同)deviance
the error of a compass due to local magnetic disturbances
a variation that deviates from the standard or norm; "the deviation from the mean" (同)divergence, departure, difference
the difference between an observed value and the expected value of a variable or function

PrepTutorEJDIC

〈人が〉…‘を'『意図する』,‘の'つもりである;…‘を'言おうとある;示そうとする / 〈単語・句・文章などが〉…‘を'『意味を持つ』,‘を'『意味する』 / (人に対して)…‘を'当てる,向ける《+『名』+『for』+『名』》 / 《受動態で》〈人・物〉‘を'(…)向きにする / 〈物事が〉…‘を'示す,表す / (結果として)〈物事が〉…‘を'もたらす,生じさせる / 《『mean』+『for』+『名』+『to』 do》《米話》(…が…するような)つもりでいる
《名詞の前にのみ用いて》(品質・価値・才能などが)『劣った』,つまらない / (人・行為などが)『卑劣な』,さもしい / 《古》(身分・地位などが)卑しい / 《まれ》(身なり・建物などが)みずぼらしい,見るもあわれな / 《米話》(動物が)御しにくい;(一般に)扱いにくい
中庸,中道 / (数学で) / 平均[値];(特に)算術平均,相加平均 / (比例式の)中項,内項 / (両極端の)中間の,中央に位置する / (大きさ・品質・程度などが)平均的な,平均の
(・・・の)『目的』,意義《+『of』+『名』》 / (吾・文の)『意味』(sense)《+『of』+『名』》 / 意味ありげな様子,意味深長 / 《名詞の前にのみ用いて》意味ありげな,意味常長な / 《複合吾を作って》・・・するつもりの,の意図をもった
〈C〉《単数・複数扱い》『方法』,『手段』 / 《複数扱い》『資力』,『財産』,富
〈C〉〈U〉(…から)外れること,(…からの)脱線,逸脱《+『from』+『名』》 / 〈C〉偏差;磁針の自差
『私を』(『に』) / 《話》『私』(=『I』)

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2012/02/23 19:52:27」(JST)

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"Mean deviation" redirects here. For the book, see Mean Deviation (book).

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 (X) |

where

D_i is the absolute deviation,

x_i 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.

Measures of dispersion

Several measures of statistical dispersion are defined in terms of the absolute deviation.

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, ..., x_n} is

The choice of measure of central tendency, $m (X)$ , has a marked effect on the value of the average deviation. For example, for the data set {2, 2, 3, 4, 14}:

Measure of central tendency $m (X)$	Average absolute deviation
Mean = 5
Median = 3
Mode = 2

The average absolute deviation from the median is less than or equal to the average absolute deviation from the mean. In fact, the average absolute deviation from the median is always less than or equal to the average absolute deviation from any other fixed number.

The average absolute deviation from the mean is less than or equal to the standard deviation; one way of proving this relies on Jensen's inequality.

For the normal or "Gaussian" distribution, the ratio of mean absolute deviation to standard deviation is . Thus if X is a normally distributed random variable with expected value 0 then

In other words, for a Gaussian, mean absolute deviation is about 0.8 times the standard deviation.

Mean absolute deviation

The mean absolute deviation (MAD), also referred to as the mean deviation, is the mean of the absolute deviations of a set of data about the data’s mean. In other words, it is the average distance of the data set from its mean during certain number of time periods.

The equation for MAD is as follows:

MAD = 1/n ∑(|e_i|) , where e_i = F_i - D_i

This method forecast accuracy is very closely related to the mean squared error (MSE) method which is just the average squared error of the forecasts. Although these methods are very closely related MAD is more commonly used because it does not require squaring.

The equation for MSE is as follows:

MSE = 1/n Σ(e_i²) , where e_i = F_i - D_i

Median absolute deviation (MAD)

Main article: Median absolute deviation

The median absolute deviation is the median of the absolute deviation from the median. It is a robust estimator of dispersion.

For the example {2, 2, 3, 4, 14}: 3 is the median, so the absolute deviations from the median are {1, 1, 0, 1, 11} (reordered as {0, 1, 1, 1, 11}) with a median of 1, in this case unaffected by the value of the outlier 14, so the median absolute deviation (also called MAD) is 1.

Maximum absolute deviation

The maximum absolute deviation about a point is the maximum of the absolute deviations of a sample from that point. It is realized by the sample maximum or sample minimum and cannot be less than half the range.

Minimization

The measures of statistical dispersion derived from absolute deviation characterize various measures of central tendency as minimizing dispersion: The median is the measure of central tendency most associated with the absolute deviation, in that

L² norm statistics: just as the mean minimizes the standard deviation,
L¹ norm statistics: the median minimizes average absolute deviation,
L^∞ norm statistics: the mid-range minimizes the maximum absolute deviation, and
trimmed L^∞ norm statistics: for example, the midhinge (average of first and third quartiles) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% have been trimmed off.

Estimation

The mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population. In order for the absolute deviation to be an unbiased estimator, the expected value (average) of all the sample absolute deviations must equal the population absolute deviation. However, it does not. For the population 1,2,3 the population absolute deviation is 2/3. The average of all the sample standard deviations of size 3 that can be drawn from the population is 40/81. Therefore the absolute deviation is a biased estimator.

External links

Advantages of the mean absolute deviation

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