|
In statistics and probability theory, a median is the number separating the higher half of a data 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 (e.g., the median of is 5). 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 〔Simon, Laura J.; ("Descriptive statistics" ), ''Statistical Education Resource Kit'', Pennsylvania State Department of Statistics〕 (the median of is (5 + 7) / 2 = 6), which corresponds to interpreting the median as the fully trimmed mid-range. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large result. A median is only defined on ordered one-dimensional data, and is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions. 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); 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 have values strictly less than the ''median'', and, at most, half have values strictly greater than the median. If each group contains 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 is ''b'', and, if ''a'' < ''b'' < ''c'' < ''d'', then the median of the list is the mean of ''b'' and ''c''; i.e., it is (''b'' + ''c'')/2. The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors. In terms of notation, some authors represent the median of a variable ''x'' either as ''x͂'' or as ''μ''1/2〔 sometimes also ''M''. There is no widely accepted standard notation for the median, so the use of these or other symbols for the median needs to be explicitly defined when they are introduced. The median is the 2nd quartile, 5th decile, and 50th percentile. ==Measures of location and dispersion== The median is one of a number of ways of summarising the typical values associated with members of a statistical population; thus, it is a possible location parameter. Since the median is the same as the ''second quartile'', its calculation is illustrated in the article on quartiles. When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation. For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient than the sample median when data are uncontaminated by data from heavy-tailed distributions or from mixtures of distributions, but less efficient otherwise, and that the efficiency of the sample median is higher than that for a wide range of distributions. More specifically, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean—see Efficiency (statistics)#Asymptotic efficiency and references therein. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「median」の詳細全文を読む スポンサード リンク
|