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. For example, in the data set, the median is 6, the fourth largest, the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is likely to fall above or below it; the basic advantage of the median in describing data compared to the mean is that it is not skewed so much by a small proportion of large or small values, so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets, which vary the mean may be skewed by a small number of high or low values. Median income, for example, may be a better way to suggest; because of this, 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 or small result.

The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers 1, 3, 3, 6, 7, 8, 9This list contains seven numbers; the median is the fourth of them, 6. If there is an number of observations there is no single middle value. For example, in the data set 1, 2, 3, 4, 5, 6, 8, 9the median is the mean of the middle two numbers: this is / 2, 4.5.. With this convention, the median can be described in a caseless formula, as follows: m e d i a n = a ⌊ l + 1 2 ⌋ + a ⌈ l + 1 2 ⌉ 2 where a is an ordered list of l numbers, ⌊ ⋅ ⌋ and ⌈ ⋅ ⌉ denote the floor and ceiling functions, respectively. Formally, a median of a population is any value such that at most half of the population is less than the proposed median and at most half is greater than the proposed median; as seen above, medians may not be unique. If each set contains less than half the population some of the population is equal to the unique median.

The median is well-defined for any ordered data, is independent of any distance metric. The median can thus be applied to ranked but not numerical classes, although the result might be halfway between classes if there is an number of cases. A geometric median, on the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid. There is no accepted standard notation for the median, but some authors represent the median of a variable x either as x͂ or as μ1/2 sometimes M. In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced; the median is a special case of other ways of summarising the typical values associated with a statistical distribution: it is the 2nd quartile, 5th decile, 50th percentile. The median can be used as a measure of location when one attaches reduced importance to extreme values because a distribution is skewed, extreme values are not known, or outliers are untrustworthy, i.e. may be measurement/transcription errors.

For example, consider the multiset 1, 2, 2, 2, 3, 14. The median is 2 in this case, it might be seen as a better indication of the center than the arithmetic mean of 4, larger than all-but-one of the values! However, the cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not true. At most, one can say; as a median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated; because the median is simple to understand and easy to calculate, while a robust approximation to the mean, the median is a popular summary statistic in descriptive statistics. In this context, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, the median absolute deviation.

For practical purposes

Hida Mountains

The Hida Mountains, or Northern Alps, is a Japanese mountain range which stretches through Nagano and Gifu prefectures. A small portion of the mountains reach into Niigata Prefecture. William Gowland coined the phrase "Japanese Alps" during his time in Japan, but he was only referring to the Hida Mountains when he used that name; the Kiso and Akaishi mountains received the name in the ensuing years. The layout of the Hida Mountains forms a large Y-shape; the southern peaks are the lower portion of the Y-shape, with the northern peaks forming two parallel bands separated by a deep V-shaped valley. It is one of the steepest V-shaped valleys in Japan; the Kurobe Dam, Japan's largest dam, is an arch dam located in the Kurobe Valley in the central area of the mountains. The western arm of mountains known as the Tateyama Peaks, are dominated by Mount Tsurugi and Mount Tate; the eastern arm, known as the Ushiro Tateyama Peaks, are dominated by Mount Shirouma and Mount Kashimayari. Although it was thought that no glaciers existed in East Asia south of Kamchatka, recent research has shown that three small glaciers still survive in Mount Tsurugi and Mount Tate owing to the wet climate of the Hokuriku region allowing for heavy snowfalls on the high peaks.

Mount Shirouma, 2,932 m Mount Kashimayari, 2,889 m Mount Tate, 3,015 m Mount Tsubakuro, 2,763 m Mount Tsurugi, 2,999 m Mount Noguchigoro, 2,924 m Mount Yari, 3,180 m Mount Hotaka, 3,190 m Mount Norikura, 3,026 m Japanese Alps Kiso Mountains Akaishi Mountains List of mountains in Japan 100 Famous Japanese Mountains Chūbu-Sangaku National Park North Alps Broad Band Network Northern Alps Lodging Cooperative Northern Alps Information for Gifu Prefecture

William Wynne (judge)

Sir William Wynne was an English judge and academic, Dean of the Arches 1788 to 1809, Master of Trinity Hall, Cambridge from 1803. The son of John Wynne and his wife Anne Pugh, he matriculated at Trinity Hall, Cambridge in 1747, graduating LL. B. in 1752, LL. D. in 1757. He became a Fellow of the college in 1755. Wynne was admitted as an advocate of the Court of Arches in 1757, where his practice was on marriage and probate matters, he contested unsuccessfully the 1764 election for the Master of his college, losing out to Sir James Marriott. In 1788 he became Dean of the Arches. In 1803 he was elected Master, made improvements in the College, he was elected a Fellow of the Royal Society in 1794