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The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If the data set has an odd number of observations, the middle one is selected (after arranging in ascending order). For example, the following list of seven numbers, 1, 3, 3, 6, 7, 8, 9
When is an odd number, the median of the collection is obtained by setting = (+) /. When n {\displaystyle n} is even, there are two choices for the median, obtained by rounding this choice of k {\displaystyle k} down or up, respectively: the lower median with k = n / 2 {\displaystyle k=n/2} and the upper median with k = n / 2 + 1 {\displaystyle ...
Firstly, computing median of an odd list is faster and simpler; while one could use an even list, this requires taking the average of the two middle elements, which is slower than simply selecting the single exact middle element. Secondly, five is the smallest odd number such that median of medians works.
The sample median may or may not be an order statistic, since there is a single middle value only when the number n of observations is odd. More precisely, if n = 2 m +1 for some integer m , then the sample median is X ( m + 1 ) {\displaystyle X_{(m+1)}} and so is an order statistic.
The five-number summary gives information about the location (from the median), spread (from the quartiles) and range (from the sample minimum and maximum) of the observations. Since it reports order statistics (rather than, say, the mean) the five-number summary is appropriate for ordinal measurements , as well as interval and ratio measurements.
If there are an odd number of data points in the original ordered data set, include the median (the central value in the ordered list) in both halves. If there are an even number of data points in the original ordered data set, split this data set exactly in half. The lower quartile value is the median of the lower half of the data.
For example, imagine throwing n balls to a basket U X and taking the balls that hit and throwing them to another basket U Y. If p is the probability to hit U X then X ~ B(n, p) is the number of balls that hit U X. If q is the probability to hit U Y then the number of balls that hit U Y is Y ~ B(X, q) and therefore Y ~ B(n, pq).
The few systems that calculate the majority function on an even number of inputs are often biased towards "0" – they produce "0" when exactly half the inputs are 0 – for example, a 4-input majority gate has a 0 output only when two or more 0's appear at its inputs. [1] In a few systems, the tie can be broken randomly. [2]
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