Everything about Median totally explained
In
probability theory and
statistics, a
median is described as the number 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's an even number of observations, the median isn't unique, so one often takes the
mean of the two middle values.
At most half the population have 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 . The median is 2 in this case, as is the
mode, and it might be seen as a better indication of
central tendency than the
arithmetic mean of 3.166….
Calculation of medians is a popular technique in
summary statistics and
summarizing statistical data, since it's simple to understand and easy to calculate, while also giving a measure that's more robust in the presence of
outlier values than is the
mean.
Theoretical properties
An optimality property
The median is also the central point which minimizes the average of the absolute deviations; in the example above this would be (1 + 0 + 0 + 0 + 1 + 7) / 6 = 1.5 using the median, while it would be 1.944 using the mean. In the language of probability theory, the value of
c that minimizes
»
is the median of the probability distribution of the
random variable X. Note, however, that c isn't always unique, and therefore not well defined in general.
An inequality relating means and medians
For continuous probability distributions, the difference between the median and the mean is less than or equal to one
standard deviation. See
an inequality on location and scale parameters.
Efficient computation
Even though
sorting n items takes in general
O(
n log
n) operations, by using a
"divide and conquer" algorithm the median of
n items can be computed with only
O(
n) operations (in fact, you can always find the
k-th element of a list of values with this method; this is called the
selection problem).
Easy explanation (Statistics)
As an example, we'll calculate the median of the following population of numbers: 1, 5, 2, 8, 7.
Start by sorting the numbers: 1, 2, 5, 7, 8.
In this case, 5 is the median, because when the numbers are sorted, it's the middle number. If there's an even amount of numbers, the median is the
arithmetic mean of the two middle numbers.
Further Information
Get more info on 'Median'.
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