Lesson WHAT IS Median

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<H2>Median</H2>

<H3>Definition of Median</H3>

<BLOCKQUOTE>If the group of numbers is given, &nbsp;then Median is the middle number of the group when they are ranked in order.
If there are an even number in the group, &nbsp;the mean of the middle two is taken.</BLOCKQUOTE>

<H3>How to find Median</H3>

To find the median of the group of numbers:

<OL>
  <LI>order the list according to its elements value. </LI>
  <LI>Then repeatedly remove the pair consisting of the smallest and biggest numbers until either one or two values are left. </LI>
  <LI>If exactly one number is left, &nbsp;this is the Median.</LI>
  <LI>If two values left, &nbsp;Median is the arithmetic mean of these two.</LI>
</OL>

<B>Example 1.</B>
Calculate Median of the group of numbers &nbsp;{1, &nbsp;7, &nbsp;3, &nbsp;13}.

First, &nbsp;we should order the group: &nbsp;{1, &nbsp;3, &nbsp;7, &nbsp;13}.
Then we have to remove the pair of smallest and highest numbers to obtain the list &nbsp;{3, &nbsp;7}. 
Since there are two elements in this remaining list, &nbsp;Median is equal to Arithmetic mean, &nbsp;(3 + 7)/2 = 5.

<B>Example 2.</B>
Calculate Median of the group of numbers &nbsp;{3, &nbsp;3, &nbsp;4}.

The group is already ordered.
Then the first &nbsp;3&nbsp; and the last &nbsp;4&nbsp; are removed, &nbsp;and only number &nbsp;3&nbsp; is remained.
So, &nbsp;Median is &nbsp;3.

If the group of numbers is arithmetic progression, &nbsp;Median of this group of numbers is the same as Arithmetic mean of the group.

<B> Example 3.</B>
Consider the group of numbers &nbsp;{2, &nbsp;5, &nbsp;8, &nbsp;11, &nbsp;14, &nbsp;17}.
This is arithmetic progression with the difference equal to 3.
Median for this group is 9.5, exactly as its Arithmetic mean.

In general case Median of the group of numbers is different from its Arithmetic mean.

<B> Example 4.</B>
Consider the group of numbers &nbsp;{2, &nbsp;5, &nbsp;8, &nbsp;11, &nbsp;14, &nbsp;47}.
This group is different from that of &nbsp;<B>Example 3</B> &nbsp;only in the last term.
Median of this group is still &nbsp;9.5, &nbsp;while Arithmetic mean is different: &nbsp;it is equal to &nbsp;14.5.

So, &nbsp;do not mess Median of the group with its Arithmetic mean.

Calculating of Median is used often in homes trading to quickly separate the whole current market into cheaper and expensive halves and to attribute 
the concrete given sample to one of these sub-sets.

<H3>Median vs. Average: what is better to use and when?</H3>

Arithmetic mean is a good estimation when data is evenly distributed and the values are relatively close. For example, &nbsp;mean &nbsp;(average) &nbsp;height of students in a class 
is a useful number that tells us how tall are the students.

The arithmetic mean may be not an adequate estimation if there are outliers in the data-set that produce significant influence on the value of average.

For example, &nbsp;suppose that Acme company has &nbsp;20&nbsp; employees who earn &nbsp;$30,000 per year and one manager who earns a million dollars per year. 
The average &nbsp;(mean) &nbsp;salary of this group is

{{{ Average_salary = (20*30000+1000000) / 21 = 76190}}}

So, &nbsp;even all employees, &nbsp;except one, &nbsp;earn &nbsp;$30,000 per year, &nbsp;the average salary is more than twice that amount!

Using a median instead of mean, &nbsp;gives us a more realistic picture of &nbsp;"what a typical employee makes".

My other lessons on &nbsp;<B>Average, &nbsp;Mean and Median</B>&nbsp; in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Geometric-Mean.lesson>WHAT IS Geometric mean</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Difference-between-Amean-and-Gmean.lesson>Difference between Arithmetic mean and Geometric mean</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Solved-problems-on-average-scores-weight-height-and-temperature.lesson>Solved problems on average scores, weight, height and temperature</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Solved-problems-on-average-scores.lesson>Solved problems on average scores</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Solved-problems-on-average-age.lesson>Solved problems on average age</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Miscellaneous-problems-on-average-values.lesson>Miscellaneous problems on average values</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Math-circle-level-problem-on-average.lesson>Math circle level problem on average</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/Entertainment-problems-on-average.lesson>Entertainment problems on average</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/The-mean-and-the-total-value-problem-for-the-International-Fools-day-of-April-1.lesson>The mean and the total value problem for the International Fools' day of April, 1</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Average/OVERVIEW-of-lessons-on-Average-Mean-and-Median.lesson>OVERVIEW of lessons on Average</A>