5.4. Median http://www.ck12.org
From the discussion, the activity with the blocks, and the activity with the desks, you and your classmates should
have an understanding of the meaning of the median with respect to a set of data. Here is another example. The
test scores for 7 students were 25, 55, 58, 64, 66, 68 and 70. The mean mark is 48.6, which is lower than all but 1
of the student’s marks. The one very low mark of 25 has caused the mean to be skewed. A better measure of the
average performance of the 7 students would be the middle mark of 64. Themedianis the number in the middle
position once the data has been organized. Organized data is simply the numbers arranged from smallest to largest
or from largest to smallest. 64 is the only number for which there are as many values above it as below it in the set
of organized data, so it is the median. The median for an odd number of data values is the value that divides the data
into 2 halves. Ifnrepresents the number of data values andnis an odd number, then the median will be found in the
n+ 1
2 position.
The median is often preferable to the mean as a summary statistic, because the mean is affected by extreme values, or
outliers. For example, if you were employed by a company that paid all of its employees a salary between $60,000
and $70,000, you could probably estimate the mean salary to be about $65,000. However, if you had to add in the
$150,000 salary of the CEO when calculating the mean, then the value of the mean would increase greatly. It would,
in fact, be the mean of the employees’ salaries, but it probably would not be a good measure of the central tendency
of the salaries. It would be much better to use the median as a summary statistic in this case.
Example A
Find the median of the following data:
a) 12, 2, 16, 8, 14, 10, 6
b) 7, 9, 3, 4, 11, 1, 8, 6, 1, 4
a) The first step is to organize the data, or arrange the numbers from smallest to largest.
12 , 2 , 16 , 8 , 14 , 10 , 6 → 2 , 6 , 8 , 10 , 12 , 14 , 16
The number of data values is 7, which is an odd number. Therefore, the median will be found in then+ 21 position.
n+ 1
2=
7 + 1
2
=
8
2
= 4
In this case, the median is the value that is found in the 4thposition of the organized data.
2 , 6 , 8 , 10 , 12 , 14 , 16
This means that the median is 10.
b) The first step is to organize the data, or arrange the numbers from smallest to largest.
7 , 9 , 3 , 4 , 11 , 1 , 8 , 6 , 1 , 4 → 1 , 1 , 3 , 4 , 4 , 6 , 7 , 8 , 9 , 11
The number of data values is 10, which is an even number. Therefore, the median will be the mean of the number
found before then+ 21 position and the number found after then+ 21 position.
n+ 1
2