Understanding Median - Measures of Central Tendency

Median is an important measure of central tendency. It is the most commonly used measure of central tendency after A.M. When there are some extremely large or small observations in comparison to the other observations in the data set, median, being a positional average, serves as a better measure of central tendency than A.M. When a frequency distribution has open-end classes median can be used as a measure of central tendency.

Median of a data set is defined to be the middlemost value when the values of the variable are arranged in order of magnitude, i.e., either in increasing or in decreasing order. The median divides the entire set of observations into two equal halves, i.e., the number values less than the median is equal to the number of values greater than the median.

Median of a data set is usually denoted by $\tilde{x}$

Median for non-frequency type data:

If the number of observations, say ‘ $n$ ’ is odd, then there will be a unique, middlemost value when the observations are arranged in order of magnitude. So, if ‘ $n$ ’ is odd we have a unique median.

$\therefore$ if ‘ $n$ ’ is odd we have,

Median $(\tilde{x})= (\dfrac{n+1}{2})th$ ordered value

Example: Find the median of 9, 7, 8, 4, 11, 3 and 10.

Solution:

First of all we arrange the given terms is ascending or descending order of their magnitude.

On arranging the given terms in ascending order, we get: 3, 4, 7, 8, 9, 10 and 11

Here, the number of observations, $n=7$ , which is odd.

$\therefore$ the median $(\tilde{x})= (\dfrac{7+1}{2})th= (\dfrac{8}{2})th$ , i.e., the 4^th ordered value.

Here, we see that the 4^th ordered value is 8 which is the middlemost value..

$\therefore$ the median $(\tilde{x})= 8$

Clearly, the number of values less than 8=the number of values more than 8=3.

However, if ‘ $n$ ’ is even, then there are two middlemost values in the ordered arrangement, viz. the $\dfrac{n}{2}th$ ordered value and the $(\dfrac{n}{2} +1)th$ ordered value. Any value in between these two middlemost values may be taken as the median. For the sake of definiteness, we take the A.M. of these two values as the median.

$\therefore$ if ‘ $n$ ’ is even we have,

Median $( \tilde{x} )=$ A.M. of the $\dfrac{n}{2}th$ and the $(\dfrac{n}{2} +1)th$ ordered value.

Example: Find the median of 22, 17, 11, 15, 28, 20, 29, 26

Solution:

First of all we arrange the given terms is ascending or descending order of their magnitude.

On arranging the given terms in ascending order, we get: 11, 15, 17, 20, 22, 26, 28, and 29

Here, the number of observations, $n=8$ , which is even.

$\therefore$ the median $(\tilde{x})= A.M. of (\dfrac{8}{2})th$ and $(\dfrac{8}{2} + 1)th$ ordered values

$=$ A.M. of the 4^th and 5^th ordered values

Here, we see that the 4^th ordered value is 20 and the 5^thordered value is 22.

So, 20 and 22 are the two middlemost values.

$\therefore$ the median $(\tilde{x})= \dfrac{20+22}{2}=\dfrac{42}{2}=21$

Clearly, the number of values less than 21=the number of values more than 21=4.

Note:

It is to be noted that, in the first example, the value of the median was 8 which in itself was present in the given data set. However, in the second example, the value of the median was 21, which itself, was not present in the given data set. We used the two middlemost values of the given data set, viz., 20 and 22 to calculate a value which we took to be the representative value for the median of the data set.
Whether the given terms are arranged in ascending or descending order, the value of median always remains the same.

Merits and Demerits of Median:

Merits:

Median is easy to comprehend and easy to calculate.
Median is rigidly defined.
It is unaffected by the presence of extreme values.
It is not affected by the presence of extreme values.
Median can be calculated for a frequency distribution with open ended classes or with unequal class widths.
Median can be used as a measure of central tendency even in the case of some qualitative data where the individual can be ranked.

Demerits:

It is not directly dependent upon all given values.
It is not amenable to further mathematical or algebraic treatment.

Exercise:

1. A student got the following marks in 9 questions of a question paper:

3, 5, 7, 3, 8, 0, 1, 4, 6

Find the median of these marks.

2. The following are the scores of a batsman in the first 10 innings of his career:

55, 20, 75, 1, 4, 101, 37, 48, 76, 0

Find the median score of the batsman in these 10 innings.

3. Suppose the blood pressure levels (in mm. Hg.) for 7 randomly selected individuals are as follows:

118.4, 132.5, 124.9, 109.8, 134.6, 113.7, 128.5

Find the median blood pressure level of these seven individuals.

4. The height (in inches) of 8 boys of a class are given below:

5.3, 5.9, 5, 6, 6.2, 5.6, 6.1, 5.6

5. Find the median of:

The first twelve natural numbers
The first nine even numbers
The first nine odd numbers
The first five multiples of 4
The first four multiples of 5

Median for non-frequency type data:

Merits and Demerits of Median:

Exercise:

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