MEDIAN

Median is a measure of central tendency which gives the value of the middle-most observation in the data. The median of a distribution divides the distribution into two equal parts.

Median of ungrouped data

For finding the median of ungrouped data follow the below steps

• Arrange the data values or the observations in ascending order. Count the number of observations. Let there be n observations.
• If n is odd, the median is the `((n+1)/2)^(th)`observation.
• if n is even, then the median will be the average of the `(n/2)^(th)`and `(n/2+1)^(th)`observations.

Example:Find the median of the following values

37, 30, 42, 43, 46, 25, 38, 44, 32

Solution:

Arranging the data in ascending order, we have

25, 30, 32, 37, 38, 42, 43, 44, 46

Here the number of observations n = 9 (odd)

`\therefore` Median = Value of `((9+1)/2)^(th)`observation

= Value of 5th observation= 38.

Example:The median of the observation 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 arranged in ascending order is 24.Find the value of x.

Solution:

Number of observations n = 10(even).

So, Median=`{(n/2)^(th)observation+(n/2+1)^(th)observation}/2`

24=`{5^(th)observation+6^(th)observation}/2`

`\Rightarrow`24=`{x+2+x+4}/2`

`\Rightarrow`24=`{2x+6}/2`

`\Rightarrow`48=`2x+6`

`\Rightarrow`48-6=`2x` `\Rightarrow`x=`42/2=21`

`\therefore` x= 21

Example: Find the median of the following data, which is about the marks, out of 50 obtained by 100 students in a test :

Solution:

Here n is even so the median will be the average of `(n/2)^(th)` and `(n/2+1)^(th)` observations. that is `50^(th)` and `51^(th)` observations.

`50^(th)` observation is 28

from 51 to 78 observations is 29

so `51^(th)` observation is 29.

`\therefore`Median=`(28+29)/2 =28.5`

Median of Grouped data

Formula:

We use the following formula for calculating the median.

where l = lower boundary of median clas,

n = number of observations,

cf = cumulative frequency of class preceding the median class,

f = frequency of median class,

h = class size (size of the median class).

Example: Find the median of below given grouped frequency distribution of marks obtained, out of 100 by 53 students in a certain examination.

Solution:

In the distribution above,

n = 53. So `n/2=26.5.`

Now 60-70 is the class whose cumulative frequency 29 is greater than (and nearest to) 26.5.

`\therefore` 60-70 is the median class.

Median = `l+[(n/2-cf)/f]times h`

l = lower boundary of median clas=60

n = number of observations=53

cf = cumulative frequency of class preceding the median class=22

f = frequency of median class=7

h = class size (size of the median class)=10

Median `= 60+[(26.5-22)/7]times10`

`= 60+45/7=60+6.4=66.4`

So, the 66.4 is median of the given data.

Example:

A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and data was obtained as shown in table. Find their median.

Number of observations, n = 51

`n/2=51/2= 21.5^(th)`observation,

which lies in the class 145 - 150.

∴ 145 - 150 is median class

Then, l (the lower boundary) = 145,

Median = `l+[(n/2-cf)/f]times h`

cf (the cumulative frequency of the class preceding 145 - 150) = 11,

f (the frequency of the median class 145 - 150) = 18,

h (the class size) = 5.

Median `= 145+[(25.5-11)/18]times 5`

`= 145+72.5/18=145+4.03=149.03`

So, the 149.03 is median of the given data.

Example:

The median of the following data is 525. Find the values of x and y, if the total frequency is 100.

Solution:

It is given that n = 100

So, 76 + x + y = 100, i.e., x + y = 24 `--\rightarrow`(1)

The median is 525, which lies in the class 500 – 600

So, l = 500, f = 20, cf = 36 + x, h = 100

Using the formula

Median = `l+[(n/2-cf)/f]times h`

525`= 500+[(50-36-x)/20]times 100`

`Rightarrow` 525 – 500 = (14 – x) × 5

`Rightarrow` 25 = 70 – 5x

`Rightarrow`5x = 70 – 25 = 45

So, x = 9

from (1), we get 9 + y = 24

∴ y= 24-9=15

Uses of Median :

• Median is the only average to be used while dealing with qualitative data which cannot be measured quantitatively but can be arranged in ascending or descending order of magnitude.
• It is used for determining the typical value in problems concerning wages, distribution of wealth, etc.

Merits of Median :

• It is strictly defined, easy to understand, and calculated.
• All of these are not affected by extreme values.
• It can be located graphically, even if the class - intervals are unequal.
• In some cases, this may also be determined by inspection.

Demerits of Median :

• In the case of an even number of observations the mediator cannot be determined with certainty
• It is not based on all observations.
• It is not subject to algebraic treatment.
• It is greatly affected by sample fluctuations.

When would you take median rather than mean?

In problems where individual observations are not important, especially extreem values, and we wish to find out a ‘typical’ observation, the median is more appropriate, for example finding the typical productivity rate of workers, average wage in a country, etc.These are situations where extreme values may exist. So, rather than the mean, we take the median as a better measure of central tendency.

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