Median of grouped data

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Median of discrete frequency distribution
If `n` is odd, then
`M=` value of `((n+1)/2)^(th)` observation

If `n` is even, then
`M=(text{Value of } (n/2)^(th) text{ observation} + text{Value of } (n/2 + 1)^(th) text{ observation})/2`

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1. Calculate Median from the following grouped data

X	Frequency
0	1
1	5
2	10
3	6
4	3

Solution:

`x` `(1)`	Frequency `(f)` `(2)`	`cf` `(5)`
0	1	1
1	5	6
2	10	16
3	6	22
4	3	25
---	---	---
	`n=25`	--

Median :
M = value of `((n+1)/2)^(th)` observation

= value of `(26/2)^(th)` observation

= value of `13^(th)` observation

From the column of cumulative frequency `cf`, we find that the `13^(th)` observation is `2`.

Hence, the median of the data is `2`.

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2. Calculate Median from the following grouped data

X	Frequency
10	3
11	12
12	18
13	12
14	3

Solution:

`x` `(1)`	Frequency `(f)` `(2)`	`cf` `(5)`
10	3	3
11	12	15
12	18	33
13	12	45
14	3	48
---	---	---
	`n=48`	--

Median :
M = value of `((n+1)/2)^(th)` observation

= value of `(49/2)^(th)` observation

= value of `24.5^(th)` observation

From the column of cumulative frequency `cf`, we find that the `24.5^(th)` observation is `12`.

Hence, the median of the data is `12`.

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Median of continuous frequency distribution
To find Median class, we find cumulative frequencies of all classes and then find `n/2`.
The class whose cumulative frequency is `>= n/2` is called Median class

Median `M = L + (n/2 - cf)/f * c`
where
`:. L = `lower boundary point of median class

`:. n = `Total frequency

`:. cf = `Cumulative frequency of the class preceding the median class

`:. f = `Frequency of the median class

`:. c = `class length of median class

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3. Calculate Median from the following grouped data

Class	Frequency
2 - 4	3
4 - 6	4
6 - 8	2
8 - 10	1

Solution:

Class `(1)`	Frequency `(f)` `(2)`	`cf` `(6)`
2-4	3	3
4-6	4	7
6-8	2	9
8-10	1	10
---	---	---
--	`n = 10`	--

To find Median Class
= value of `(n/2)^(th)` observation

= value of `(10/2)^(th)` observation

= value of `5^(th)` observation

From the column of cumulative frequency `cf`, we find that the `5^(th)` observation lies in the class `4 - 6`.

`:.` The median class is `4 - 6`.

Now,
`:. L = `lower boundary point of median class `=4`

`:. n = `Total frequency `=10`

`:. cf = `Cumulative frequency of the class preceding the median class `=3`

`:. f = `Frequency of the median class `=4`

`:. c = `class length of median class `=2`

Median `M = L + (n/2 - cf)/f * c`

`=4 + (5 - 3)/4 * 2`

`=4 + (2)/4 * 2`

`=4 + 1`

`=5`

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4. Calculate Median from the following grouped data

Class	Frequency
0 - 2	5
2 - 4	16
4 - 6	13
6 - 8	7
8 - 10	5
10 - 12	4

Solution:

Class `(1)`	Frequency `(f)` `(2)`	`cf` `(6)`
0-2	5	5
2-4	16	21
4-6	13	34
6-8	7	41
8-10	5	46
10-12	4	50
---	---	---
--	`n = 50`	--

To find Median Class
= value of `(n/2)^(th)` observation

= value of `(50/2)^(th)` observation

= value of `25^(th)` observation

From the column of cumulative frequency `cf`, we find that the `25^(th)` observation lies in the class `4 - 6`.

`:.` The median class is `4 - 6`.

Now,
`:. L = `lower boundary point of median class `=4`

`:. n = `Total frequency `=50`

`:. cf = `Cumulative frequency of the class preceding the median class `=21`

`:. f = `Frequency of the median class `=13`

`:. c = `class length of median class `=2`

Median `M = L + (n/2 - cf)/f * c`

`=4 + (25 - 21)/13 * 2`

`=4 + (4)/13 * 2`

`=4 + 0.6154`

`=4.6154`

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5. Calculate Median from the following grouped data

Class	Frequency
10 - 20	15
20 - 30	25
30 - 40	20
40 - 50	12
50 - 60	8
60 - 70	5
70 - 80	3

Solution:

Class `(1)`	Frequency `(f)` `(2)`	`cf` `(7)`
10 - 20	15	15
20 - 30	25	40
30 - 40	20	60
40 - 50	12	72
50 - 60	8	80
60 - 70	5	85
70 - 80	3	88
---	---	---
	`n = 88`	-----

To find Median Class
= value of `(n/2)^(th)` observation

= value of `(88/2)^(th)` observation

= value of `44^(th)` observation

From the column of cumulative frequency `cf`, we find that the `44^(th)` observation lies in the class `30 - 40`.

`:.` The median class is `30 - 40`.

Now,
`:. L = `lower boundary point of median class `=30`

`:. n = `Total frequency `=88`

`:. cf = `Cumulative frequency of the class preceding the median class `=40`

`:. f = `Frequency of the median class `=20`

`:. c = `class length of median class `=10`

Median `M = L + (n/2 - cf)/f * c`

`=30 + (44 - 40)/20 * 10`

`=30 + (4)/20 * 10`

`=30 + 2`

`=32`

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6. Calculate Median from the following grouped data

Class	Frequency
20 - 25	110
25 - 30	170
30 - 35	80
35 - 40	45
40 - 45	40
45 - 50	35

Solution:

Class `(1)`	Frequency `(f)` `(2)`	`cf` `(7)`
20 - 25	110	110
25 - 30	170	280
30 - 35	80	360
35 - 40	45	405
40 - 45	40	445
45 - 50	35	480
---	---	---
	`n = 480`	-----

To find Median Class
= value of `(n/2)^(th)` observation

= value of `(480/2)^(th)` observation

= value of `240^(th)` observation

From the column of cumulative frequency `cf`, we find that the `240^(th)` observation lies in the class `25 - 30`.

`:.` The median class is `25 - 30`.

Now,
`:. L = `lower boundary point of median class `=25`

`:. n = `Total frequency `=480`

`:. cf = `Cumulative frequency of the class preceding the median class `=110`

`:. f = `Frequency of the median class `=170`

`:. c = `class length of median class `=5`

Median `M = L + (n/2 - cf)/f * c`

`=25 + (240 - 110)/170 * 5`

`=25 + (130)/170 * 5`

`=25 + 3.8235`

`=28.8235`

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