3. Calculate Five number summary from the following grouped data

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

Solution:
Five number summary :

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

Minimum value `=2`

Maximum value `=10`

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First quartile `Q_1` :

Here, `n = 10`


`Q_1` class :

Class with `(n/4)^(th)` value of the observation in `cf` column

`=(10/4)^(th)` value of the observation in `cf` column

`=(2.5)^(th)` value of the observation in `cf` column

and it lies in the class `2 - 4`.

`:. Q_1` class : `2 - 4`

The lower boundary point of `2 - 4` is `2`.

`:. L = 2`

`Q_1 = L + (( n)/4 - cf)/f * c`

`=2 + (2.5 - 0)/3 * 2`

`=2 + (2.5)/3 * 2`

`=2 + 1.6667`

`=3.6667`

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Median `Q_2` :


`Q_2` class :

Class with `((2n)/4)^(th)` value of the observation in `cf` column

`=((2*10)/4)^(th)` value of the observation in `cf` column

`=(5)^(th)` value of the observation in `cf` column

and it lies in the class `4 - 6`.

`:. Q_2` class : `4 - 6`

The lower boundary point of `4 - 6` is `4`.

`:. L = 4`

`Q_2 = L + ((2 n)/4 - cf)/f * c`

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

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

`=4 + 1`

`=5`

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Third quartile `Q_3` :


`Q_3` class :

Class with `((3n)/4)^(th)` value of the observation in `cf` column

`=((3*10)/4)^(th)` value of the observation in `cf` column

`=(7.5)^(th)` value of the observation in `cf` column

and it lies in the class `6 - 8`.

`:. Q_3` class : `6 - 8`

The lower boundary point of `6 - 8` is `6`.

`:. L = 6`

`Q_3 = L + ((3 n)/4 - cf)/f * c`

`=6 + (7.5 - 7)/2 * 2`

`=6 + (0.5)/2 * 2`

`=6 + 0.5`

`=6.5`

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Thus Five number summary is
1. Minimum value `=2`

2. First quartile `Q_1=3.6667`

3. Median `Q_2=5`

4. Third quartile `Q_3=6.5`

5. Maximum value `=10`

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