Mean of grouped data

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Mean of discrete frequency distribution
If `x_1,x2,...,x_n` are observations with respective frequencies `f_1,f_2,...,f_n` then,
the sum of values of all observations `=f_1x_1+f_2x_2+...+f_n x_n`
and number of observations `=f_1+f_2+...+f_n`

Mean `bar x = (f_1x_1+f_2x_2+...+f_n x_n)/(f_1+f_2+...+f_n)`

Mean `bar x = (sum f_i x_i)/(sum f_i)`

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1. Calculate Mean from the following grouped data

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

Solution:

`x` `(1)`	Frequency `(f)` `(2)`	`f*x` `(3)=(2)xx(1)`
0	1	0
1	5	5
2	10	20
3	6	18
4	3	12
---	---	---
	`n=25`	`sum f*x=55`

Mean `bar x = (sum fx)/n`

`=55/25`

`=2.2`

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2. Calculate Mean from the following grouped data

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

Solution:

`x` `(1)`	Frequency `(f)` `(2)`	`f*x` `(3)=(2)xx(1)`
10	3	30
11	12	132
12	18	216
13	12	156
14	3	42
---	---	---
	`n=48`	`sum f*x=576`

Mean `bar x = (sum fx)/n`

`=576/48`

`=12`

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Mean of continuous frequency distribution

we use formula

Mean `bar x = (sum f_i x_i)/(sum f_i)`, where `x_i` is the mid-value of the class

Mid-value `=("Upper class limit + Lower class limit")/2`

For eg, Mid-value of class 10-20 is `=(10+20)/2=15`

This method is also called Direct Method

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3. Calculate Mean 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)`	Mid value `(x)` `(3)`	`f*x` `(4)=(2)xx(3)`
2-4	3	3	9
4-6	4	5	20
6-8	2	7	14
8-10	1	9	9
---	---	---	---
--	`n = 10`	--	`sum f*x=52`

Mean `bar x = (sum fx)/n`

`=52/10`

`=5.2`

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4. Calculate Mean 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)`	Mid value `(x)` `(3)`	`f*x` `(4)=(2)xx(3)`
0-2	5	1	5
2-4	16	3	48
4-6	13	5	65
6-8	7	7	49
8-10	5	9	45
10-12	4	11	44
---	---	---	---
--	`n = 50`	--	`sum f*x=256`

Mean `bar x = (sum fx)/n`

`=256/50`

`=5.12`

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Step deviation method (Assumed mean(A) Method)

Mean `bar x = A + (sum f_i d_i)/(sum f_i) * h`

5. Calculate Mean 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)`	Mid value `(x)` `(3)`	`d=(x-A)/h=(x-45)/10` `A=45,h=10` `(4)`	`f*d` `(5)=(2)xx(4)`
10 - 20	15	15	-3	-45
20 - 30	25	25	-2	-50
30 - 40	20	35	-1	-20
40 - 50	12	45=A	0	0
50 - 60	8	55	1	8
60 - 70	5	65	2	10
70 - 80	3	75	3	9
---	---	---	---	---
	`n = 88`	-----	-----	`sum f*d=-88`

Mean `bar x = A + (sum fd)/n * h`

`=45 + (-88)/88 * 10`

`=45 + (-1) * 10`

`=45 -10`

`=35`

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6. Calculate Mean 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)`	Mid value `(x)` `(3)`	`d=(x-A)/h=(x-37.5)/5` `A=37.5,h=5` `(4)`	`f*d` `(5)=(2)xx(4)`
20 - 25	110	22.5	-3	-330
25 - 30	170	27.5	-2	-340
30 - 35	80	32.5	-1	-80
35 - 40	45	37.5=A	0	0
40 - 45	40	42.5	1	40
45 - 50	35	47.5	2	70
---	---	---	---	---
	`n = 480`	-----	-----	`sum f*d=-640`

Mean `bar x = A + (sum fd)/n * h`

`=37.5 + (-640)/480 * 5`

`=37.5 + (-1.3333) * 5`

`=37.5 -6.6667`

`=30.8333`

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