2. Mean Example
Mean of grouped data
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)`
1. Calculate Mean from the following grouped data
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`
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`
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
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`
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`
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`
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`
This material is intended as a summary. Use your textbook for detail explanation.
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