Formula
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1. Mean deviation of Mean `delta bar x = (sum f*|x - bar x|)/n`
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2. Mean deviation of Mean `delta bar x = (sum f*|x - M|)/n`
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3. Mean deviation of Mode `delta bar x = (sum f*|x - Z|)/n`
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Examples
1. Calculate Mean deviation about median from the following grouped data
| Class-X | Frequency |
| 2 - 4 | 3 |
| 4 - 6 | 4 |
| 6 - 8 | 2 |
| 8 - 10 | 1 |
Solution:
Class
`(1)` | `f`
`(2)` | `cf`
`(3)` | Mid value (`x`)
`(4)` | `|x-M|=|x-5|`
`(5)` | `f*|x-M|`
`(6)=(2)xx(5)` |
| 2 - 4 | 3 | 3 `3=0+3`
`(3)=`Previous `(3)+(2)` | 3 `3=(2+4)/2` | 2 `|3-5|=2`
`|x - 5|` | 6 `6=3xx2`
`(6)=(2)xx(5)` |
| 4 - 6 | 4 | 7 `7=3+4`
`(3)=`Previous `(3)+(2)` | 5 `5=(4+6)/2` | 0 `|5-5|=0`
`|x - 5|` | 0 `0=4xx0`
`(6)=(2)xx(5)` |
| 6 - 8 | 2 | 9 `9=7+2`
`(3)=`Previous `(3)+(2)` | 7 `7=(6+8)/2` | 2 `|7-5|=2`
`|x - 5|` | 4 `4=2xx2`
`(6)=(2)xx(5)` |
| 8 - 10 | 1 | 10 `10=9+1`
`(3)=`Previous `(3)+(2)` | 9 `9=(8+10)/2` | 4 `|9-5|=4`
`|x - 5|` | 4 `4=1xx4`
`(6)=(2)xx(5)` |
| --- | --- | --- | --- | --- | --- |
| -- | `n=10` | -- | -- | -- | `sum f*|x-M|=14` |
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`
Mean deviation of Median
`delta bar x = (sum f*|x - M|)/n`
`delta bar x = 14/10`
`delta bar x = 1.4`
Coefficient of Mean deviation `=(delta bar x)/(bar x)`
`=1.4/5`
`=0.28`
2. Calculate Mean deviation about median from the following grouped data
| Class-X | 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)` | `f`
`(2)` | `cf`
`(3)` | Mid value (`x`)
`(4)` | `|x-M|=|x-32|`
`(5)` | `f*|x-M|`
`(6)=(2)xx(5)` |
| 10 - 20 | 15 | 15 | 15 | 17 | 255 |
| 20 - 30 | 25 | 40 | 25 | 7 | 175 |
| 30 - 40 | 20 | 60 | 35 | 3 | 60 |
| 40 - 50 | 12 | 72 | 45 | 13 | 156 |
| 50 - 60 | 8 | 80 | 55 | 23 | 184 |
| 60 - 70 | 5 | 85 | 65 | 33 | 165 |
| 70 - 80 | 3 | 88 | 75 | 43 | 129 |
| --- | --- | --- | --- | --- | --- |
| -- | `n=88` | -- | -- | -- | `sum f*|x-M|=1124` |
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`
Mean deviation of Median
`delta bar x = (sum f*|x - M|)/n`
`delta bar x = 1124/88`
`delta bar x = 12.7727`
Coefficient of Mean deviation `=(delta bar x)/(bar x)`
`=12.7727/32`
`=0.3991`
This material is intended as a summary. Use your textbook for detail explanation.
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