1. Formula & Example-1
Formula
1. Regression coefficient
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Regression coefficient `y` on `x`
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Regression coefficient `x` on `y`
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1. |
`byx = (sum (x-bar x)(y-bar y))/(sum (x-bar x)^2)`
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`bxy = (sum (x-bar x)(y-bar y))/(sum (y-bar y)^2)`
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2. |
`byx = (n sum xy - (sum x)(sum y))/(n sum x^2 - (sum x)^2)`
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`bxy = (n sum xy - (sum x)(sum y))/(n sum y^2 - (sum y)^2)`
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3. |
`byx = (n sum dx dy - (sum dx)(sum dy))/(n sum dx^2 - (sum dx)^2)`
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`bxy = (n sum dx dy - (sum dx)(sum dy))/(n sum dy^2 - (sum dy)^2)`
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4. |
`byx = r * (sigma y)/(sigma x)`
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`bxy = r * (sigma x)/(sigma y)`
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2. Regression Line y on x
`y - bar y = byx (x - bar x)`
3. Regression Line x on y
`x - bar x = bxy (y - bar y)`
Example-1
1. Find Regression line equations from the following data
| Class-X | Y | | 2 - 4 | 3 | | 4 - 6 | 4 | | 6 - 8 | 2 | | 8 - 10 | 1 |
Solution:
| Class-X | Mid value `x` | `y` | `x^2` | `y^2` | `x*y` | | 2 - 4 | 3 | 3 | 9 | 9 | 9 | | 4 - 6 | 5 | 4 | 25 | 16 | 20 | | 6 - 8 | 7 | 2 | 49 | 4 | 14 | | 8 - 10 | 9 | 1 | 81 | 1 | 9 | | --- | --- | --- | --- | --- | --- | | | `sum x=24` | `sum y=10` | `sum x^2=164` | `sum y^2=30` | `sum xy=52` |
Mean `bar x = (sum x)/n`
`=24/4`
`=6`
Mean `bar y = (sum y)/n`
`=10/4`
`=2.5`
`byx = (n sum xy - (sum x)(sum y))/(n sum x^2 - (sum x)^2)`
`=(4 * 52 - 24 * 10 )/(4 * 164 - (24)^2)`
`=(208 - 240 )/(656 - 576)`
`=-32/80`
`=-0.4`
Regression Line y on x
`y - bar y = byx (x - bar x)`
`y - 2.5 = -0.4 (x - 6)`
`y - 2.5 = -0.4 x + 2.4`
`y = -0.4 x + 2.4 + 2.5`
`y = -0.4 x + 4.9`
`bxy = (n sum xy - (sum x)(sum y))/(n sum y^2 - (sum y)^2)`
`=(4 * 52 - 24 * 10 )/(4 * 30 - (10)^2)`
`=(208 - 240 )/(120 - 100)`
`=-32/20`
`=-1.6`
Regression Line x on y
`x - bar x = bxy (y - bar y)`
`x - 6 = -1.6 (y - 2.5)`
`x - 6 = -1.6 y + 4`
`x = -1.6 y + 4 + 6`
`x = -1.6 y + 10`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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