Slope and Y-intercept of The Regression Line | 회귀선의 기울기와 y절편

In a simple linear regression line (LMS), the regression line can be expressed as following equation:

y = ax + b

where

• y = The variable that you want to predict (예측하고 싶은 값) | Dependent variable (종속 변수)
• x = The variable that you are using to predict (예측에 사용하는 값) | Independent variable (독립 변수)
• a = Slope (기울기)
• b = y-intercept (y 절편)

 

그렇다면, y = ax+b 에서 slope(a)와 y-intercept(b)는 어떻게 구하는지 알아보겠습니다. 

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Recall, 

r (correlation coefficient) = Average of product of (x in standard units) and (y in standard units)

or in python <datascience>:

standard_units converts an array of numbers into standard units

 

Then, the slope of the best fit line is: 

Slope (a) = r * (standard deviation of y) / (standard deviation of x)

 

That is,

a = r * SD_y / SD_x          (*)

where

• r = correlation coefficient
• SD_y = standard deviation of y
• SD_x = standard deviation of x

 

And, the y-intercept of the regression line is:

y-intercept (b) = (average of y) - (slope)(average of x)

 

That is,

b = y_average - a * x_average

where

• y_average = average of y 
• x_average = average of x
• a = slope = r * SD_y / SD_x          (*)

 

In summary, the equations for slope and y-intercept are in the following image: