Compute the least squares line from summary statistics

Least-squares line from summary statistics

With x as the explanatory variable and y as the response variable, the slope uses the correlation and both spreads. Find b before a, and keep unrounded values until the final reporting step

1. Assign the roles Use x for the explanatory variable and y for the response variable \(\displaystyle x=\text{EV},\quad y=\text{RV}\)
2. Calculate the slope Multiply r by the response standard deviation divided by the explanatory standard deviation \(\displaystyle b=(-0.8)\frac{15}{10}=-1.2\)
3. Calculate the intercept Subtract b times the explanatory mean from the response mean \(\displaystyle a=42-(-1.2)(10)=54\)
4. Round at the end Use the exact slope when finding the intercept, then round the reported coefficients as requested \(\displaystyle \widehat{y}=54-1.2x\)

Worked example

For a data set, x is the explanatory variable and y is the response variable. Given r = −0.6, x̄ = 20, sₓ = 5, ȳ = 50 and sᵧ = 10, find the least-squares line

1. \(\displaystyle b=r\frac{s_y}{s_x}=(-0.6)\frac{10}{5}=-1.2\)Use the response spread in the numerator and the explanatory spread in the denominator
2. \(\displaystyle a=\bar{y}-b\bar{x}=50-(-1.2)(20)=50+24=74\)Use the unrounded slope in the intercept formula
3. \(\displaystyle \widehat{y}=74-1.2x\)Substitute the intercept and slope into the least-squares form y-hat = a + bx

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