correlation and regression

S2

symbol for variance

S

symbol for standard deviation

Simplest general pattern

linear

Direction

positive (uphill), negative (downhill)

Strength

how closely the data follows the pattern (the line)

r

symbol for correlation coefficient, between -1 and 1

Correlation

describes the strength and direction of the linear relationship between x and y

Perfect positive linear relationship

+1 = r

Perfect negative linear relationship

-1 = r

Weak correlation

anything between -0.3 and 0.3

Strong correlation

anything between -1 and -0.7 and 0.7 and 1

Moderately weak

anything between -0.3 and -0.5 and 0.3 and 0.5

Moderately strong

anything between -0.5 and -0.7 and 0.5 and 0.7

Properties of correlation

2 quantitative variables only, linear relationship only, r has no units, switching x and y doesn’t affect r, and r is affected by outliers and skewness

b0

symbol for y-intercept

b1

symbol for slope

SSE

sum of squared errors

Ybar

mean of all y values

Coefficient of determination

% of variability in y that is explained by x. Notation: R2 = r2

Y hat

predicted y in terms of x

Notes about interpreting y-intercept

X = 0 needs to make sense, need data near where X = 0

Extrapolation

no data out there, think about temperatures

How to calculate residuals

observed y – predicted x

If line fits well

residual should have no pattern (should have random scatter about the regression line), no systematic change as X increases, no unusually large values of a residual (outlier in y direction), no influential points (outlier in x direction)