Big Query

Simple Linear Regression

Dataset

Number of Students	Sales by Quarter
x	y
2	58
6	105
8	88
8	118
12	117
16	137
20	157
20	169
22	149
26	202

Slop

$$b_1=\frac{\sum{(x_i-\bar{x})(y_i-\bar{y})}}{\sum{(x_i-\bar{x})^2}}=\frac{2840}{568}=5$$

Intercept

$$b_0=\bar{y} - b_1 \bar{x} = 130 - 5(14) = 60$$

Equation

$$\hat{y} = 60 + 5x$$

DECLARE XBar, YBar, B1, B0 FLOAT64;
SET (XBar, YBar) =  (SELECT STRUCT(AVG(NumberOfStudents) , AVG(SalesByQuarter)) FROM Shop.Sales);
SET B1 = (SELECT SUM( (NumberOfStudents - XBar)* (SalesByQuarter - YBar)) / SUM(POWER(NumberOfStudents - XBar, 2)) FROM Shop.Sales);
SET B0 = YBar - B1 * XBar;
CREATE TABLE Shop.LG AS
  SELECT B1, B0;

Predict Procedure

$$\hat{y} = b_1 * x + b_0$$

CREATE PROCEDURE Shop.PredictLG(X INT64)
BEGIN
  SELECT X * B1 + B0 FROM Shop.LG;
END;

Score R²

$$R^2=1-\frac{\sum{(y-\hat{y})^2}}{\sum{(y-\bar{y})^2}}$$

DECLARE YBar FLOAT64;
SET YBar = (SELECT AVG(SalesByQuarter) FROM Shop.Sales);
SELECT 1 - SUM(POWER(SalesByQuarter - (NumberOfStudents * B1 + B0), 2))/
  SUM(POWER((SalesByQuarter - YBar), 2))
 FROM Shop.Sales, Shop.LG;

Adjusted R²

$$Ajusted \space R^2 = 1 - \frac{n-1}{n-2}(1 - R^2)$$

Standard Error

$$S_{est}=\sqrt{\frac{\sum{(y-\hat{y})^2}}{n-2}}$$