You want to calculate the least-squares regression line for two variables or the correlation coefficient that expresses the strength of the relationship between them.

Apply summary functions to calculate the necessary terms.

When the data values for two variables X and Y are stored in a database, the least-squares regression for them can be calculated easily using aggregate functions. The same is true for the correlation coefficient. The two calculations are actually fairly similar, and many terms for performing the computations are common to the two procedures.

Suppose that you want to calculate a least-squares regression using the age and test score values for the observations in the testscore table:

mysql>SELECT age, score FROM testscore;
+-----+-------+
| age | score |
+-----+-------+
|   5 |     5 |
|   5 |     4 |
|   5 |     6 |
|   5 |     7 |
|   6 |     8 |
|   6 |     9 |
|   6 |     4 |
|   6 |     6 |
|   7 |     8 |
|   7 |     6 |
|   7 |     9 |
|   7 |     7 |
|   8 |     9 |
|   8 |     6 |
|   8 |     7 |
|   8 |    10 |
|   9 |     9 |
|   9 |     7 |
|   9 |    10 |
|   9 |     9 |
+-----+-------+

The following equation expresses the regression line, where a and b are the intercept and slope of the line:

Y = bX + a

Letting age be X and score be Y, begin by computing the terms needed for the regression equation. These include the number of observations; the means, sums, and sums of squares for each variable; and the sum of the products of each variable:^[17]

mysql>SELECT
    -> @n := COUNT(score) AS N,
    -> @meanX := AVG(age) AS 'X mean',
    -> @sumX := SUM(age) AS 'X sum',
    -> @sumXX := SUM(age*age) AS 'X sum of squares',
    -> @meanY := AVG(score) AS 'Y mean',
    -> @sumY := SUM(score) AS 'Y sum',
    -> @sumYY := SUM(score*score) AS 'Y sum of squares',
    -> @sumXY := SUM(age*score) AS 'X*Y sum'
    -> FROM testscoreG
*************************** 1. row ***************************
               N: 20
          X mean: 7.000000000
           X sum: 140
X sum of squares: 1020
          Y mean: 7.300000000
           Y sum: 146
Y sum of squares: 1130
         X*Y sum: 1053

From those terms, calculate the regression slope and intercept as follows:

mysql>SELECT
    -> @b := (@n*@sumXY - @sumX*@sumY) / (@n*@sumXX - @sumX*@sumX)
    -> AS slope;
+-------------+
| slope       |
+-------------+
| 0.775000000 |
+-------------+
mysql> SELECT @a := (@meanY - @b*@meanX) AS intercept;
+----------------------+
| intercept            |
+----------------------+
| 1.875000000000000000 |
+----------------------+

The regression equation then is:

mysql>SELECT CONCAT('Y = ',@b,'X + ',@a) AS 'least-squares regression';
+-----------------------------------------+
| least-squares regression                |
+-----------------------------------------+
| Y = 0.775000000X + 1.875000000000000000 |
+-----------------------------------------+

To compute the correlation coefficient, many of the same terms are used:

mysql>SELECT
    -> (@n*@sumXY - @sumX*@sumY)
    -> / SQRT((@n*@sumXX - @sumX*@sumX) * (@n*@sumYY - @sumY*@sumY))
    -> AS correlation;
+------------------+
| correlation      |
+------------------+
| 0.61173620442199 |
+------------------+