Mathematics for Collegiate Students of Agriculture and General Science

239. The Regression Coefficient. The regression coefficient is a fixed ratio between deviations of correlated characters, so that, knowing how much one of the characters differs from its mean in any unit of measurement, say inches, we are enabled to predict how much the associated character may be expected to depart from its mean in its unit of measurement, say in pounds.

The regression of a character y with respect to a character x is given by the formula

the regression of a character x with respect to a character y is given by the formula

where σ is the standard deviation of the population with respect to the x character, o, is the standard deviation with respect to the y character, and r is the correlation coefficient given by (2), § 238.

The regression coefficient is useful for prediction. That is to say, if we know the deviation of one character from its mean, this coefficient will enable us to predict what will probably be the deviation of the correlated character from its mean. It should be noted that (4) cannot be obtained from (3) by solving for x.

In this connection let us observe that we find from the correlation table in § 237, that the average length of ears of weight 9 ounces is 7.5 inches, while the average weight of ears of length 7.5 inches is 8 ounces.

Either of the equations (3) or (4) represents a straight line which is called the line of regression. This line may be obtained graphically as follows. Arrange a population with respect to

two characters x and y. For each value of x obtain the mean of the y values. Plot these means as ordinates with the corresponding value of x as abscissas. If correlation exists, these means do not lie at random over the field, but arrange themselves more or less in the form of a smooth curve called the curve of regression.

This curve is a crude picture of the function which defines the correlation of the y-character relative to the x-character. Experience has shown that, in many sets of measurements, this curve is approximately a straight line.

EXERCISES

1. Find, for the correlation table in § 237:
(a) the regression of weight relative to length;
(b) regression of length relative to weight.

Ans. (a) 2.03 (b) 0.38

2. Find the equation of the line of regression in both cases of Ex. 1. 3. Plot the line of regression in Ex. 2 from the equation found there and then again plot the line from the data as suggested in the last paragraph of this chapter.

4. From Table II, p. 321, which gives the correlation of circumference and number of rows of kernels of ears of corn, compute the mean circumference, the mean number of rows of kernels, the standard deviation with respect to circumference, the standard deviation with respect to rows of kernels, the correlation coefficient, and regression coefficients. 5. Examine Table IV, p. 322, which gives the number of children of various statures born of 205 mid-parents of various statures. From this table compute:

Mp Mc

бр

= mean height of mid-parents,

= mean height of adult children,

standard deviation of height of mid-parents,

σc = standard deviation of height of adult children,

r = the correlation coefficient, and both regression coefficients. 6. For Ex. 4 plot the lines of regression (a) from their equations, (b) from the data directly.

7. For Ex. 5 plot the lines of regression (a) from their equations, (b) from the data directly.

8. From the following table find a measure of the effectiveness of vaccination against smallpox.

9. Construct a correlation table from your own observations on length and breadth of leaves. (a) Use 30 classes for length. (b) Use 15 classes for length, thus making the class interval twice as large. Compute in each case the correlation coefficient.

10. From Table I, p. 320, which gives the correlation of lengths and weights of ears of corn, compute the mean length, the mean weight, the standard deviation with respect to length, the standard deviation with respect to weight, the correlation coefficient, and both regression coefficients.

11. The same as Ex. 10 after writing circumference in place of weight, using Table II, p. 321, in place of Table I.

I. CORRELATION OF LENGTH AND WEIGHT OF EARS OF CORN