Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors

Figures & data

Figure 1 Connecting the means of the individual columns of data provides a crude approximation of the regression line. The slope is exactly 0.50 and the correlation is approximately r = 0.51. Many, though not all, of the points are closer to the offspring pea size mean of 9 on the Y‐axis than to the parental pea size mean of 10 on the X‐axis. The numeric data appear in Appendix B.

Figure 2. Correlation remains constant even though the slope of the line changes as a result of differences in variability between the two variables. In all three graphs, the correlation is identical at approximately r = 0.64. In 2A the variability in X is identical to the variability in Y. In 2B the variability in X was magnified to be larger than the variability in Y. In 2C the variability in Y was magnified to be larger than the variability in X. The numeric data appear in Appendix B.

Table 1. Numeric Example Demonstrating that Adding or Subtracting an Offset from the Product-Moment Worsens the Prediction of Y from X.

Deviation Data	Product	No Offset (Optimal)	Positive Offset	Negative Offset
	x	y	x*y	2.0x^Table Footnoteb	Squared Errors^Table Footnotec	2.1x	Squared Errors	1.9x	Squared Errors
	−1	−3	3	−2	1	−2.1	0.81	−1.9	1.21
	−1	−2	2	−2	0	−2.1	0.01	−1.9	0.01
	−0.5	−1	0.5	−1	0	−1.05	0.025	−0.95	0.025
	−0.5	−1	0.5	−1	0	−1.05	0.025	−0.95	0.025
	0.5	1	0.5	1	0	1.05	0.025	0.95	0.025
	0.5	3	1.5	1	4	1.05	3.8025	0.95	4.2025
	2	3	6	4	1	4.2	1.44	3.8	0.64
Sum	0	0	14		6		6.1375		6.1375
Mean	0	0	2.0^Table Footnotea

^a The mean of the deviation cross-products, that is, the product-moment.

^b This column uses 2.0 as the value of the slope in the equation = 2.0 x. The raw product-moment value can be used here as the slope without correcting for the variability in x because the variance of x was programmed to be exactly equal to 1. Note that the general equation for regression slope is .

^c The squared errors of prediction

Figure 3. Simplified diagram showing influences of three preceding generations of ancestors on the present day individual.

Table 2. Raw Data on Diameters of Parent and Daughter Seeds Generated from Galton (1894) Table 2 in Natural Inheritance.

Diameter of Parent Seed (0.01 inch)	Diameter of Daughter Seed (0.01 inch)	Frequency
21.00	14.67	22
21.00	15.67	8
21.00	16.67	10
21.00	17.67	18
21.00	18.67	21
21.00	19.67	13
21.00	20.67	6
21.00	22.67	2
20.00	14.66	23
20.00	15.66	10
20.00	16.66	12
20.00	17.66	17
20.00	18.66	20
20.00	19.66	13
20.00	20.66	3
20.00	22.66	2
19.00	14.07	35
19.00	15.07	16
19.00	16.07	12
19.00	17.07	13
19.00	18.07	11
19.00	19.07	10
19.00	20.07	2
19.00	22.07	1
18.00	14.35	34
18.00	15.35	12
18.00	16.35	13
18.00	17.35	17
18.00	18.35	16
18.00	19.35	6
18.00	20.35	2
17.00	13.92	37
17.00	14.92	16
17.00	15.92	13
17.00	16.92	16
17.00	17.92	13
17.00	18.92	4
17.00	19.92	1
16.00	14.28	34
16.00	15.28	15
16.00	16.28	18
16.00	17.28	16
16.00	18.28	13
16.00	19.28	3
16.00	20.28	1
15.00	13.77	46
15.00	14.77	14
15.00	15.77	9
15.00	16.77	11
15.00	17.77	14
15.00	18.77	4
15.00	19.77	2

Galton, F. (1894), Natural Inheritance (5th ed.), New York: Macmillan and Company.

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