and the standard deviation of the errors is σ = 0.378.

least squares the errors are

or

0%, +20%, -19%, +11%

with a standard deviation of the errors of σ = 0.368.

EXERCISES

1. If the pairs of values (x1, 1), (X2, Y2),

For the method of

(xn, Yn) of an empirical table are connected by the relation y = mx show by the method of least squares that the most probable value of m is m = Zxy/Zx2.

2. In a psychological experiment to determine the ability of an individual to determine variation in pressure, the following table was obInitial weight in grams 10, 20, 30, 40,

tained. Find the law, and

Just perceptible weight in grams 5. 15, 7.8, 9.9, 13.3, the change in

weight which is just perceptible when the initial weight is 50 grams. 3. A steel bar was stretched by attaching weights, and the measure

Tension in pounds
Stretching in inches

150, 400, 500, 700,

.32, .80, .98, 1.44,

ments made as given in the table. Determine the law, and find how much the bar

would be stretched by a weight of 800 pounds. 4. The table gives the horse power required for a given load in the

Load in tons | 50, 70, 90, 100,

Horse power load of 68 tons.

81, 107, 127, 142,

case of a locomotive running 40 miles an hour. Determine the law, and find the horse power required for a

•

(Xn, Yn) of an empirical table

5. If the values (x1, 1), (x2, Y2), are connected by the relation y = mx + b, show that the most probable values of m and b are given by the equations

6. Use the results of the preceding exercise to calculate the coefficients of the linear function P = mW +b in Example 2, Section 27, arranging the work with columns headed x, y, x2, xy. Compare the results with those obtained in Section 27.

7. If the values (X1, Y1), (X2, Y2) ・ ・ ・ (Xn, Yn) of an empirical table are connected by a relation of the form y ax2 + bx + c show that the most probable values of the coefficients a, b, c, are given by the equations aZ(x4) + bZ(x3) + cZ(x2) = Z(x2y),

8. Use the results of the preceding exercise to calculate the most probable values of the coefficients of the quadratic function representing the

empirical table in the example in Section 35. Arrange the work with columns headed x, y, xy, x2, x3, x2y, xa. Compare the results with the values obtained for the coefficients by the method of Section 35.

9. If the values (xi, y1), (X2, Y2) ... connected by a relation of the form y of the coefficients, given that Dx loge u form y = keb find the most probable values of the coefficients.

(Xn, Yn) of an empirical table are kxm find the most probable values Dau/u. If the relation is of the

10. Determine the most probable values of the constants in an equation of the form y = kxm connecting the following values of x and y, (5, 11), (10, 31), (15, 59), (20, 88). Suggestion: Find log x and log y, rounded off to two figures, and then proceed as for a linear function.

11. The temperature of a body cooling in air at zero temperature was 78°, 60°, 48°, 36° at the end of 1, 2, 3, 4, minute intervals respectively. Determine the most probable values of the rate of cooling and the original temperature.

12. Six measurements of a line with a steel tape were in feet 639.15 639.08, 639.11, 639,19, 639.10, 639.14, and with a chain 639.2, 639.1, 639.1, 639.4, 639.3, 639.0. Find the means of the two sets of measurements and their weights and the weighted mean.

13. Find the most probable value of the velocity of light from the following determinations in kilometers per second: 298,000 1000; 298,500 1000; 299,990 ± 200; 300,100 ± 1000; 299,930 ± 100.

14. A line is measured 10 times and the probable error of the mean is 0.012 foot. How many additional measurements of the same precision are required to reduce the probable error of the mean to 0.005 foot?

15. The parallax of the sun by two determinations is 8.883 ± 0.034 and 8.943 ± 0.051. What are the weights of the two observations? What is their weighted mean?

144. Correlation. In the experiment to determine the coefficient of friction of wood on wood, discussed in Section 27, the causal relation between the two weights is apparent. To a change in one weight corresponds a change in the other. An arbitrary value can be given to either weight and the corresponding value of the other determined, and the pairs of values found by the two methods are the same. That is, the direct and inverse relations are given by the same equation. In short, one variable is a function of the other.

When some variables are compared, for example, the price of roses and the amount of salt mined, no causal relation can be detected.

Between the extremes of two variables which are causal dependents or absolutely unrelated, there are varying degrees

of dependence. In this section we shall consider the kind of relation given in the

DEFINITION. Two variables are said to be correlated if an increase in one is accompanied either by an increase or by a decrease in the other. A measure of the degree of dependence is called a coefficient of correlation.

The coefficient of correlation measures the probability with which the value of one variable may be predicted from an assumed value of the other.

The correlation is said to be perfect if a causal relation is established between the two variables, that is, if one is a function of the other. The correlation is said to be positive or negative, according as an increase in one variable is accompanied by an increase or decrease in the other.

The values of two associated variables may be conveniently arranged in rows and columns. Such an array is called a correlation table.

EXAMPLE 1. Represent the following correlation table graphically.

Correlation between July precipitation and yield of corn in Ohio for the years 1854-1913 inclusive.

Each row is a frequency distribution for the class interval

of y given at the left, and each column for the class interval of x given at the top.

The class interval for the precipitation is one inch, and for the corn yield five bushels per acre.

The number 8 in the row and column headed respectively 37.5 and 4.5 means that in 8 different years a yield of 35 to 40 bushels of corn per acre was associated with a precipitation of 4 to 5 inches of rain. The other frequencies have similar meanings. The column on the right and the row at the bottom of the table give respectively the sums of the frequencies for each class interval of y and x. The sum of the right-hand column, or of the bottom row, is the total number of items, n = 60. A graphical representation of the table is obtained by plotting the tables below. Table 1 gives the mid-values of the x classes and the mean values of the corresponding vertical columns. The pairs of values in this table are plotted as small circles in Fig 224. In this instance, as in many cases, the circles lie approximately on a straight line l1.

Table 1

Table 2

Mean y

У

Mean x

1.5 30

22.5

2.5 31.7

3.5 34.4

4.5

34.4

5.5

38

6.5

37.5

4.5 27.5 2.9 32.5 3.6 37.5 4.5 42.5 5.5 521.0

8.5 42.5

7244.5

M2 = 4.2

Table 2 gives the mid-values of the y classes and the mean values of the corresponding horizontal rows. The pairs of values are plotted as small squares, and

a straight line l1⁄2 may be fitted to them approximately. In this case the small square at the point (22.5, 4.5) represents but one of n = 60 items, and may be neglected in drawing l2.

slopes, to an increase in either variable corresponds an increase in the other. Hence the correlation is positive.

The mean values of x and y for the entire correlation table are given at the ends of tables 1 and 2. These values may be obtained also as the means of the row and column headed fr and f. In practice, they are computed in connection with other quantities, as in Example 3 below. Notice that the lines l1 and l2 appear to intersect at the point M (Mx, My).

As in Example 1, the direct and inverse relations between two correlated variables are usually represented by two distinct lines. These lines coincide if the correlation is perfect, while l1 and l1⁄2 are parallel to the x and y-axes respectively if the variables are unrelated. The correlation is positive or negative according as the slopes of these lines are positive or negative.

The line representing the means of the columns always makes a smaller angle with the x-axis than the line l2 representing the means of the rows. If x is the height of fathers and y that of all their sons, the sons of men who are taller or shorter than the average tend to approach the average height more closely than their fathers. On this account Galton called the falling back of the lines toward the axes regression, and hence the lines are generally called lines of regression.

We shall assume the fact that the lines of regression pass through the point M whose coördinates M, and M, are the mean values of the x and y distributions (see Example 1). Let x' and y' be new variables referred to axes through M parallel to the old axes. Let the equations of l and l2 be

* If P1 and P2 are two points on l2, and if 02 is the inclination of l1⁄2, then