included between the limits ∞ and +∞, the area between the curve and its asymptote * must represent certainty, and hence be equal to unity. It can be shown that the area between the curve

and its asymptote is σ√2π, a result which we assume. Then the area between the curve

and its asymptote is σV2 C, since the latter curve may be obtained from the former by multiplying the ordinates by C.

* Let A denote the area bounded by the probability curve, the x-axis,

and the ordinates x a and x = a.

a definite limit as a becomes infinite. the curve and its asymptote.

It can be shown that A approaches
This limit is called the area between

In the theory of measurements it is customary to set h2

1

202

(3)

EXAMPLE. Find the equation of an ideal frequency curve (probability curve) which fits approximately the upper polygon in the figure at the beginning of the section. Test the accuracy with which the curve fits the polygon by finding the values of y given by the equation for the values of x given in the table and the amounts by which these values differ from the given values of y.

saw that σ = Ax, and here Ax = 1. The equation required is found by substituting this value of σ in (2) which gives

Substitution of the given values of x in this equation gives the numbers in the third column of the table. The computation requires the use of a table of values of the exponential function, and is effected by the use of logarithms.

EXERCISES

1. Construct the center frequency polygon in the figure at the beginning of Section 141, find the equation of an ideal frequency curve which approximates the polygon, and plot the curve on the same axes as the polygon. Find the difference between the ordinate of each vertex of the polygon and the corresponding ordinate of the curve.

2. In determining the constant of integration in equation (1) of the preceding section the area between the curve and its asymptote is some

times taken equal to n, the sum of the frequencies. Show that in this case the equation becomes

What would equation (3) become in this case?

3. If à coin is tossed eight times, find the frequencies of each of the various ways it can fall. Plot these frequencies as ordinates at a unit distance from each other, and draw the frequency polygon. Find the equation of the smoothed frequency curve by means of the preceding exercise.

Frequency

4. Find the arithmetic mean and the standard deviation for the distribution of grades in trigonometry given in the table, where the midGrade values of the class intervals for the grades are given. Plot the frequency polygon. Find the equation of the smoothed frequency curve using Exercise 2. Plot the curve on the same figure as the polygon, using as origin the point on the x-axis which represents the mean, and computing the values of y for the values of x corresponding to the mid-values of the class intervals given in the table.

95

85

22

75

24

65

17

55

6

45

3

AGRNAG

5. Find the arithmetic mean and the standard deviation of the heights of 12-year old boys given in Exercise 6, page 19. Find the equation of the smoothed frequency curve, and plot the frequency polygon and curve in the same figure.

142. Probable Error. The value r of x such that half the errors lie between the limits - r and

error.

+

r is called the probable

of the deviation of the Since in a symmetrical

The probable error is a measure measurements from the true value. distribution the mean and median coincide, in this type of distribution the probable error is the same as the quartile deviation (Q3 - Q1)/2. Graphically it is the abscissa on the x-axis whose ordinate bisects the area of one branch of the

given in the following table for different values of hx.

From the table it is seen that the area is A =

when hx is

between 0.4 and 0.5. The calculation of hr to four places, which is beyond the scope of this course, gives

The probable error is here expressed in terms of the unknown errors x. In a particular case only the deviations from the mean can be calculated. If we change from errors to deviations in the formula, the numerator is diminished, since Zd2 <Zx2 (the sum of the squares of the deviations from the mean is smaller than from any other value). It is assumed in practice that the best value of the denominator to conform to this change is n - 1.

Hence

and

YA

A

If two sets of measurements of unequal precision are com‐ pared it will be found that while the areas under the probability curves are the same, viz., unity, the measurements in the more precise set will cluster more closely about the mean, the value of σ for this set will be smaller, and since h = 1/σ√2, the value of

FIG. 223.

B

h will be larger. In the figure the curve A represents the set with greater precision.

The value h is called the measure of precision of a set of measurements. Since the intercept on the y-axis is h/√π, this intercept is proportional to the measure of precision.

In comparing different sets of measurements the weight attached to the arithmetic mean of any set (Theorem 3, page 418) is given by h2, whose value, from (3), is

Theorem 1. If y = f(x), if the probable error in x is r, and if the probable error in y is R, then R = D2y r.

Squaring each of these equations and adding the results, Σ(dy)2 = [DayZ (dx)2.

Extracting the square root of both sides of this equation and multiplying both sides of the result by 0.6745/Ö,

Theorem 2. If y is a function of several variables, y f(x, z, w) and if the probable errors in x, z, w, and y are respectively r1, r2, r3, and R, then

R2 = (Day)2rı2 + (Dzy)2r22 + (Dwy)2rz2.