sum, the present value will be the mathematical expectation of receiving $12,220 contingent upon the probability that he will live 10 years.

The probability that a person 10 years old will live at least 10 years is

Hence the present value of the inheritance is (12,220) (.92637)

EXERCISES

1. Plot the graph of the mortality table given in Section 137. By means of the graph estimate your own chance of living to the age of 75. 2. A man is 45 years of age and his son is 15. What is the probability that both will be alive 10 years hence? What is the probability that at least one will be alive?

3. A man and his wife are 40 and 35 years old respectively, when their child is 10 years old. What is the probability that all will be alive until the 20th anniversary of the child's birth? What is the probability that at least one will survive?

4. What should be the minimum cost of insuring the life of a person 20 years old for $1000 for five years? For insuring a couple against the death of either or both for the same sum and period if the husband is 30 and the wife 25? (Neglect interest, etc.)

5. A man makes a will leaving $40,000 to his wife in case she survives him. A son is to inherit the money if he survives both parents. If the ages of husband, wife, and son are 60, 50, 25 respectively, and if money is worth 5% compounded semi-annually, what is the present value of the expectation (a) that the wife will inherit the money in 10 years? (b) that the son will inherit the money in 15 years?

6. In each of the following exercises plot a graph with the probabilities as ordinates at arbitrary equal intervals along the x-axis.

(a) A coin is tossed six times. Find the probabilities of the various ways in which it can turn up heads.

(b) Find the probabilities for the various ways in which two dice can fall in one throw.

(c) In the long run A wins 3 games out of 4 from B at chess. Find the probabilities of the various numbers of games which A might win in 8 successive games.

(d) If a die is tossed six times, find the probabilities of the various ways in which an ace can turn up.

(e) If the quantity of a trait in an individual of a group is the result of a chance combination of seven causes, determine the probabilities of the ways in which the trait may occur.

138. Frequency Distributions. At an agricultural experimental station 110 apples were classified with respect to the number of seeds each contained, and the number of apples in each class was determined.

Number of seeds

46678-8

Number of
apples

9

5

4

14

21

24

9

25

10

13

The results are given in the table.

Thus 9 apples had 4 seeds each (the minimum number found in this investigation), 13 apples had 10 seeds each (the maximum number), while 25 apples had 9 seeds each (the most frequent number occurring).

Such an arrangement of the individuals of a group, classified with respect to some characteristic which gives the number of individuals in each of the classes is called a frequency distribution and the table in which the classes and frequencies are given a frequency table.

25

A graphical representation of this analysis, called a frequency polygon, is obtained by plotting the magnitudes of the classes as abscissas, the frequencies as ordinates, and connecting the points. by straight lines as in the figure.

The magnitude of each class in this case is an integer and the class intervals are said to vary discretely. In case the

Number of Apples

220

15

9

10

characteristic measured varies continuously, as for instance the stature of a group of men, the size of the class intervals and their mid-points are chosen arbitrarily.

The magnitude measured may vary discretely but by such small amounts that the number of classes is so great that the variation of the group with respect to the characteristic cannot be easily determined. In such a case the class interval is enlarged by grouping the frequencies in two or more adjacent

classes and associating the resulting frequency with the midvalue of the resulting larger class.

EXAMPLE. The grades in geometry of 30 students were 30, 42, 48, 55, 60, 64, 68, 71, 72, 74, 75, 76, 77, 77, 78, 78, 78, 79, 80, 82, 82, 83, 84, 85, 86, 87, 88, 88, 91, 95. Collect the data in frequency tables with class intervals of 5% and 10%.

Tables 1 and 2 show the data collected in class intervals of 5% with different mid-points. In table 3 the class interval is 10%.

The class interval of 1% is too small for an adequate presentation of the data. Even class intervals of 5% leave some classes empty.

The class interval should be chosen so as to avoid empty classes. The smaller the number of measurements the larger the class interval should be, and vice versa. The starting points of the intervals are not so material, but it is convenient to take them so that the mid-points of the intervals are integers. In age distributions the returns usually cluster about the multiples of 5, which are taken as the mid-points of the intervals.

A second method of representing frequency tables graphically is indicated in the Figs. 210-212 representing the tables above. Rectangles are constructed on the class intervals as bases with altitudes equal to the frequencies. Such diagrams are called histograms. The area of a histogram is the sum of the frequencies times the class interval.

If the number of observations be increased and the class interval decreased the frequency polygon or histogram will

approach more and more closely to a smooth curve. curve is called a frequency curve.

Such a

A frequency histogram is sometimes smoothed by drawing a curve through the mid-point of the upper base of each rectangle in such a way that the area under the curve is the same as the area of the histogram.

A frequency table is smoothed by replacing the middle frequency of three adjacent frequencies by the average of the three frequencies. The two end values are counted twice and averaged with the adjoining value.

If the polygons or histograms corresponding to successive smoothings of the table are plotted we can approximate closely to the frequency curve which best

represents the data.

The types of frequency distributions which are most common are: (a) The symmetrical distribution in which the frequencies decrease to zero symmetrically on either side of a central magnitude.

Frequencies

Magnitudes

FIG. 213.

It is found in the distribution of errors in chemical and physical measurements and in biological measurements, particularly the measurements of anthropology.

Frequencies

Magnitudes

FIG. 214.

(b) The moderately asymmetrical distribution in which the frequencies decrease more rapidly on one side of the value of maximum frequency than on the other.

It is the most

common of all the distributions occurring

in all forms of statistics.

Frequencies

Magnitudes

FIG. 216.

(c) The J-shaped

distribution in which

the frequency con

Frequencies

Magnitudes

FIG. 215.

stantly increases or decreases. It is found in economic statistics and is characteristic of the distribution of wealth.

(d) The U-shaped distribution in which the frequency decreases to a minimum value and then increases. It is rare. It occurs in meteorological statistics and in statistics pertaining to heredity.