EXERCISES 1. The following grades were obtained in a spelling test given to thirdgrade pupils; 65, 85, 55, 60, 100, 23, 92, 74, 73, 75, 76, 94, 82, 17, 10, 63, 97, 77, 96, 75, 90, 85, 90, 75, 86, 82, 94, 100, 90, 100, 74, 100, 100. Construct the frequency polygon, giving frequencies for class intervals of 5% arranged along the x-axis with multiples of 5 (a) at the first points of the intervals, (b) at the middle points of the intervals. Construct the smoothed curves. Were the words used a good test of the ability of the pupils in spelling? 2. Construct the histogram for the data in Exercise 1, with class intervals of 10%, (a) with mid-points at 55, 65, etc.; (b) with mid-points at 60, 70, etc. Calculate the smoothed values in the first table, and draw a curve through the ordinates resulting. 3. The number of seeds per apple in normal and aphis-injured Rome apples, as determined by an investigation, are given in the table. Number of seeds per apple.. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Frequencies of normal apples 0, 0, 0, 4, 31, 30, 38, 63, 64, 37, 31, 1, 1 Frequencies of aphis-injured apples... 5, 14, 20, 29, 44, 37, 28, 44, 38, 25, 13, 3, 0 Plot the histograms based on these samples. What is the most probable number of seeds in the normal and in the aphis-injured apple? State two differences between the two variations. 4. The following table gives the degrees of cloudiness of the sky observed at Breslau during the years 1876-1885, on a scale of 10. Degrees Number of days, 0, ང་ 2, ཉ་ 9, 10 21, 71, 194, 117, 2089 4, 5, 6, 7, 8, 751, 179, 107, 69, 46, 9, Plot the histogram and the smoothed curve. distribution? Would the arithmetic average of the measurements represent the variation well? What is the probability that a day at Breslau will be clear? Very cloudy? What is the type of the 5. Individuals A and B were tested by hearing a series of 10 letters read at the rate of 1 per second and being required to write as many as they could remember in the proper order of the letters, as soon as the reading was finished. Their scores were: Letters correct A frequencies B frequencies 4, 5, 6, 7, 8, 9, 10, Draw the histograms and the smoothed curves. Which is the better performance? What would be the probability that A and B could each get 8 correct in a particular trial? 6. The following table gives the age distributions for deaths from typhoid, measles, scarlet fever, influenza, tuberculosis, circulatory and respiratory diseases in New York state in 1916. Plot the curves given by these tables, identify the types of curves, and state characteristics of the diseases. 7. The errors of observation in 471 astronomical measurements made by Bradley were distributed as in the table, in which the mid-values of the magnitudes of the errors are given in decimals of a second of arc. Mid-values of errors |0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6. 5, 7.5, 8.5, 9.5, 10. 5 ́ Frequencies 94, 88, 79, 58, 51, 36, 26, 14, 10, 7, 8 Assuming that positive and negative errors occur with equal frequency, plot the histogram and the frequency curve. Discuss their forms. 8. Discuss the form of the frequency curve obtained by plotting the probabilities given by the terms of the expansion (p + q)", for p = }, q = {, n = 6, at equal intervals along the x-axis and drawing a smooth What is the most probable number of times the event will occur in 6 trials? If the distances along the x-axis are multiples of unity, what will the area under the curve from the maximum ordinate to the right hand extent of the curve represent? curve. 9. Plot the graph of the frequency curve y = e-2. Determine approximately by counting squares the abcissa of the point whose ordinate bisects the area of the right branch of the curve. 8 - 10(1+) (1) from )( 10. Plot the graph of the frequency curve y = 10( 1 + x = 3 to x 4. What is the type? = 11. Plot the graph of the frequency curve y = 10x0.5 -0.2 from x to x = 10, and discuss the type. 139. Averages. The frequency gives the variation of the magnitude measured but it is convenient in most cases to have a single value to represent the table. The representative values or averages which are commonly used for various kinds of frequency distribution are (a) the arithmetic mean, (c) the mode, (d) the geometric mean, (e) the harmonic mean. The arithmetic mean. The word mean or average alone is generally understood to denote the arithmetic mean. It is defined by the equation. where the symbol Em is used to denote the sum of the m's. If a measurement m1 is obtained fi times, then the sum of these fi measurements is fimi. If the measurements mi, mn occur with the frequencies f1, f2, spectively, then the sum of all the measurements is m2, fim1+f2m2 + ... + ƒnmn = Σfm, and the total number of measurements is fi + f2+...+fn = Σf. 2 fn, re Hence the arithmetic mean of the measurements is A = Σfm This equation defines the weighted arithmetic mean of the numbers mi, m2, . . ., mn with the weights f1, f2, fn. It is used whenever the numbers to be averaged are not of equal importance. EXAMPLE. If the averages of the weekly wages of three factories are 14, 17, 18 dollars and the number of employees are 450, 360, 670, respectively, then the average weekly wage for all three factories is The calculation of the arithmetic mean of a frequency distribution is simplified by considering the measurements of a class as concentrated at the center of the class interval, assuming the class interval as a unit and applying an extension of the rule on page 83, given by the Theorem. If m1, m2, mn are measurements with frequencies f1, f2, . . ., fn, if A is their true mean, E an estimated value of the mean, and d1, d2, dn deviations of the measure ments from E, then We have m1 = Edi, so that fim1 = fiE +fıdı, Corollary. The sum of the deviation from the arithmetic mean is zero. For if E A, then the correction c = Efd, the sum of the deviation from A The computation of the correction c = Στα = is zero. Hence Σα EXAMPLE 1. Find the arithmetic mean of the incomes of the group of families in the first two columns of the table. In the computation it is assumed that the income of the families in each class is the income at the middle of the class |