6. What is the median of the distribution in Problem 41 7. Find the medians of Tables I, II, IV, counting down from the top of the distributions.

8. What is the mode of the distribution shown in Table II of Bulletin No. 6?

9. Find the median of Table II of Bulletin No. 6.

10. Find the medians of the three distributions shown in Table III of Bulletin No. 6.

3. The A B C of Educational Statistics; Measures of Type: The Mean

["Educational Measurements for the Class Teacher." Courtesy of Board of Education, City of New York, Bureau of Reference, Research and Statistics, EUGENE A. NIFENECKER, Director, No. 8, March, 1923.]

II-How To Obtain a Measure of Type (Continued from Bulletin No. 7)

3. By Taking into Account Not Only the Position But Also the Value of Each Measure We Get the Arithmetic Mean.— The first method of indicating the central tendency, by means of the mode, does not take into account the values of the measures. It is determined by the number of measures that happen to be concentrated at a given point in the distribution. The mode is a "position" average.

The second method of indicating the central tendency by means of the median, likewise does not take into account the values of the measures. It is true that in finding the median we first arrange the measures according to their values, but subsequent steps regard all measures of equal value. The first measure has no more weight than the last.

The arithmetic mean is the third method of indicating the central tendency of a distribution. It takes into account not only the position of each of the measures but also their values. The arithmetic mean (commonly called the "average" or "arithmetic average") is the sum of the values of all the scores in the distribution divided by the number of scores.

The arithmetic mean is the average that has been used most frequently by teachers in summarizing test results. Its computation is described in the following illustrations.

ILLUSTRATION ONE-ARITHMETIC MEAN WITH SCORES UNGROUPED

To find the arithmetic mean of the following scores in spelling obtained by 40 pupils of a class in a fifty word spelling test (scores expressed in number of words correct).

42 - 45 45 45 - 44 44 43 44 44 43

(a) Find the sum of the scores.

It is 1800.

(b) Divide this sum by the number of scores 1800 ÷ 40 = 45 = the arithmetic average.

It is customary to save labor by first arranging the scores into a series as follows:

TABLE I.-DISTRIBUTION OF SPELLING SCORES

The Values or Each Score Scores Times The (f)

No. Getting

Score

(m)

Tabulate by
Strokes

Frequency

To find the arithmetic average or mean

(a) we multiply each score or measure by the number of times

it occurs (m times f);

(b) we find the sum of these products;

(c) we divide this sum by the number of scores or cases (N) Expressing the above as a formula we get

The Mean=

The sum of the products of each value times its frequency
number of scores

or employing symbols, M

Σf m

N

In this formula M represents the arithmetic mean
Σ means the "sum of"

m measure or score

f frequency or number of times each measure

occurs

N total number of scores or cases

ILLUSTRATION TWO-ARITHMETIC MEAN OF A DISTRIBUTION WITH SCORES GROUPED

Bulletin No. 6, in describing the first steps to be taken in the statistical treatment of test data, called attention to the necessity, for ease in handling, of reducing the number of classes or intervals in a frequency distribution to about twenty class intervals. This is usually done before the mean is computed.

Take for illustration the scores obtained by 40 pupils in 8B in rate of writing as shown in problem 4 of page 7 of Bulletin No. 7 (p. 790). The highest score is 116, the lowest is 64. If we arrange the scores in unit steps we would have 52 steps or intervals. By grouping we can reduce the number of steps to a number easier to handle. We may group in intervals of 3, 5 or 10 units.

TABLE II.-SCORES IN RATE OF WRITING GROUPED IN CLASS INTERVALS OF THREE UNITS

3557 + 40 = 88.9, the approximate arithmetic mean of the distribution.

Grouping into intervals of 3 units condenses the score into 19 class intervals. The actual values of the scores are included within the class intervals and in computing the average we assume the value of each score in a class interval to be the same as the midvalue of the interval.

The computation of the arithmetic mean is the same as in illustration one.

(a) Multiply the value of the midpoint of each interval by the number of scores in such interval.

(b) Sum up these products.

(c) Divide by the number of scores.

The result is the approximate arithmetic mean.

Grouping the scores in class intervals and regarding all the scores of any given class interval as equal to the midpoint value of the interval introduces a certain amount of error into the computation so that the result is called the "approximate" mean to distinguish it from the "true" mean. To illustrate this-take

the 7 scores in the interval 83 to 85.99. We assume each of these scores to be equal to 84.5, the value of the midpoint of the interval. By reference to the actual scores we see that they are 83, 83, 84, 84, 85, 85, 85. The assumed value of the 7 scores equals (7 x 84.5) 591.5. The actual sum of the 7 scores is 589 and the average value of the seven scores is 84.1+. The error is slight and does not change our average very much. The true average of the 40 scores in Table II is 88.5.

TABLE III.-SCORES IN RATE OF WRITING GROUPED IN CLASS INTERVALS OF FIVE UNITS

TABLE IV.-SCORES GROUPED IN CLASS INTERVALS OF FIVE UNITS WITH THE VALUE OF MID-POINT OF EACH INTERVAL AN INTEGER

TABLE V.-SCORES IN RATE OF WRITING GROUPED IN CLASS INTERVALS

Grouping reduces the amount of labor.-If the reader actually does the computation in the above illustrations he will readily see that the process of grouping the scores reduces the amount of computation involved. By using a class interval of 3 units we reduce the intervals to 19. By using an interval of 5 units we reduce the number of intervals to 12, and if we use an interval of 10 units we obtain only 5 class intervals.

The arithmetic average obtained varies with the grouping. With intervals of 3 units our obtained average is 88.9, with 5 units it is 89.0 and 88.6 and with 10 units it is 89.0. The true average is 88.5.