6. What is meant by a frequency table or frequency distribution? It is the tabular arrangement (Table II) that results from our grouping of measures into classes. It consists, as we have seen, 1st of a serial list of the classes or class intervals arranged preferably with the smaller measures at the lower end, and 2d of a column of frequencies.

7. Into how many class intervals shall we divide or group our scores?

The object of our grouping is to classify our data in such way that if we subsequently compute an average for the data in its classified or condensed form, it will agree closely with the true average obtained from the entire list of the original scores.

The larger we make the interval or class size, that is the more the measures are condensed, the less will be our labor in computation. At the same time this will introduce errors that become greater with the size of the interval. Obviously it is much more accurate to say that (see Table I) two pupils spelled 40 words, one 38, two 37, and one 36 words than to say six pupils spelled from 36 to 40 words correctly.

In general when the units of the scale covered by the range are as few as 10, 15 or 20, nothing is gained by grouping into fewer classes and we may let each unit stand for a class interval. In our table the range is 35 units and must be grouped. A practical rule is to condense to not more than twenty intervals and to choose a size that gives ease of tabulation. In this case class intervals of five units convert a range of 50 units into 10 class intervals, a good working number.

8. What is the difference between a discrete and a continuous series?

If we measure the number of pupils on register in the classes of a school we get a series of integral or whole numbers. One class has 40, another 41, a third 42, and a fourth has 43 and so on. There are gaps between 40 pupils and 41 pupils, between 41 pupils and 42, since four-tenths of a pupil or seventy-five hundredths of a pupil do not exist. Such a series, where the different items are separate or where there are gaps between them, is said to be discrete. On the other hand if we measure the ages of pupils in a class we arrange them in a series which is capable of any degree of subdivision. In this case, if we use the class intervals of 6 to 7, 7 to 8, 8 to 9 and so on, the pupils in the 6 to 7 class, for instance, mean pupils who are 6 but not 7. Some may be six years and one month, some six years and two months, some six years and five months, and so on. Such a series, where the items are strung along running into each other, is said to be continuous. Almost all mental traits are quantities in continuous series, and in educational measurements series, although discrete, are regarded as continuous. For instance, in tabulating a frequency table for the scores in an

arithmetic test (see Table III) we tabulate according to the number of examples correct. Suppose we found 9 pupils who obtained a score of 18. This would not mean that all 9 were precisely at the 18 example point on the scale. They would be regarded as distributed between the 18th point on the scale and the 19th. Some would have just finished 18 examples, others would be one quarter through the nineteenth example, others almost done with the nineteenth example. Because our scale is not refined enough to indicate the exact status of their work (18 examples, 18.25 examples, 18.50, 18.75 or 18.9 examples finished), we classify their scores according to the last point passed.

In Table III, showing a distribution of pupils according to ages, the ten-year-olds may mean those who are ten years and up to eleven, or it may mean those who are between 9.5 and 10.5 years old.

9. How should the class intervals be defined?

In order to safeguard against errors in tabulation it is very necessary that the numerical limits of the class intervals to be used are clearly defined or marked off from each other. While there are several methods that may be used, the safest method is to indicate in the frequency table the limits of the class interval. If the interval contains ten units the class intervals should be indicated as follows:

10 to 19.99

20 to 29.99

30 to 39.99 etc.

pupil's

If we employ the limits 10 to 20, 20 to 30, 30 to 40, etc., errors are apt to creep into our tabulation.

10. What is assumed in all grouping of measures?

In all grouping of measures in a frequency distribution we assume that all values in any class interval are concentrated at the mid-point of the interval and may be represented by the values of this point.

In the class interval 36 to 40, in Table II, for instance, where there are six cases, we represent these six cases by the mid-point of the interval or 38. The six scores, however, were two at 40, one at 38, two at 37, and one at 36. The sum of these six scores is 228. Representing the six cases by a score of 38 gives us the same total of 228.

IV.—What Will Aid in the Interpretation of Our Data?

The graphic representation of the data in question will aid considerably in their interpretation.

1. How can the results of our tests be represented concretely?

They may be represented by means of (a) a frequency polygon, (b) a column diagram, (c) a bar graph as illustrated in Diagrams A, B and C.

(a) The horizontal base line represents the scale. Along this scale the class intervals (see Table II) of the frequency distribution are laid off, i.e., 0 to 5, 6 to 10, 11 to 15, 16 to 20, 21 to 25, etc.

(b) The vertical line at the left represents a scale of the number of pupils whose scores occur in a particular class interval or at a given point on the scale.

2. In plotting the curves the steps are as follows:

(a) Note the number of class intervals in the range of the frequency distribution. Table II shows ten intervals included.

(b) Mark off the ten units of the frequency table on the base line from left to right, making the units as large as the width of the paper permits.

(c) At the left end of the horizontal line draw a vertical line upward and mark off on the vertical line the number of units needed to represent the number of cases or pupils. In order to tell how far apart to put these marks find out the largest number of cases in any one interval of the frequency column. In the case of Table II the largest num

ber is 17 and as there is sufficient room our vertical scale
will be one pupil to a square.

(d) Plot the number of cases for each class interval on the
scale by locating points at the proper vertical distance
above the mid-point of each unit on the base line.
(e) Draw lines connecting these points.

2. The A B C of Educational Statistics: Measures of Type ["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. 7, November, 1922.]

I.-What Is the Second Step in the Statistical Treatment of Data Resulting from Educational Measurements?

The first step, that of classification of data resulting in an orderly arrangement of scores by serial classes was described in Bulletin No. 6. Thus far while we have introduced a few technical statistical terms, we have done nothing with our data that every teacher has not done numerous times. From such arrangement the teacher finds out how many pupils obtained each score, what the highest score was, what the lowest score was, what was the range of scores, that is, how far apart the highest and lowest pupils are.

In addition to the above the teacher may desire to compare the results of a test in her class with the results obtained in a previous test of a similar kind. She may wish to compare the achievements of her class with those of other classes if the latter have been given the same test, or she may want to see how her class compares with the scores set up as standards for her grade. For such purposes she needs more than a listing of scores she must summarize them in some way.

The second step, therefore, is that of condensation and involves the selection of one measure to stand for or represent the entire distribution. Such a measure summarizing or typifying the whole series is called a measure of type or an average. An average gives a concise picture of a large group and it permits the comparison of different groups.

II.-How to Obtain a Measure of Type of a Frequency Distribution

There are several kinds of measures of type or averages, obtained in different ways: