If we count out the five who scored 3 or higher, we shall still require half of the distance represented by the next highest individual in order to have counted out 5.5 (half of 11). If our assumption is true, then, we shall need to count half way down from 3.00 to 2.67 in order to find the median point, 2.83. The calculation of the median point is not necessary, however, unless there is a very large number of cases in the distribution and unless very accurate comparisons must be made. In passing it may be said that the calculation of the median point at 2.83 is just as sensible and just as accurate as the calculation of the average point at 3.18, and that the median point is a much more useful measure of the distribution than the more commonly used average.
The user of the Mentimeter tests will not, under ordinary circumstances, be satisfied with interpreting an individual’s score merely by indicating its direction from the median, mode or average of a group. It will not usually be sufficient to say “He made the modal or most popular score,” or “His score was lower than the average,” or even “His score was higher than the median.” Some indication will be desired as to how much better or poorer a given score is than the median, or just what percentage of the standard group made better scores. An illustration of the method to be employed in such calculations and a review of the method of finding the median is given below in connection with a distribution of scores on one of the Mentimeter tests. (See Mentimeter No. 24, page [234].)
| I | II | III | IV |
|---|---|---|---|
| SIZE OF SCORE ANALOGIES TEST | FREQUENCY: NO. OF COLLEGE GRADUATES | TOTAL NO. FROM LOWEST SCORES | TOTAL % FROM LOWEST SCORES |
| 30 | 2 | 129 | 100 |
| 29 | 4 | 127 | 98.5 |
| 28 | 10 | 123 | 95.3 |
| 27 | 22 | 113 | 87.6 |
| 26 | 32 | 91 | 70.6 |
| 25 | 20 | 59 | 45.8 |
| 24 | 18 | 39 | 30.3 |
| 23 | 8 | 21 | 16.3 |
| 22 | 4 | 13 | 10.1 |
| 21 | 2 | 9 | 7.0 |
| 20 | 1 | 7 | 5.4 |
| 19 | 2 | 6 | 4.7 |
| 18 | 1 | 4 | 3.1 |
| 17 | 1 | 3 | 2.3 |
| 16 | ... | ... | ..... |
| 15 | 1 | 2 | 1.6 |
| 14 | ... | ... | ..... |
| 13 | ... | ... | ..... |
| 12 | 1 | 1 | .8 |
| 11 | ... | ... | ..... |
| Total | 129 |
Having distributed the scores obtained by a group of college graduates on the Analogies test, the next important step toward their interpretation is the totaling of the frequencies up to and including those of each possible size, as shown in the third column of the accompanying table. The fourth column is then prepared showing the corresponding percentages of the total number (129) of persons tested, for each of the total frequencies shown in column III. The table as a whole is then to be read from left to right. As an example, one may begin at 20 in the first column and read as follows: “1 college graduate made a score of exactly 20 points, making in all 7 individuals who obtained a score of 20 points or less, which (7) is 5.4 per cent. of the 129 individuals tested.” Dropping the eye to the next percentage below this line in column IV, one can interpret the score of the individual who made a score of 20 as follows: “This is a poor showing for a college graduate, for of 129 college graduates tested only 4.7 per cent. made a lower score.”
A very popular method of interpreting a score is to tell in what quarter or, as the statisticians would say, in what “quartile” of the distribution a given score is found. The upper or first quartile of a distribution is the range of scores below which 75 per cent. of those tested have fallen. The second quartile is the range of scores below which 50 per cent. are found but above which 25 per cent. of those tested are found. The third quartile is the range below which only 25 per cent. are found and above which 50 per cent. are found, and the fourth or lowest quartile is the range of scores in which are found the lowest 25 per cent. of the scores made. The first and second quartiles are above the median, while the third and fourth quartiles are below the median. Obviously the individual who scored 20 points in the Analogies test, and is included in the lowest 5.4 per cent. is also in the lowest quartile of the college graduate scores. The point dividing the first and second quartiles is called the 75 percentile, while the point dividing the third and fourth quartiles is called the 25 percentile. As was stated above, the median or 50 percentile divides the second and third quartiles.
Columns III and IV in the foregoing table assist one quite materially in calculating the median and the other percentile points. To find the median, one will need to count half way through the distribution, in this case to count out 64.5 scores (129
2 = 64.5). The 20 persons who scored on 25, in the above distribution, are shown by column III to be included in the lowest 59 scores and by column IV to be in the lowest 45.8 per cent. To include 64.5 (or 50 per cent.) of the scores, 5.5 of the 32 individuals who scored on 26 will need to be taken (64.5 − 59 = 5.5); 5.5 is .17 of 32, so it will be necessary to take .17 of the distance (26.0 up to 27.0) represented by a score of 26. This places the 50 percentile or median point at 26.17, if we assume that the 32 individuals obtaining a score of 26 were evenly distributed in their exact values between 26.0 and 27.0, which is the safest assumption one can make about these scores.
The 25 percentile is found by counting out one fourth of the frequencies, beginning with the low-score end of the distribution. In the case of the college graduates’ distribution on the Analogies test, the 25 percentile is 24.63. The 75 percentile, which is found by counting out three fourths of the frequencies from the low-score end or one fourth from the high-score end of the distribution, is 27.26 in the case of the analogies distribution shown above. The “middle 50 per cent.” of the distribution, or the second and third quartiles, lie between 24.6 and 27.3 according to these calculations. One may therefore assert that the typical college graduate, meaning one who is within the two middle quartiles of the college graduate distribution, should be expected to make a score of 24, 25, 26, or 27 on the Analogies test in the Mentimeter series.
Occasionally intellectual measurements are reported by tenths, the first tenth being the tenth of the distribution having the highest scores, just as the first quartile is the quarter containing the highest scores. For practical purposes with the Mentimeter tests, however, it is recommended (1) that the score made on each test be recorded, (2) that the median score of the standard group, with which each individual’s score is to be compared, be calculated, and (3) that the percentage of the standard group making lower scores than that individual’s score be used as an interpretation. For these simple interpretations, a table, such as that shown on page [102] for college graduates in the Analogies tests, practically completes the necessary calculations,[[2]] except for the calculation of the median score. It will be fairly intelligible to describe Henry Smith’s score as follows: “Smith has a score of 24 points as compared with the median score of 26.2 points for his group. Only 16.3 per cent. of the college graduates make a poorer score than Smith, but 69.7 per cent. make a better score.”
[2]. For the purpose of assisting the reader in keeping and interpreting records of the Mentimeter tests, the authors have prepared a record booklet which may be used with the tests to excellent advantage. It will be found economical to use this booklet because of the guide lines, headings, and practical suggestions which it contains, reducing copying and memory work in the calculations to a minimum. It is recommended also that calculating tables or a slide rule be used to calculate the percentages called for in the final column of the distribution tables. Such aids are very desirable because of their contribution to the accuracy of results and to economy of time.
Assuming now that the reader has a fairly clear idea of how to administer and record the results of the Mentimeter tests, the next question to be answered is: “What shall be done about these test records?” Measurement in any field does not change to any appreciable degree the material which has been measured. The surveyor, for example, who measures the area of a field makes very little impression upon the soil over which he passes. A physician who measures the weight of an infant does not thereby increase that weight or diminish it. In the same way the psychologist who applies a Mentimeter test to a filing clerk, does not by that act increase the efficiency of that clerk. Measurements, of themselves, are of no value. Something must be done about the result which is obtained or all of the expense in time and money is of no avail.