The greatest danger to be guarded against in considering all scales as continuous rather than discrete, is that careless thinkers may refine their calculations far beyond the accuracy which their original measurements would warrant.  One should be very careful not to make such unjustifiable refinements in his statement of results as are often made by young pupils when they multiply the diameter of a circle, which has been measured only to the nearest inch, by 3.1416 in order to find the circumference.  Even in the ordinary calculation of the average point of a series of measures of length, the amateur is sometimes tempted, when the number of measures in the series is not contained an even number of times in the sum of their values, to carry the quotient out to a larger number of decimal places than the original measures would justify.  Final results should usually not be refined far beyond the accuracy of the original measures.

It is of utmost importance in calculating medians and other measures of a distribution to keep constantly in mind the significance of each step on the scale.  If the scale consists of tasks to be done or problems to be solved, then “doing 1 task correctly” means, when considered as part of a continuous scale, anywhere from doing 1.0 up to doing 2.0 tasks.  A child receives credit for “2 problems correct” whether he has just barely solved 2.0 problems or has just barely fallen short of solving 3.0 problems.  If, however, the scale consists of a series of productions graduated in quality from very poor to very good, with which series other productions of the same sort are to be compared, then each sample on the scale stands at the middle of its “step” rather than at the beginning.

The second kind of scale described in the foregoing paragraph may be designated as “scales for the quality of products,” while the other variety may be called “scales for magnitude of achievement.”  In the one case, the child makes the best production he can and measures its quality by comparing it with similar products of known quality on the scale.  Composition, handwriting, and drawing scales are good examples of scales for quality of products.  In the other case, the scales are placed in the hands of the child at the very beginning, and the magnitude of his achievement is measured by the difficulty or number of tasks accomplished successfully in a given time.  Spelling, arithmetic, reading, language, geography, and history tests are examples of scales for quantity of achievement.

Scores tend to be more accurate on the scales for magnitude of achievement, because the judgment of the examiner is likely to be more accurate in deciding whether a response is correct or incorrect than it is in deciding how much quality a given product contains.  This does not furnish an excuse for failing to employ the quality-of-products scales, however, for the qualities they measure are not measurable in terms of the magnitude of tasks performed.  The fact appears, however, that the method of employing the quality-of-products scales is “by comparison” (of child’s production with samples reproduced on the scale), while the method of employing the magnitude-of-achievement scales is “by performance” (of child on tasks of known difficulty).