Percentile scaling
Direct Calculation of Percentiles
A child who earns a certain score on
a test can be assigned a percentile rank (PR) of 27, 42 or 77, say, depending
upon his position in the score distribution. Percentile rank locates a child on
a scale of 100 and tells us immediately what proportion of the group has
achieved scores lower than he. Moreover, when a child has taken several tests,
a comparison of his PR's provides measures of relative achievement, and these
may be combined into a final total score. As a method of scaling test scores,
PR's have the practical advantage of being readily calculated and easily
understood. But the percentile scale also possesses marked disadvantages which
limit its usefulness.
Percentile scales assume
that the difference between a rank of 10 and a rank of 20 is the same as the
difference between a rank of 40 and a rank of 50, namely, that percentile
differences are equal throughout the scale. This assumption of equal percentile
units holds strictly only when the distribution of scores is rectangular in
shape; it does not hold when the distribution is bell-shaped, or approximately
normal. Figure 60 shows graphically why this is true. In the diagram we have a
rectangular distribution and a normal curve of the same area plotted over it.
When the rectangle is divided into 5 equal segments, the areas of the small
rectangles are all the same (20%) and the distances from 0 to 20, 20 to 40, 40
to 60, 60 to 80,
and 80 to 100 are all equal. These percentiles, P20, P40, etc., have been marked off along the top of the rectangle.
Now let us compare the
distances along the base line of the normal curve when these are determined by
successive 20% slices of area. These base-line intervals can be found in the
following
way. From Table A we read that the 30% of area to the left of the mean extends
to -.84
. The first 20% of a
normal distribution, therefore, falls between -3.00 and, -.84: covers a distance of
2.16 along the base line. The second 20% (P20
to P40) lies between -.84 and -.25 (since -.25 is at a distance of 10% from the mean); and
covers a distance of .59 along the base line. The third 20% (P40
and P60) lies between -.25 and.25: straddles the
mean
and covers .50 on the base line. The fourth and fifth 20%'s
occupy the same relative positions in the upper half of the curve as the second
and first 20%'s occupy in the lower half of the curve. To summarize:
First 20% of area covers a distance of 2.16
Second 20% of area covers a distance of .59
Third 20% of area covers a distance of .50
Fourth 20% of area covers a distance of .59
Fifth 20% of area covers a distance of 2.16
It
is clear (1) that intervals along the base line from the extreme left end (0
to P20, P20
to P40, etc.) to the extreme right end of the normal curve
are not equal when determined by successive 20% slices of area; and (2) that
inequalities are relatively greater at the two ends of the distribution, so
that the two end fifths are 4 times as long as the middle one.
Distributions
of raw scores are rarely if ever rectangular in form. Hence equal percent's of N (area) cannot be taken to
represent equal increments of achievement and the percentile scale
does
not progress by equal. steps. Between Ql and Q3, however, equal percent's of
area are more nearly equally spaced along the base line (see Fig. 1), so that
the PR's of a child in two.
or
more tests may be safely combined or averaged if they fall within these limits.
But high and low PR's (above 75 and below 25) should be combined, if at all,
with full knowledge of their limitations.
FIG.1 To illustrate the position of the same five percentile in rectangular and normal distribution
Percentile
In case of median, total frequency is divided into two
equal parts; in case of quartiles, total frequency is divided into four equal
parts; similarly in case of percentiles, total frequency is divided into 100
equal parts. We have learned that the median is that point in a frequency
distribution below which lie 50% of the measures or scores; and that Q1 and Q3
mark points in the distribution below which lie, respectively, 25% and 75% of
the measures or scores
Using the same method by which the median and the
quartiles were found, we may compute points below which lie 10%, 43%, 85%, or
any percent of the scores. These points are called percentiles, and are
designated, in general, by the symbol PP, the p referring to the
percentage of cases below the given value
Percentile Ranks from The Normal Curve
Differences between
points on the percentile scale may be allowed for by proper spacing when scores
are to be represented by a profile. Table 47 shows the PR’s of
various CT scores in the normal curve and their
scores and as T scores
PR Sigma Score T Score
99
2.33 73
95 1.64 66
90 1.28 63
80 .84 58
70 .52 55
60 .25 53
50 .00 50
40 -.25 47
30 -.52 45
20 -.84 42
10 -1.28 37
5 -1.64 34
1 -2.33 27
corresponding T scores.
Unequal gaps between PR's when compared with T score intervals at the-middle
and the ends of the scale are clearly apparent. Figure 61 shows graphically the
performance of a twelfth-grade boy on the Differential Aptitude Tests. Percentile
ranks on the chart have been marked off in such a way (larger at extremes,
smaller at middle) as to accord with the normal form of distribution. T scores
(along the Y axis) may be compared with PR's. Note that James is very high in
the mechanical and spatial tests, average in numerical and abstract, and low in
verbal and spelling.
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