======================================================
STEP No.|VALUE or SAMPLE|RANGE OF STEP |LENGTH OF STEP
--------+---------------+--------------+--------------
0. . . .| 0 | 0- .91[32] | .91
1. . . .| 1.83 | .92-2.21 | 1.30
2. . . .| 2.60 |2.22-3.14 | .93
3. . . .| 3.69 |3.15-4.21 | 1.07
4. . . .| 4.74 |4.22-5.29 | 1.08
5. . . .| 5.85 |5.30-6.30 | 1.00
6. . . .| 6.75 |6.30-7.23 | .93
7. . . .| 7.72 |7.24-8.05 | .81
8. . . .| 8.38 |8.05-8.87 | .82
9. . . .| 9.37 |8.88- |
======================================================
From the above table we find that step 3 has a length of 1.07 units. If
we count out 35 of the 39 papers, or, in other words, if we pass upward
into the step .90 of the total distance (1.07 units), we shall arrive at
a point .96 units (.90 × 1.07 = .96) above the lower limit of step 3,
which we find from the table is 3.15. Adding .96 to 3.15 gives 4.11 as
the median point of this eighth grade distribution.
The median and the percentiles of any distribution of scores on the
Hillegas scale may be determined in a manner similar to that illustrated
above, if the scores are assigned to the individual papers according to
the directions outlined above.
A similar method of calculation is employed in discovering the limits
within which the middle fifty per cent of the cases fall. It often seems
fairer to ask, after the upper twenty-five per cent of the children who
would probably do successful work even without very adequate teaching
have been eliminated, and the lower twenty-five per cent who are
possibly so lacking in capacity that teaching may not be thought to
affect them very largely have been left out of consideration, what is
the achievement of the middle fifty per cent. To measure this
achievement it is necessary to have the whole distribution and to count
off twenty-five per cent, counting in from the upper end, and then
twenty-five per cent, counting in from the lower end of the
distribution. The points found can then be used in a statement in which
the limits within which the middle fifty per cent of the cases fall.
Using the same figures that are given above for scores in English
composition, the lower limit is 2.64 and the limit which marks the point
above which the upper twenty-five per cent of the cases are to be found
is 5.08. The limits, therefore, within which the middle fifty per cent
of the cases fall are from 2.64 to 5.08.
It is desirable to measure the relationship existing between the
achievements (or other traits) of groups. In order to express such
relationship in a single figure the coefficient or correlation is used.
This measure appears frequently in the literature of education and will
be briefly explained. The formula for finding the coefficient of
correlation can be understood from examples of its application.
Let us suppose a group of seven individuals whose scores in terms of
problems solved correctly and of words spelled correctly are as
follows:[33]
======================================
INDIVIDUALS|No. OF |No. OF WORDS
MEASURED |PROBLEMS|SPELLED CORRECTLY
CORRECTLY | |
-----------+--------+-----------------
A | 1 | 2
B | 2 | 4
C | 3 | 6
D | 4 | 8
E | 5 | 10
F | 6 | 12
G | 7 | 14
======================================
From such distributions it would appear that as individuals increase in
achievement in one field they increase correspondingly in the other. If
one is below or above the average in achievement in one field, he is
below or above and in the same degree in the other field. This sort of
positive relationship (going together) is expressed by a coefficient of
+1. The formula is expressed as follows:
(Sum x · y)
r = ------------------------------
(sqrt(Sum x^2))(sqrt(Sum y^2))
Here _r_ = coefficient of correlation.
_x_ = deviations from average score in arithmetic (or difference between
score made and average score).
_y_ = deviations from average score in spelling.
Sum = is the sign commonly used to indicate the algebraic sum (_i.e._
the difference between the sum of the minus quantities and the plus
quantities).
_x · y _= products of deviation in one trait multiplied by deviation in
the other trait with appropriate sign.
Applying the formula we find:
===================================================================
|ARITH-| | | SPEL- | | | |
|METIC | x | x^2 | LING | y | y^2 | x·y |
--+------+---+------------+-------+---+-------------+-------------+
A | 1|-3 | 9| 2|-6 | 36| +18|
B | 2|-2 | 4| 4|-4 | 16| +8|
C | 3|-1 | 1| 6|-2 | 4| +2|
D | 4| 0 | 0| 8| 0 | | |
E | 5|+1 | 1| 10|+2 | 4| +2|
F | 6|+2 | 4| 12|+4 | 16| +8|
G | 7|+3 | 9| 14|+6 | 36| +18|
| ___| | __| ___| | ___| __|
| 7 |28| |Sum x^2 = 28| 7 |56| |Sum y^2 = 112|Sum x·y = +56|
|Av. =4| | |Av. =8 | | | |
===================================================================
Sum x · y +56 +56
r = ---------------------------- = --------------------- = ---- = +1
(sqrt(Sum x^2)(sqrt(Sum y^2) (sqrt(28))(sqrt(112)) 56
If instead of achievement in one field being positively related (going
together) in the highest possible degree, these individuals show the
opposite type of relationship, _i.e.,_ the maximum negative relationship
(this might be expressed as opposition--a place above the average in one
achievement going with a correspondingly great deviation below the
average in the other achievement), then our coefficient becomes -1.
Applying the formula:
===================================================================
|ARITH-| | | SPEL- | | | |
|METIC | x | x^2 | LING | y | y^2 | x*y |
--+------+---+------------+-------+---+-------------+-------------+
A | 1|-3 | 9| 14|+6 | 36| -18|
B | 2|-2 | 4| 12|+4 | 16| -8|
C | 3|-1 | 2| 10|+2 | 4| -2|
D | 4| 0 | | 8| 0 | | |
E | 5|+1 | 2| 6|-2 | 4| -2|
F | 6|+2 | 4| 4|-4 | 16| -8|
G | 7|+3 | 9| 2|-6 | 36| -18|
| ___| | __| ___| | ___| __|
| 7 |28| |Sum x^2 = 28| 7 |56| |Sum y^2 = 112|Sum x·y = -56|
|Av. =4| | |Av. =8 | | | |
===================================================================
It will be observed that in this case each plus deviation in one
achievement is accompanied by a minus deviation for the other trait;
hence, all of the products of _x_ and _y_ are minus quantities. (A plus
quantity multiplied by a plus quantity or a minus quantity multiplied by
a minus quantity gives us a plus quantity as the product, while a plus
quantity multiplied by a minus quantity gives us a minus quantity as the
product.)
(Sum x·y) -56 -56
r = ------------------------------ = ------------------- = ---- = -1.
(sqrt(Sum x^2))(sqrt(Sum y^2)) (sqrt(28)sqrt(112)) = 56
If there is no relationship indicated by the measures of achievements
which we have found, then the coefficient of correlation becomes 0. A
distribution of scores which suggests no relationship is as follows:
=================================================================
|ARITH- | | | | | |
|METIC | x | x^2 |Spelling | y | y^2 | x.y
--+-------+----+-----------+---------+----+-------------+--------
| | | | | | | - +
A | 2 | -2 | 4 | 12 | +4 | 16 | -8 +6
B | 1 | -3 | 9 | 8 | 0 | | 0 +4
C | 4 | 0 | | 2 | -6 | 36 | 0 +4
D | 5 | +1 | 1 | 14 | +6 | 36 | -6
E | 3 | -1 | 1 | 4 | -4 | 16 | -14 +14
F | 7 | +3 | 9 | 6 | -2 | 4 |
G | 6 | +2 | 4 | 10 | +2 | 4 |
| ____| | | ___ | | |
| |28 | |Sum x^2=28 | 7|56 | | Sum y^2=112 | x·y=0
| AV.=4 | | | AV.=8 | | |
===================================================================
(Sum x·y) 0
r = ---------------------------- = ------------------- = 0.
(sqrt(Sum x^2)sqrt(Sum y^2)) (sqrt(28)sqrt(112))
In a similar manner, when the relationship is largely positive as would
be indicated by a displacement of each score in the series by one step
from the arrangement which gives a +1 coefficient, the coefficient will
approach unity in value.
===============================================================
ARITHMETIC| x | x^2 |SPELLING| y | y^2 |
---+------+----+-----------+--------+----+------------+--------
A |1 | -3 |9 |4 | -4 | 16 |+ 12
B |2 | -2 |4 |2 | -6 | 36 |+ 12
C |3 | -1 |1 |8 | 0 | |+ 4
D |4 | 0 | |6 | -2 | 4 |+ 4
E |5 | +1 |1 |12 | +4 | 16 |+ 18
F |6 | +2 |4 |10 | +2 | 4 |Sx·y=50
G |7 | +3 |9 |14 | +6 | 36 |
|Av. =4| |Sum x^2 =28|Av. = 8 | |Sum y^2= 112|
===============================================================
Sum x·y +50
r= -------------------------- = ---- = +.89.
sqrt(Sum x^2)sqrt(Sum y^2) 56
Other illustrations might be given to show how the coefficient varies
from + 1, the measure of the highest positive relationship (going
together) through 0 to -1, the measure of the largest negative
relationship (opposition). A relationship between traits which we
measure as high as +.50 is to be thought of as quite significant. It is
seldom that we get a positive relationship as large as +.50 when we
correlate the achievements of children in school work. A relationship
measured by a coefficient of ±.15 may _not_ be considered to indicate
any considerable positive or negative relationship. The fact that
relationships among the achievements of children in school subjects vary
from +.20 to +.60 is a clear indication of the fact that abilities of
children are variable, or, in other words, achievement in one subject
does not carry with it an _exactly corresponding_ great or little
achievement in another subject. That there is some positive
relationship, _i.e.,_ that able pupils tend on the whole to show
all-round ability and the less able or weak in one subject _tend_ to
show similar lack of strength in other subjects, is also indicated by
these positive coefficients.