-------------------- = ---- = +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 relatio