|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         |