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PDFSum = 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:
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===============================
|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 | | | |
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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:
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============ |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|
