=================================================================== |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: