═════════╤═════════╤═════════╤═════════╤═════════╤═════════ 1 │ 2 │ 3 │ 4 │ 5 │ 6 ─────────┼─────────┼─────────┼─────────┼─────────┼───────── │ A │ B │A squared│B squared│A times B ─────────┼─────────┼─────────┼─────────┼─────────┼───────── 1901 │ −3│ −5│ 9│ 25│ +15 1902 │ −1│ +1│ 1│ 1│ −1 1903 │ +2│ +3│ 4│ 9│ +6 1904 │ +2│ +1│ 4│ 1│ +2 Sum │ │ │ 18│ 36│ +22 ─────────┴─────────┴─────────┴─────────┴─────────┴───────── The standard deviation of A is the square root of the sum of the A squares, or 18, divided by 4. The square root of 18 divided by 4 is 2.1. Standard deviation of B, in like manner, is 3. The sum of AB divided by 4, or +22 divided by 4, equals +5.5. The correlation coefficient is +5.5 divided by the standard deviation of A multiplied by the standard deviation of B, or 5.5 divided by 6.3, which gives +.87. A correlation coefficient of .87 is very high, perfect correlation being 1. Correlation over .5 is considered fairly good, especially if there is a long list (fifty or more) of figures in each series. The formula for determining A in terms of B is: A equals r((σ_{a})/(σ_{b}))B In this formula, r is the correlation coefficient and σa is the standard deviation of A, and σb is the standard deviation of B. Substituting for the specific problem, we get: A equals .87((2.1)/(3.0))B or A equals .609 B When B is −5 we would expect A to be 3.05; when B is +1 we would expect A to be +.609; when B is +3, we would expect A to be 1.827. Suppose now, in addition, that there are three series: A, B and C, and that the object is to determine A in terms of B and C. The three series stand: ════╤════════════╤════════════╤════════════ │ A │ B │ C ────┼────────────┼────────────┼──────────── 1901│ −3│ −5│ +2 1902│ −1│ +1│ +3 1903│ +2│ +3│ −3 1904│ +2│ +1│ −2 ────┴────────────┴────────────┴──────────── We already know that the standard deviation of A is 2.1, and of B is 3.0, and that the correlation coefficient between A and B is +.87. Using the customary method, we find that the standard deviation of C is 2.55 and that the correlation coefficient of A and C is −.89, and of B and C −.59. To find A in terms of B and C, we use the following formula: A equals ((r_{ab} − r_{ac}r_{bc})/(1 − r^2_{bc}))((σ_{a})/(σ_{b}))B + ((r_{ac} − r_{ab}r_{bc})/(1 − r^2_{bc}))((σ_{a})/(σ_{c}))C