# The correlation coefficient is a quantitative measure of the

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```							The correlation coefficient is a quantitative measure of the strength of the linear relationship
between two variables.

The sample correlation coefficient:

SS xy
r
SS x SS y

where:

SS x   ( x  x ) 2                                  (sum of squares for X)

SS y  ( y  y) 2                                    (sum of squares for Y)

SSxy  ( x  x )( y  y)                      (sum of squares for XY)

Correlation coefficient is a numerical value in the range –1 to +1. If r<0 there is a
negative relationship between two variables, if r>0 there is a positive relationship between
two variables.

Range of value (absolute value):

0,2 – 0,4             low correlation
0,4 – 0,7             moderate correlation
0,7 – 0,9             strong correlation
> 0,9                 very strong correlation

EXAMPLE

Assume a real-estate developer is interested in determining the relationship between
family income (X, in thousand of dollars) of the local resident and the square footage of their
homes (Y, in hundreds of square feet). A random sample of ten families is obtained with the
following results:

X         22       26       45      37         28       50     56      34       60       40
Y         16       17       26      24         22       21     32      18       30       20
Scatter diagram plots a series of X-Y data pairs in two-dimensional space:

35

Square footage (Y, in
30
25

hundreds)
20
15
10
5
0
0     20       40       60     80
Income (X, in thousands)

Our calculations for correlation coefficient can be done as follows:

Family         x        y                    x  x y  y ( x  x ) 2 ( y  y ) 2 ( x  x )( y  y )
1          22       16                             -17,8      -6,6          316,84          43,56   117,48
2          26       17                             -13,8      -5,6          190,44          31,36   77,28
3          45       26                             5,2         3,4           27,04          11,56   17,68
4          37       24                             -2,8        1,4           7,84           1,96    -3,92
5          28       22                             -11,8      -0,6          139,24          0,36     7,08
6          50       21                             10,2       -1,6          104,04          2,56    -16,32
7          56       32                             16,2        9,4          262,44          88,36   152,28
8          34       18                             -5,8       -4,6           33,64          21,16   26,68
9          60       30                             20,2        7,4          408,04          54,76   149,48
10          40       20                             0,2        -2,6           0,04           6,76    -0,52
Total        398      226                                                     1489,60     262,40      527,20
Mean         39,8     22,6                            Square root             38,595          16,2

Thus we have:

527 ,2       527 ,2
r                          0,843
38,595  16 ,2 625 ,2

There is a strong positive relationship between the family income and the square footage.
Larger incomes (X) are associated with larger home sizes (Y).

```
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