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Coefficient, r
                                           Strength of Association  Positive    Negative
                                           Small                     .1 to .3  –0.1 to –0.3
                                           Medium                    .3 to .5  –0.3 to –0.5
                                           Large                    .5 to 1.0  –0.5 to –1.0

              Note that the strength of the association of the variables depends on what you are measuring and sample sizes.

              Example 1: The ages and incomes of five people are given below. Calculate the Pearson Coefficient. What does it depict?
                                                      Age (x)     Income (y)
                                                        20           2000
                                                        30          40000
                                                        40          49000
                                                        50          61000
                                                        60          75000
              Solution: To calculate the coefficient, we need to calculate the following values.
                               x             y              xy              x 2                y 2
                               20          2000           40000            400              4000000
                               30          4500          135000            900             20250000
                               40          5700          228000           1600             32490000
                               50          6800          340000           2500             46240000
                               60          8000          480000           3600             64000000
                                                                                           2
                                                                           2
                            ∑x=200       ∑y=27000      ∑xy=1223000      ∑x =9000         ∑y =166980000
              Putting the values in the formula,
                             5 (1223000) – (200) (27000)
              r  =     [(5) 9000 – (200) ] [(5) (166980000) – (27000) 2
                                    2

                       715000
                  =
                    727667.5065
                  =   0.98
              This value represents a positive strong relationship between the two variables. As the age of a person increases, the
              person’s salary also goes up!
              Example 2: Amit is a model student good in both academics and sports. However, after some time, he reduced his
              sports activity and thus observed that he is scoring lesser marks in tests. To investigate his hypothesis, he noted down
              how he scored in his tests, based on how many hours he played any sport before appearing in the school tests. He
              gathered this data to check the correlation between hours of sports he played and his tests scores. He thus, calculated
              the Pearson Correlation Coefficient = 0.95. Explain what this value means.

              Solution: 0.95 shows a positive and strong strength of association between the two variables. This means that Amit scored
              better marks if he played before his exams. If Amit reduced, his playing hours, the marks he scored also reduced.

              Assumptions
              There are four assumptions for Pearson's correlation coefficient which are as follows. If any of these four requirements
              are not met, analysis of data using Pearson's correlation coefficient might not yield a valid result:

              1.   The  data  type  of  the  two  variables  should  be  continuous.  Examples  of  such  continuous  variables  include
                  height (measured in feet and inches), temperature (measured in °C), salary (measured in INR), study time (measured
                  in hours), intelligence (measured through IQ score), exam performance (measured from 0 to 100), sales (measured
                  in number of transactions every month), etc.

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