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Question Video: Finding Spearman's Correlation Coefficient From a Given Table of Ordinal Data Involving Tied Ranks Mathematics

The following table represents the relation between the results of employees’ appraisals this year and last year. Find the Spearman′s correlation coefficient between the results of the last year and current year.

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Video Transcript

The following table represents the relation between the results of employees’ appraisals this year and last year. Find the Spearman′s correlation coefficient between the results of the last year and current year.

We can see from the table that for the five employees, there are different appraisal results for last year and this year. For example, employee A met expectations last year and exceeded expectations this year. We also see from the table that there are two other appraisal results: that of exceptional performance and needs improvement. As each of the five employees has two appraisal results, we can say that each of the data points is bivariate; that is, it depends on two variables. And as the data is set up this way, we can calculate Spearman′s correlation coefficient. This coefficient gives us a numerical measure of the similarity not of the data itself, but rather of what′s called the rank of the data relative to other values.

To understand this further, we will add some extra rows to our table. Firstly, we have a rank representing last year′s appraisals and secondly a rank of this year′s appraisals. When we talk about rank, it means applying a number to represent the different results in our original data table. We will let lower ranks one, two, and so on correspond to lower, or worse, employee appraisal results. In last year′s results, there was one example, employee B, who needed improvement. As this is the lowest appraisal result, we will assign it a rank of one.

The next lowest appraisal was meets expectations, and this was achieved by employee A and employee D. Instead of assigning both of these a rank of two or assigning the ranks of two and three, what we′ll do is calculate the average of ranks two and three. This is equal to 2.5, so we assign employee A and employee D a rank of 2.5. The next lowest result is exceeds expectations. This was achieved by employee E, and we will therefore assign them a rank of four. Finally, employee C attained an appraisal result of exceptional, so we will assign them a rank of five.

We can then repeat this process for this year′s appraisals. Employee D had the lowest appraisal result of needs improvement. Next, we have employee B who met expectations. We then have employee A and employee E who exceeded expectations. This time we find the average of three and four, giving both of these a rank of 3.5. Employee C once again had the highest performance and therefore has a rank of five.

As already mentioned, when it comes to Spearman′s correlation coefficient, this uses the rankings and not the original data. And this equation is as follows. It is equal to one minus six multiplied by the sum of 𝑑 sub 𝑖 squared divided by 𝑛 multiplied by 𝑛 squared minus one, where 𝑑 sub 𝑖 represents the difference between individual ranks within one column of data, and the coefficient tells us how closely the ranks of one row agree with those of the other.

We will now add another row to our table to calculate 𝑑 sub 𝑖 where this is equal to the rank from last year minus the rank from this year. For our first column, we have 2.5 minus 3.5, which is equal to negative one. Next, we have one minus two, which is also equal to negative one. Repeating this for the other three columns gives us values of zero, 1.5, and 0.5. At this stage, we note that it′s the values of 𝑑 sub 𝑖 squared that we need to sum. We will therefore add a further row to calculate the corresponding values of 𝑑 sub 𝑖 squared. Squaring negative one gives us one. Zero squared is zero. 1.5 squared is 2.25. And 0.5 squared is 0.25.

We can now find the sum of these five values. One plus one plus zero plus 2.25 plus 0.25 is equal to 4.5. We can now substitute this value into our formula together with the value of 𝑛 of five as there are five employees. 𝑟 sub 𝑠 is therefore equal to one minus six multiplied by 4.5 divided by five multiplied by five squared minus one, and this is equal to one minus 27 over 120, which gives us an exact value of 0.775. The Spearman′s correlation coefficient between the results of last year and the current year is therefore equal to 0.775.

Whilst it is not required for this question, it is worth noting that Spearman′s correlation coefficient can take a value from negative one to positive one, where a value of positive one indicates a perfect association of ranks, a value of zero indicates no association between ranks, and a value of negative one indicates a perfect negative association of ranks. In summary, the closer 𝑟 sub 𝑠 is to zero, the weaker the association between the ranks. And in the context of this question, a value of 0.775 indicates a relatively strong association between this year and last year′s performance.

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