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Question Video: Identifying the Correct Formula for Spearman’s Rank Correlation Coefficient Mathematics

Which of the following is the formula of Spearman’s rank correlation coefficient? [A] 1 βˆ’ (6βˆ‘π‘‘Β²/(𝑛² βˆ’ 1)) [B] 1 βˆ’ (6βˆ‘π‘‘Β²/𝑛(𝑛³ βˆ’ 1)) [C] 6βˆ‘π‘‘Β²/𝑛(𝑛² βˆ’ 1) [D] 1 βˆ’ (6βˆ‘π‘‘Β²/𝑛(𝑛² βˆ’ 1)) [E] 1 βˆ’ (βˆ‘π‘‘Β²/𝑛(𝑛² βˆ’ 1))

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Video Transcript

Which of the following is the formula of Spearman’s rank correlation coefficient?

And then we have five possible options to choose from. Remember, Spearman’s rank correlation coefficient is a test that examines the degree to which two data sets are correlated, if at all. Rather than just trying to infer this information from a scatter graph, Spearman’s rank correlation coefficient is a numerical value which tells us the degree of correlation or, indeed, noncorrelation.

So let’s suppose we have paired sets of data. After arranging the data sets in a relevant table, we begin by ranking each data point. For instance, imagine we had 10 pairs of data. We will take the first element and rank by assigning each one a number from one to 10. If there are, however, two or more values that are equal, we average those ranks. We then repeat this process with the second set of data points. Once we have that, we can find the difference between the ranks, and we call that 𝑑. Let’s say we’ve ranked the first data set, and we call that π‘Ÿ sub one, and then we’ve ranked the second and call that π‘Ÿ sub two. The difference is π‘Ÿ sub one minus π‘Ÿ sub two.

The next step is to square our values for 𝑑 and find the sum. And we use βˆ‘ notation to represent this. We now notice that every single formula that we’ve been given has the βˆ‘ of 𝑑 squared in it. So which is it? Well, one of the formulae we can use to calculate Spearman’s rank correlation coefficient π‘Ÿ for 𝑛 pairs of data points is one minus six times the βˆ‘ of 𝑑 squared over 𝑛 cubed minus 𝑛. But in fact this formula does not look like any of the formulae we’ve been given. We can, however, factorize the denominator of this fraction. Taking out a common factor of 𝑛, we rewrite it as 𝑛 times 𝑛 squared minus one.

And so the formula is π‘Ÿ equals one minus six times the βˆ‘ of 𝑑 squared over 𝑛 times 𝑛 squared minus one. We notice then that this is option (D).

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