### Video Transcript

The following table represents the relation between sale and profit for six models of televisions. Find Spearman’s correlation coefficient between television sale and profit. Round your answer to three decimal places.

The first row of our table tells us the cost of these six different models of televisions. The second row shows the corresponding profit for each sale of these models. We’re asked to find the Spearman’s correlation coefficient between sale and profit. Interestingly, this coefficient doesn’t directly have to do with these data values. Instead, it describes how well the ranking of television sale price correlates with the ranking of television profit per sale.

Our first step then in solving for the Spearman’s correlation coefficient is to add two rows to our table where the first row, we’ve called it 𝑅 sub 𝑠, describes the ranking of the television sale values and the second, 𝑅 sub 𝑝, ranks the television profit values. When we talk about rankings, we mean that in each of our original data rows, we see that in a given row, some values are lower and some are higher. As we rank these values within a row relative to one another, we’ll say that the lower values have the lower ranking.

So, for example, in our first row, we see that the lowest value is 100. And therefore this will have a ranking of one. Then the next lowest value is 400. So this will get a ranking of two. Before we continue on though, note that we could’ve chosen the opposite way to rank these values. That is, we could’ve said low values of television sale correspond to higher rankings. And this would’ve been fine to do so long as we took the same approach when we ranked our television profits. But in any case, we’ve chosen the lowest values to have the lowest rankings.

Continuing to look at the first row of our data values, after 400 the next lowest value is 480. So this will have a ranking of three. Next comes 500, which will therefore have a ranking of four. And then 550 and 600 will have a ranking of five and six, respectively. And we’ll now follow the same process for our second row, the row of television profit values. The lowest value here is 90, so that will get a ranking of one. Then the next lowest is 200. So that has a ranking of two. Then comes 250 with a ranking of three, and 300 which has a ranking of four.

Now notice that we then have two identical values of 400. These would represent the fifth and sixth highest values. But since they’re the same, rather than assign them at different rankings, we can take the average of the rankings they would have, five and six, and assign them both that average ranking of 5.5. It’s the values in these last two rows of our table that Spearman’s correlation coefficient addresses. In general, the closer 𝑅 sub 𝑠 and 𝑅 sub 𝑝 are for all the different points in our table, the more nearly the correlation coefficient will be equal to positive one.

The next metric we want to calculate then is the difference between 𝑅 sub 𝑠 and 𝑅 sub 𝑝. We’ll call this 𝑑 sub 𝑖. For our first data point, four minus four gives us zero. Next, we have six minus 5.5, or 0.5. Then five minus 5.5 is negative 0.5, and then one minus one, which is zero; three minus three, which is zero; and two minus two, zero. As a final step for our table, we’ll add a row where we square these difference values. In other words, we calculate 𝑑 sub 𝑖 squared. This ensures that all of our results are nonnegative.

For our first point, zero squared is zero. 0.5 squared is 0.25. Negative 0.5 squared gives us that same result. And then we have zero squared, zero squared, and zero squared, resulting in zeros. We can now clear some space at the top of our screen and write down the mathematical equation for Spearman’s correlation coefficient. It’s equal to one minus six times the sum of 𝑑 sub 𝑖 squared all divided by 𝑛, where 𝑛 is the number of data points in our data set, times 𝑛 squared minus one.

To calculate this coefficient then, there are two things we need to know. We’ll need to know the sum of all of our different values squared. And we’ll also need to know the total number of points in our data set. Since we’ve calculated 𝑑 sub 𝑖 squared for each of the points individually, we can solve for this sum by just adding together all the values in this last row of our table. All of these values are zero except for two values which are 0.25, so their sum is 0.5. Next, regarding the number of points in our data set, we can count one, two, three, four, five, six such points. This tells us that for us, 𝑛 is equal to six.

Substituting these values into our Spearman’s correlation coefficient equation, note that a factor of six in numerator and denominator will cancel out. And since six squared is 36, we can simplify this expression to be one minus 0.5 over 36 minus one, or 35. Calculating this value gives us a result of 0.98571 and so on. We can recall, though, that our question statement asked us to give an answer rounded to three decimal places. So if we look at the fourth decimal place, we recognize that this is greater than or equal to five, which means that the value in the third decimal place will be rounded up.

To three decimal places then, this is our answer. The Spearman correlation coefficient for these data is 0.986. Because this result is so close to one, we can say that the ranking of a television sale price agrees very nearly with its corresponding profit ranking. This is what the Spearman correlation coefficient tells us.