Comparing Variation in Different Data Sets In...

Question

Comparing Variation in Different Data Sets In Exercises 45–50, find the coefficient of variation for each of the two data sets. Then compare the results.

Annual Salaries Sample annual salaries (in thousands of dollars) for entry level architects in Denver, CO, and Los Angeles, CA, are listed.

Table comparing annual salaries (in thousands) for entry-level architects in Denver and Los Angeles.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The following are sample test scores out of 100 for two different classes. Calculate the coefficient of variation for each class and determine which class has more relative variation. Awesome. So it appears for this particular problem, we're asked to solve for three separate answers. Our first two answers are we're asked to calculate the coefficient of variation for Class A and Class B. Those are our first two calculations that we're asked to solve for for our final answers. And our last, our third and last final answer we're asked to solve for is we're asked to determine. Whether Class A or Class B has more relative variation. So now that we know what we're ultimately trying to solve for, we need to note and recall once again that we are trying to determine which class has more relative variation. And in order to do that, we need to calculate the coefficient of variation, CV for short, for each class. And in order to do that, our first major step to solve for this problem is we need to recall and use our coefficient of variation formula, which states that CV or coefficient of variation. is equal to our coefficient of variation is equal to the standard deviation divided by the mean multiplied by 100 because we want to convert our percentage. We need to convert to a percentage, so we need to take our decimal because that's our standard deviation divided by the mean will give us a decimal, and then we need to multiply it by 100 to convert it to a percentage. So, moving right along here, I'll call it CA for short, but let's focus on our Class A scores first. So we're told that our Class A scores are 72. 75, 78, 80, and 85. So as you can see, I'll highlight it in the prom itself. So we're told that Class A once again is test scores are 72, 75, 78, 80, and 85. And for Class B, which I'll call CB for short, our Class B scores are as follows. Which are 68, 70, 75. 80 and 87. Awesome. So moving right along here. So now, it's important to note That our next step is we need to confine the mean values. So we need to find the mean of our Class A scores and our class B scores. So, focusing our attention on solving for the mean for the Class A scores first, which we'll call MA for short. MA is gonna be equal to 72 + 75 plus 78 plus 80 + 85. All divided by there's 5 values total. So when we plug that into our calculator, we will find that it's simply equal to 78. So now we need to find the mean. For our class B scores, which we'll call this MB and Or before we do that, before we calculate the mean for our Class B scores, let's just take a moment to do the standard deviation of Class A first. So solving for the standard deviation. We can create a table to help us solve for the standard deviation. So we can say that SD is equal to the square root of 98 divided by 5 minus 1, which will mean that it's approximately equal to 4.94974746831. And we can get the value of 98 from our data table, which I've gone ahead and used an Excel spreadsheet to help us compile our data for the standard deviation, which once again is denoted as SD. So for our different values for class A, we have 72 through 85. And then our values for x minus X bar will be in our middle column. And then finally, X minus X to the power of 2 is going to be on our far right column. And when we sum those values together, we get our value of 98, and that's what we use for our standard deviation that like the plug into our standard deviation equation to solve for our answer for the standard deviation. So, With that said, now, Our next step now is we need to calculate the coefficient of variation for class A. So we'll call this CVA and CVA is equal to parentheses 4.9. 497-474-6831. Divided by 78. Multiplied by 100, and when we plug that into our calculator. And we round to two decial places, we will find that it's approximately equal to 6.35%. Awesome. So now switching gears to solve for our Class B scores now. Which, as we should recall, our class B scores are 68. 70 75, 80 and 87. So now we can solve for the mean for our class B scores, which the mean, we'll call it MB is equal to 68 plus 70 + 75 plus 80 + 87. Which there are 5 values total, so we need to divide by 5. So when we plug that into our calculator, we will get a value of 76. And now we need to make sure we create a table for our standard deviation. So when we do that, we will find that our standard deviation. For our Class B scores will be, which our far left column is our Class B scores, which is 68 through 87. Then we have our middle column which is X minus X bar, and then finally we have X minus X squad, which sums to a total of 238. Fantastic. So that will mean that Our standard deviation for our class B scores, which we'll call SDB for short, is equal to the square root of 238 divided by 5 minus 1, which will be approximately equal to 7.71362431027. Fantastic. Now, we can solve for the coefficient of variation for our class B scores, which we call CVB for short, which is gonna be equal to parentheses 7.71362431027. Divided by 76, then don't forget to multiply by 100 to convert to a decimal. So when we round to two decimal places, we will find that it's approximately equal to 10.15%. And that's it. We've officially solved for our first two answers. So to quickly recap, our coefficient of variation for our Class B scores is 10.15%. And our coefficient of variation for our class A scores was 6.35%. So now, to solve for our third and final answer, we need to recall and note that since Class B has a higher coefficient of variation, which means that it is more relative variation, so it has a more relative variation in test scores compared to Class A. So that means when we go to look at our original problem to make our final conclusions, our final set of answers without a doubt, have to be once again. Class A is 6.35%. Class B is 10.15%, and Class B has a higher coefficient of variation than Class A. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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Comparing Variation in Different Data Sets In Exercises 45–50, find the coefficient of variation for each of the two data sets. Then compare the results.
Annual Salaries Sample annual salaries (in thousands of dollars) for entry level architects in Denver, CO, and Los Angeles, CA, are listed.

Key Concepts

Coefficient of Variation

Standard Deviation

Mean

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Comparing Variation in Different Data Sets In Exercises 45–50, find the coefficient of variation for each of the two data sets. Then compare the results.Annual Salaries Sample annual salaries (in thousands of dollars) for entry level architects in Denver, CO, and Los Angeles, CA, are listed.