Large Data Sets from Appendix B. In Exercises 25–28, refer to ...

Question

Large Data Sets from Appendix B. In Exercises 25–28, refer to the indicated data set in Appendix B. Use software or a calculator to find the range, variance, and standard deviation. Express answers using appropriate units, such as “minutes.”

Earthquakes Use the magnitudes (Richter scale) of the 600 earthquakes listed in Data Set 24 “Earthquakes” in Appendix B. In 1989, the San Francisco Bay Area was struck with an earthquake that measured 7.0 on the Richter scale. If we add that value of 7.0 to those listed in the data set, do the measures of variation change much?

Accepted Answer

Welcome back, everyone. In this problem, we want to use the given mammal lifespan data in years to find a range, variance, and a standard deviation. We also want to express our answers using appropriate units such as years. And if we add a new mammal with a lifespan of 65 years to those listed in the data set, do the measures of variation change much? In our data set, the lifespan of mammals, mammals in years are 1215, 1822, 25, 30, 35, 40, 50, and 60. For our answer choices, A says the range variance and standard deviation all increased, indicating a greater spread in the data. B says the mean increased, but the variance and standard deviation remained unchanged. C says the range increased, but the variance and standard deviation decreased while D says there was no significant change in any of the statistical measures. So what are we trying to figure out here? Well, essentially we're trying to figure out if adding a new mammal with a lifespan of 65 years is going to make much of a difference if that 65 years was not in our data set. So to figure that out, we can try to find a range of variance and standard deviation of the old data set and the new data set. That is the one that includes the 65 year old mamma. Let's find the, the, so let's find the statistical measure for each, OK? And then let's see if they are different. Let's try to figure out what the differences are. So let's start with the range, OK? I know for the range, OK. We're gonna keep the old data set on the left, OK. And let's put the new data set in red on the right, OK? Now for the old data set, the range, we know the range is equal to the minimum minus the maximum value, or sorry, the difference between the maximum and minimum value. So that is going to be equal to for the older data set 60 minus 12, which equals 48 years, OK. And now If we think about this for the new data set, the range then is gonna be equal to our new maximum value 65 minus 12, which equals 53 years. With the old data set, it's 48 years. With the new data set, it's 53 years. So that means the range has increased when we add uh our mammal of 65 years, OK? Next, let's move on to the mean. How does the mean change? Well, we know that to find the mean we sum all the data points and divide by the number of points. So for the original data set that's going to be equal to 12 + 15 + 18 + 22 + 25 plus 30 plus 35 + 40 + 50 plus 60 and all of this is gonna be divided by 10. Which gives us a mean age of 30.7 years, OK, for the old data set. Now, for the new data set, OK, it's essentially all of those values, OK. So let me just copy and paste them here. It's gonna be all of these values plus our new value 65, OK? And all of that will be divided by 10. Now when we calculate here, the new mean will be 33.82 years. So what do we notice here? Well, when we oops, this should be divided by 11, my apologies because since we've added a new value, we instead of 10 in the data set, it would be 11. But the 33.82 is correct. So now, when we add 65 to our data set, our mean has also increased. Next, let's try to figure out what happens to the variance and the standard deviation, OK? I know to do that, it might be helpful for us to create some tables, OK? And no. OK, so let's go ahead here. Variance and standard deviation. OK, recall that the variance. is equal in both cases is equal to the sum of the difference between each data point and the mean squared, all divided by N minus 1, OK? So now for the. All the data set. We're gonna take each data point, OK. Find the difference between it and the mean. OK. Square that difference, OK? And then sum all of those squares, OK? And we're gonna do that both for the, for the old data set and the new data set. So now for the old data set, we know we had 1215, 18. 22, 25, 30, 35, 40, uh 50, and 60, OK. So now we're gonna find the values for all of these and then for our new data set, OK OK. We know that essentially it's the same thing. But no, we're just gonna add 65, OK? We have a new value. But again, remember that the the the mean would be different, which is why we have to do a different table, OK? So now, for example, uh, for the old data set, recall that our mean is 30.7. So the difference between 12 and 30.7 is negative 18.7. So the square of that is 349.69. Essentially, we're going to repeat the same process for each data point, OK? For 15, the difference is -15.7 and the square is 246.49. For 18, it's -12.7 and 161.29. For 22, it's -8.7 and 75.69. For 25, it's -5.7 and 32.49. But 30, negative 0.7 and 0.49, for 35, 4.3, and 18.49, for 4, 9.3 and 86.49, I think I'm running out of space here. For 50, it's 19.3 and 372.49. And for 60, it's 29.3 and 858.49. Now when we take all of these values. All of our squared values and some of them, I think I might uh try to make some space here actually. Let me just bring this up a bit. Then the sum of all these values would be 2,202.1. So the variance, OK, for all data set. OK. It's gonna be equal to 2,202.1 divided by 10 minus 1, which equals 244.68 years squared, OK? And thus, that means then, That our standard deviation. Let me put this in a, a box here. Let me make some space here. S is gonna be equal to the square root of that, which is equal to 15.64 years. OK. So, our standard deviation is 15.64 years for the old data set. Now what about the new data set? Let me highlight that. What about our new data set? Well, no, we're gonna do the same thing, but remember we're using a different mean. No, we're using the mean of 33.82 years. And now, when X is 12, then the difference is -21.82 and the square deviation is 476.03. That is the new square deviation. When it's 15, it's negative 18.82 and 354.12. When it's 18, negative 15.82 and 250.21, 22-11.82 and 139.67. 25 negative 8.82. And the 77.76. 30 3.82 and 14.58. 35 is 1.18 and 1.4, 46.18, and 38.21. 50, 16.18 and 261.85, 60, 26.18 and 685.49, and 65, 31.18 and 972.31. Now in this case again. To find the variance, we will sum all of those squared values. So now the variance is gonna be equal to the sum of all these values, which you should get 3,271. 0.64 divided by the number of values 11 minus 1 and you should get the variance to be 327.16 years squared. And if that's the variance, then let me put this in a box here, then that means the standard deviation is the square root of the variance would be equal to 18.09 years. So now let's compare both of these measures. Notice that the variance for the new data set is higher than the variance for the old data set, and the standard deviation for the new data set is higher than the standard deviation for the old data set. So adding the new data point increased the range, variance and standard deviation and the mean indicating a well. The range variance and standard deviation which indicates a greater span spread in life span values while the mean also shifted upward. Therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Range

Variance and Standard Deviation

Impact of Outliers

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