In Exercises 25 and 26, find the range, mean, variance, and st...

Question

In Exercises 25 and 26, find the range, mean, variance, and standard deviation of the population data set.

The mileages (in thousands of miles) for a rental car company’s fleet.

4 2 9 12 15 3 6 8 1 4 14 12 3 3

Accepted Answer

Welcome back, everyone. A sample of laptop batterizing orders is 759, 10, 867, 12, 8, 11, 97, 10, and a 6. Find the range, mean, variance and standard deviation for this population. Now, in this problem, let's first start like it says by finding the range, OK? Recall that the range is the difference between the minimum and the maximum value from our data set. In our data set, we can tell here that the minimum value is 5 and the maximum value is 12. So that means the range equals 12 minus 5, which equals 7, OK? In other words, the batch the laptop battery lives ranges by 7 hours. Next, let's calculate the population mean. Now recall that to find the meaning of the data set, all we need to do is to add all of the values and divide by the number of values we have. So in this case, that's gonna be 7 + 5 + 9 + 10 + 8 + 6 + 7 + 12 + 8. Plus 11 + 9 + 7 + 10 + 6. OK. And no, that's gonna be all divided by the number of values we have, which we can tell is 14. When we add all of these values up, we get 115 and 115 divided by 14 is approximately 8.2, OK? So that is the mean of our population. Now, let's try to calculate our population variants. What do we know about variants? Well, recall that the variance came. For the population, sigma squared. Is going to be equal to the sum of the squared deviations, so that's x minus the mean mu squared all divided by the number of values n. OK, and it would do well here for me to just show here that when I talk about the mean, we can also write it as mu. So to find our variants we'll need to find all of our deviations, square them, and then add them up to divide the result by N. So let's do, let's let's make a table to help us to, uh, be able to track that a little better, OK? So we have our X values in our table, we have our deviations X minus mu, and then we have the squares of our deviations, OK? So now let's do a table, OK. And now, let's do our values. We know we have 14 values here. So that's 759, 10, 8. 67, OK. 12, 8, 11, 97, 10, and then we have 6. OK. So these are all our X values. Now if we're going to find our deviations, we need to subtract our mean from each of these values. So starting with 7, the deviation is going to be 7 minus 115 divided by 14. That's our mean, which is going to be equal to -1.214 approximately. And when we square that. And then it becomes 1.474. So now, if we continue this trend, I'll, I'll allow you to do the calculations on your own, but you should find that when X is 5, or deviation is negative 3.214, and that squared is 10.332. When it's 9, we get 0.786 and 0.617. When it's 10, 1.786. And 3.189. When it's 8, 0.214 and 0.046. 6-2.214 and 4.903, um 7-1.214 and 1.474. 12 is 3.786, and 14.332.8, again negative 0.214 and 0.046. 11, we get 2.786 and 7.760. 9, we get 0.786 and 0.617. 7-1.214, 1.474. 10 is 1.786, and 3.189. And then 6 is -2.214 and 4.903. So now that we have all of these, we can sum them up, OK? So now when we do that, the sum of our squared deviations from the mean is going to be approximately 54.357. And now if we take this value back to our formula for the variance, that means it will be equal to or approximately equal to 54.357. Divided by N, which is 14, and that is approximately equal to 3.9. So our variance is approximately 33.9. Finally, for the population standard deviation, which is sigma. Then that standard deviation is equal to the square root of the variance, OK? So in this case, that's gonna be the square root of 54.357 divided by 14, which is gonna be approximately 2.0, OK? So, to recap, for our data set, its range is 7, mean is 8.2, variance is approximately 3.9, and standard deviation approximately 2 hours. Thanks a lot for watching everyone. I hope this video helped.

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In Exercises 25 and 26, find the range, mean, variance, and standard deviation of the population data set.

The mileages (in thousands of miles) for a rental car company’s fleet.
4 2 9 12 15 3 6 8 1 4 14 12 3 3

Key Concepts

Descriptive Statistics

Range

Variance and Standard Deviation

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In Exercises 25 and 26, find the range, mean, variance, and standard deviation of the population data set.The mileages (in thousands of miles) for a rental car company’s fleet.4 2 9 12 15 3 6 8 1 4 14 12 3 3