In Exercises 13 and 14, find the range, mean, variance, and st...

Question

In Exercises 13 and 14, find the range, mean, variance, and standard deviation of the population data set.

Drunk Driving The number of alcohol-impaired crash fatalities (in thousands) per year from 2010 through 2019 (Source: National Highway Traffic Safety Administration)

10.1 9.9 10.3 10.1 9.9 10.3 11.0 10.9 10.7 10.1

Accepted Answer

Welcome back, everyone. A class of 12 students took a math test and received the following scores out of 100. 85, 90, 78, 92, 88, 76, 95, 89, 84, 91, 87, and 80. Calculate the range, mean, variance and standard deviation for this population data set. Now, let's start by calculating the range. Remember that the range is the difference between the minimum and the maximum values in our data set. Notice here that the minimum value is 76 while the maximum value is 95. So that means the range. Is going to be equal to 95 minus 76, which equals 19. So our scores have a range of 19 points. Next, recall that the mean. OK, the mean, the mean that we can call mu, the population mean, will be equal to the sum of data points divided by the number of data points. So in this case, that would be 85 + 90 + 78 plus 92 + 88 plus 76 + 95 plus 89 + 84 + 91 + 87 + 80, OK. It would be the sum of all those values. Divided by 12. Now, when we add those values, we get 1,0035 and 1,0035 divided by 12 is going to be approximately 86.3. So our mean is approximately 86.3 to 3 significant figures. Now what about our population variance? How do we find the variant? Well, recall that our variance, sigma squared is going to be the sum of the squared deviations. OK. So that's the, the, each data point minus the mean squared, OK, divided by the number of data points N. OK. So in this case, if we're going to find the, the, the variants, we need to find the sum of the squared deviations from the mean. So let's set up a table to help us do that, OK? So we're gonna have each data point X, and then we're gonna have the difference between each data point and the mean, and then we're going to square that value. OK? So it's X minus mu squared. So let's set that table up. Uh let me make some space here. And, you know, I think I could be a little more, make some more space in the middle, just to at least show you how we do one or two of these, OK? I know for data points, we know that it goes 85, 90. 78 92 88. 76 9589 84, OK, 91. 87 and 80, OK. So now these are all our data points. So let's go ahead and find these differences. For example, In the first one, OK. X1 minus mu. So no, we're gonna have X 85 minus or approximate value of 86.3 or 10035 divided by 12. And that would be equal to 5/4 or -1.25. And when we square that, -1.25 squared is equal to 1.5625. Essentially, we're going to do the same thing for the rest of the data values. So I'll allow you to work each of those out, but in the interest of time, when X is 90, then our difference is 3.75 and the squared difference is 4.0625. When it's 78, we get negative 8.25 and 68.0625. When it's 92, we get 5.75 and 33.0625. When it's 88, 1.75 and 3.0625. 76 is negative 10.25 and 105.0625. 95 is 8.75 and 76.5625. 89 is 2.75 and 7.5625. 84 is -2.25 and 5.0625. For 91, we get uh 41, sorry, no, 4, 4.75. And uh 22.5625 means 87, we have 0.75 and 0.5625. And then when it's 80, we have negative 6.25 and 39.0625. So now all we need to do is to sum these values, OK? So now when we do that, OK. Then we should find that the sum or the total of the squared deviations from the mean by summing all of these values is going to be equal to 376.25. OK? So that's the sum of our squared deviations. I know that we have that sum, if we go back to the formula for the variance, let me just uh put a line over here to separate. Then this would imply that the variance is going to be equal to 376.25 divided by 12, which is approximately 31.4. Now, the final thing we need is the population standard deviation. And again, for this one, recall that the standard deviation is really the square root of the variance, OK? And since we already have the variance, that means it's going to be equal to. Eh? The square root, let's use our exact value of 376.25 divided by 12, which is approximately 5.6. So what do we know? Here we figured out that our standard deviation. It is approximately 5.6. The variance is approximately 31.4. The mean is approximately 86.3, and the range is 19. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Descriptive Statistics

Variance and Standard Deviation

Population vs. Sample Statistics

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