In Exercises 37– 40, use the data set, which represents the mo...

Question

In Exercises 37– 40, use the data set, which represents the model 2020 vehicles with the highest fuel economies (in miles per gallon) in the most popular classes. (Source: U.S. Environmental Protection Agency)

36 30 30 45 31 113 113 33 33 33 52 141 56 117 58

118 50 26 23 23 27 48 22 22 22 121 41 105 35 35

Find the five-number summary of the data set.

Accepted Answer

Welcome back everyone. In this problem, a teacher records the scores out of 100 for 12 students on a quiz. 56, 72, 68, 90, 85, 77, 62, 80, 74, 88, 95, and 70. What is the 5 number summary for this data set? Now, when the problem refers to the 5 number summary, OK, it's really referring to the maximum, minimum value of this data set and its quarters. OK, that is the 1st, 2nd, and 3rd quarter. But before we find any of this information, we'll need to sort this data set in order. OK? So if we were to start it in ascending order. Then our data set, OK, is going to be 56, 62, 68, OK, 70. 72, 74, 77, 80, 85, 88, 90, and then 95. Let me just move the data set to the left a little bit just to make some space. But now this is our data set, OK? So now, how can we find the 5 number summary? Well, after we've ordered the data, now we can tell that the minimum value is 56, and we can also tell that the maximum value is 95. So now we just need the other three values. Now, let's talk about the quarters. First, let's start with the 2nd quartile, because the 2nd quartile is the median of the data. Now, the median, OK, let me just make a line here. The median OK. Which is Q2 is going to be the value that is halfway between the data set. And in this case, because we have 12 values, that means the median is going to be the average of the 6th and 7th values. If we start to count, starting from 56, this is the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th value. So 74 is the 6th value and 77 is the 7th value. So that means the median is going to be equal to 74. Plus 77 divided by 2, which is going to be equal to 75.5. Thus that's the median value. Next, if we're talking about our first quartile, Q1, it's the lower quartile, and it's going to be the median of the lower half. Now, if we look at our data set here, our lower half goes from 56 to 74, so what's the median value? Well, again, because we have even an even number of values, the median value here Q1. It's going to be between the 3rd and the 4th value. So for our data set, our median is going to be, or Q1 or value Q1 is going to be between 68 and 70. Now If we find their average 68 plus 70 divided by 2 will be equal to 69. Thus Q1 is 69. Finally, let's find the 3rd quartile, and that is the upper quartile or the median of the upper half of our values. Now if we go to the other side for the upper half, that goes from 77 to 95. So that means Q3 will be between the 9th and the 10th value, OK. That is, it is going to be between 85 and 88. This is where Q3 is gonna be. And now, 85 plus 88 divided by 2 is equal to 86.5. Therefore, the 5 numbers summary for our data set is that the minimum is 56, the maximum is 95, Q1 is 69, Q2 is 7775.5, and the Q3 is 86.5. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Five-Number Summary

Quartiles

Median

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