Constructing a Frequency Distribution and a Frequency Polygon ...

Question

Constructing a Frequency Distribution and a Frequency Polygon In Exercises 35 and 36, construct a frequency distribution and a frequency polygon for the data set using the indicated number of classes. Describe any patterns.

Declaration of Independence

Number of classes: 5

Data set: Number of children of those who signed the Declaration of Independence (Source: The U.S. National Archives & Records Administration) 5 2 12 18 7 4 10 8 16 3 3 7 3 1 2 7 13 0 8 3 7 5 2 6 0 6 7 9 0 11 9 10 7 8 13 5 8 3 5 0 3 13 3 15 5 6 3 2 5 2 0 3 7 12 4 1

Accepted Answer

Welcome back, everyone. In this problem, we want to construct a frequency distribution and a frequency polygon for the given data set using 6 classes. Describe any patterns in the distribution. The data set shows the scores achieved by 50 students on a recent standardized history exam with a maximum possible score of 100. A says the distribution is perfectly symmetrical. The distribution is positively skewed or right skewed, meaning the tail extends longer towards the higher scores. C says the distribution is nearly bell shaped, which is slightly skewed to the left, and the D says the distribution is uniform with all six classes having roughly the same frequency showing no significant peaks. Now let's focus on the first part of our problem. Let's try to construct the frequency distribution. To do that, we'll need to calculate the class with using the range and number of classes. So what do we know for our range? Well, if we take a look at our data set here, you may notice that our minimum value. OK, is equal to 55. And now when you look through the data set for our maximum value. You'll probably notice that it is 98. So that means the range. Is going to be equal to 98 minus 65, which equals 43. And no, we can use that to calculate the class width. OK. Because it's equal to the range divided by the number of classes. Which our range is 43 and in our problem we're told we want to construct the frequency distribution using 6 classes. So the number of classes is 6. 43 divided by 6 is approximately 6.167, and since the class width must be a whole number, we round up to the next integer 8. Therefore, our class width is 8 after we round up. Know that we have that, we can construct or frequency distribution. Now, let's go ahead and create a table then of our class with, OK. Our midpoint. And the class with frequency, OK? And the midpoint is going to be very helpful when we're ready to construct our frequency polygon. So now that we know that our class weight is 8 and our minimum value is 55, it means we add 8 to 55 and continue to add 8 until we get over 6 classes, OK? So our first class will be 55 to 62. Next, it will be 63 to 70. OK. Next it's gonna be 71 to 78, uh 79 to 86, 87 to 94, and 95 to 102, OK? Thus these are our classes. Uh, they're, they're 8 units wide basically. So now in this case. We can go ahead and get our mid points, and our mid points will be our halfway value. So for example, for 55 to 62, the midpoint will be 59, for 63 to 70, it will be 67, 71 to 78, it will be 75, 79 to 86 will be 83, 87 to 94 will be 91, and 95 to 102 will be 99. And now we can go ahead and count up all our values that should be in that class. So for example, to find the frequency of the class with 55 to 62, we would go back to our data set and look for all the values between 55 to 62. In the interest of time, I'll allow you to do that one on your own, but you should find that our frequencies are 5. 9, 1211, 9, and 4. So in total we have over 50 values. Know that we have the class with the midpoint and the frequency that is we have our frequency distribution. We can use it to plot our frequency polygon, and a frequency polygon is created by plotting the frequency against the class midpoint. And for it we'll add anchor points with a frequency of 01 class width below the first midpoint, and 1 class width above the last midpoint. Now, I've, I've already constructed my frequency polygon, so I'll just put it on our screen here and you can see if you got the same thing. OK? So when you plot, your frequency polygon should look something like this. So you notice here we have our anchor 51 with a frequency of 0 on our sorry below our lowest value and then we have our anchor of 107 at our highest value also with an anchor of 0. And then if we look at our frequencies for our midpoint 59, the frequency is 5, the midpoint 67, it's 9. For 75 it's 1283 is 1191 is 9. 99 is 4, and then like we said, we have our anchor of 107. So this would be the frequency polygon of history exam scores. Now remember the remaining part of our question. What pattern do you observe? Well, when we look at our polygon, notice that it is a bell shaped or mound shaped distribution that is slightly skewed to the left, OK? Slightly skewed. To the left In other words, we say here that it is negatively skewed. So since it is bell shaped, OK, I can note that here, and it's slightly skewed to the left. If we look back at our answer choices, C would be the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Frequency Distribution

Frequency Polygon

Class Intervals and Number of Classes

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