Measures of Central Tendency and Variability
Lesson 40
In lesson 40 our warm up will review division. Your students will need to know how to divide in order to find the mean of a set of data.
Matt is a football player at Mooresville High. He plays outside linebacker. The number of tackles he has for his last 9 games are listed below, he needs to tell a colleges coach how many tackles he usually has per game. What measure of central tendency should he use. If we “S” the problem, or study the problem we will first underline our question. What measure of central tendency should he use? We will then answer the question, what is this problem asking you to find, this problem is asking me to find whether should use the mean, median, or mode.
We will use our yellow counters to represent the numbers we have . As your students if any of the number repeat, or if any of the groups are exactly the same. In this case the 2 numbers that are the same are 3. WE have two groups of 3 which are the same. If you are identify the mode of a set of data, it is the one that occurs the most, or the number that occurs the most. So for this set of data, our mode is 3. If we have all this numbers in order from least to greatest, which is what we have in this case, 3,3,5,9,10 or from greatest to least so we can find the median. The median is the number in the middle, when the numbers are in order from least to greatest or greatest to least. In this case to find the number in the middle we remove the numbers one at a time and we see that 5 is our number in the middle, so the median is 5. In order to find the mean, we actually combine our entire group, so we push all these together. And then we separate them out one at the time into 5 equal groups. We are trying to even everything out; each group will have 3 , each group will have 4, each group will have 5, and there there’s enough for each group to have 6, so we because we have created 5 equal groups and there are 6 in each group our mean is 6. We pushed everything to together which was the process of addition, and then we split into equal groups, 5 equals groups because we had 5 numbers to begin with and that is division.
We are going to look at our measures of variability for a set of data, in order to do this. The first thing we must do is put all of our data in order from least to greatest. Our lowest number is 17 we can cross that out, after 17 our next number that I see is 19, and I only see one 19, and I move to 20 and I do have a 20, I only see one 20 so then I move to 21, I do not have 21, 22, I do not have 22, 23, I do have a 23, so I cross it out. I do not have another 23 or 24 or 25 so then I look at 26, then after 26, I look for 27, there is none, 28, none, 29 I see and I actually have 2 of those so I have to list both of them. Just because I have 2 does not mean you can only list them once, then there is 30, I have 2 31’s, so I have to list both of them, and 34. Have your student’s count what was the total number I had in my data set, 1,2,3,4,5,6,7,8,9,10,11 do I still have 11, 1,2, 3,4,5,6,7,8,9,10,11. Our lower extreme is the lowest number in our data set or 17; our upper extreme is our largest number in our data set or 34. and if we want to find our range, we actually subtract our lower extreme from the upper extreme. So you say 34 minus 17 so our range is 17. In order to find the first in order to find the first quartile. We must first find our median or our second quartile so we can separate of data into the lower half and the upper half. We have 11 pieces of data which means in order to find the middle, we will go one by one moving towards the middle, and we see that 29, is our median, or second quartile. Once we have identified 29 we can look at our lower half of data and our upper half of the data. Our lower half of the data consists of 5 points. We have 17, 19, 20, 23, and 26. If we are to find the first quartile is the median of the 5 numbers. You can go towards the middle and see that our first quartile is 20. Our third quartile is the median of the upper half of the data, we have all of our numbers and if we want to find this median we can see that 31 is in the middle. So our third quartile is 31; but from here we want to find our inter-quartile range, our inter quartile range is actually the third quartile minus the first quartiles. So we say 31 minus 20 and our inter-quartile range is 11. We are now going to make a box and whisker plot which represents our data above. We will first use a number line and we will first start with 15 and we have to use equal intervals so I am going to do my best to make these equal and I am going to use a scale of 5. I’m going to go from 15 to 35 because my lower extreme is 17 and my upper extreme is 34 so all of the data will fall within 15 and 35. We will first begin by plotting the median of the data, we said that our median, or the number in the middle was 29. SoI’m going to say that 29 is gong to be up here close to 30. So I’m going to put a point right there, we will then plot our points for the first quartile and the third quartile. Our first quartile is at 20, and our third quartile is at 31, it’s is actually going to be about the same distance on the other side of here. This creates our box. We will have the lower end of the box at our lower, or first quartile, we will the middle at our median, and we have our box and then we will find our lower extreme which is at 17 and that is going to be down here. And our upper extreme at 34, in these actually are going to be our whiskers. Please notice on my box and whisker plot that my median is not always going to be right in the middle of the box, it will not always divide box evenly; so your students need to be aware that that’s a possibility. We also could have whisker that extended very far to the right or the left, it’s just depending on our data. To create our foldable, we are going to create a 8 page booklet. We will begin by doing our hotdog folder, folding our paper vertically, you will then fold it, our hamburger in half, and hamburger fold in half again. When you open it back up you can see that you have created, you have 2 rectangles open in at just one time, you have a open part and folded part you going to go to the side with the folded, and then you are going to go to the corner which is also folded. And you are going to trim just a very top, just barely cut that top part off, from the folded corner to the middle crease. You don’t want to go all the way to the end you just from the folded corner to the middle crease. Then when you open back up you’re going to go to the middle and you pull out so that you create a star, or a plus. And these pages just lay down so that you have a little booklets. 1,2,3 pages and with those pages are we are going to have our measures of central tendency and variability, we show how to find, we give a definition of mean, media, and mode; and an example and we give our definition or our variability and we give an example of our box and whiskers shows all of our measures of variability and our median.
We have already “S” the problem we know that this problem is asking me to find whether Matt should use the mean, media, or mode. In “O” organize our facts we will identify our facts. Matt is a football player at Mooresville High, he plays outside linebacker, the number of tackles for his last 9 games are listed below. So these are all kind of one stat.; he has to tell a college coach how many tackles he has in a game, this is a fact. We have to decide whether these facts are necessary or unnecessary. Matt is a football player, probably not a necessary fact, so we can cross that out, He plays outside linebacker, is not necessary, the number of tackles he has for the last 9 games are listed below, these are necessary so we are going to list those. He needs to tell a college coach how may tackles he usually has per game. Some students will say that his is a very necessary fact it is important because he wants to tell the college coach, his best measure of central tendency, or his best number, so we could say; from here in “L” line up a plan we will choose our operations and write in words what our plan of action will be. Because we are trying to find the measure of central tendency’s. The median and the means, the mean will require us to add and divide. So we will use adding and division and our plan of action will be , we will find the mean, mean and modes. In “V” verify your plan with action, our mean, median, and mode should all be less than 11. Now we are going to carry out our plan by adding the numbers, you get a total of 72, and we divide 72 by the 9 games to give us a mean of 8. You want to find the median when we switch the order from least to greatest or greatest to least, going one by one, we see that the number in the middle is 7, that our median is 7, and our mode is the number that shows up the most or the most occurring, and that is also 7. If he wanted to tell the largest number, or the best number then he should say his mean of 8, is what he wants to tell the college coach. 8 the mean which the largest measure of central tendency. In “E” examine your results, does your answer makes sense we are were looking for a central tendency, where we should use the mean, median, or mode, and yes we decided we should use the mean. So yes our answer makes sense. Is our answer reasonable; we said that all 3 should be less than 11, and our answer is less than 11 so yes our answer is reasonable. And is your answer accurate, if you have your student rework their answer on another sheet of paper, or let them use a calculator to make sure they have the right number. We will now write our answer as a complete sentence. Matt should tell his new coach the mean of his past 9 tackles because it is the largest of the 3 measures of central tendency.
We will close our lesson by looking at the essential questions. Number 1, what is the different between measures of central tendency and measures of variability? Central tendency tells where the middle of the data set is. And variability tells how our data varies or is distributed. Question 2, How do you find the mean of data? To find the mean you will add the terms and then the divide by the number of terms. How do you find the median of the data, question 3? The median is the middle number when the data is in order from least to greatest or greatest to least. How do you find the range? You subtract the lower extreme from the upper extreme. Questions 5, how do you find the inner quartile mean of data? You subtract the first quartile from the third quartile. And questions 6, how do you find the quartiles of the set of data? The first quartile is the in the middle of the lower half of the data and the third quartile is the middle of the upper half of the data.