Note to CCSD HS Pre-Algebra Teachers: 3 rd quarter benchmarks begin with the last 2 sections of Chapter 6 (probability, which we will refer to as 6B), and then address Chapter 11 benchmarks (which will be divided between permutations/combinations, which we will refer to as 11A, and data analysis which we will call 11B). We have probability, permutations and combinations to create a first unit test; data analysis concepts (which are 11B) are in the second unit test. See below. Practice Test Unit 06B 11A: Probability, Permutations and Combinations Pre-Algebra HS, Unit 6B 11A: Probability, Permutations, Combinations Specification Sheet CCSD Syllabus Objective McDougal Littell Reference The student will: 6.7 Find the probabilities of an event and its complement. 6.7 6.9 Differentiate between the probability of an event and the odds of an event. 6.7 6.8 Find the odds in favor of and odds against an event. 6.7 6.11 Use permutations and combinations to count possibilities of an event. 11.6 6.10 Distinguish between permutations and combinations. 11.6, 11.7 6.12 Determine the number of outcomes using Venn diagrams. 11.8 Practice Test Unit 11B: Data Analysis Pre-Algebra, Unit 11B: Data Analysis Specification Sheet CCSD Syllabus Objective McDougal Littell Reference The student will: 6.1 Construct and analyze stem-and-leaf plots, box-and-whisker plots, and histograms, pictographs, circle graphs, bar graphs, scatterplots, and line 11.1, 11.2 graphs. 6.2 Apply appropriate measures of data distribution, including interquartile range and central tendency. Pg 39-40 6.4 Formulate inferences and predictions through interpolation and extrapolation of data to solve practical problems. 11.3,11.5 6.5 Evaluate statistical arguments that are based on data analysis for accuracy and validity. 11.3,11.5 6.3 Formulate questions that will guide the collection of data. 11.4 6.6 Design data analysis projects. Pg 646-647
Pre-Algebra, Unit 11B: Data Analysis Notes Stem-and-Leaf Plots Objectives: (6.1) The student will construct and analyze stem-and-leaf plots. Let s use the following test scores to construct a stem and leaf plot. 82, 97, 70, 72, 83, 75, 76, 84, 76, 88, 80, 81, 81, 82, 82 We first determine how the stems will be defined. In our case, the stem will represent the tens column in the scores, the leaf will be represented by the ones column. When we present our information, it will be in two parts, the stem and the leaf. Let s say I had this: 5 7 4. The way I would read that is by knowing the stem represents fifty, and the leaf has two scores, 7 and 4. Reading that information, I have a 57 and a 54. Knowing that, let s arrange our data in a stem-and-leaf plot. Knowing our lowest score is in the 70 s and the highest is in the 90 s, our stem will consist of 7, 8, and 9. Usually, the smaller stems are placed on top. You can make the decision for yourself. Another decision you can make is whether or not you put the scores in order in the leaf portion. As you can see, I didn t. 7 0 2 5 6 6 8 2 3 4 8 0 1 1 2 2 9 7 Notice that leaf part of the graph did not have to be in any particular order. So a person reading this plot would know the scores are 70, 72, 75, 76, 76, 82, 83, 84, 88, 80, 81, 81, 82, 82 and 97. What could be easier? Histogram Objectives: (6.1) The student will construct and analyze histograms. A histogram is made up of adjoining vertical rectangles or bars. If we rotated the last stem-andleaf graph 90 degrees and made the rectangles as high as the leaf portion, we would have a histogram. A histogram looks like a bar graph, except the rectangles are connected. Let s actually do a problem using the information from the previous example. McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 2 of 9
A histogram would typically identify what you are talking about on the horizontal axis. The vertical axis describes the frequency of those observations. One problem you might encounter on a histogram is when data falls on the line that divides two rectangles. In which rectangle do you count the data? Another problem is the width of the rectangles: how wide do you want them? Both of these problems are easily overcome. To determine the width, first find the range, which is the difference in the largest score and the smallest. 70, 72, 75, 76, 76, 82, 83, 84, 88, 80, 81, 81, 82, 82 and 97 Using the data from the example we have: 97 70 = 27 If you wanted three categories, you divide 27 by three; then each width would be about nine. If you wanted four categories, you d divide 27 by 4; then the width would be a little bigger than 6. It s your decision. No big deal. That takes care of the width problem. Now what about if something falls on a line that separates the rectangles? Do we count it in the left or right rectangle? Well, we just won t let that happen. We ll expand the range by one half then no score can fall on a line. Don t you just love how easy that was to take care of? So, I m deciding to have four groups, the width is a little more than 6 I ll say seven. And I m going to begin at 69.5 rather than 70. That should result in all my data falling within a rectangle.. Let s see what it looks like. Box-and-Whisker Plot Objectives: (6.1) The student will construct and analyze box-and-whisker plots. (6.2) The student will apply appropriate measures of data distribution, including interquartile range. Let s take a look at what might be a way of giving notes to your students to help them learn the steps for creating a box-and-whisker plot. See the next page! McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 3 of 9
Making a Box & Whisker Plot You try: 10 12 8 14 16 16 11 13 11 15 8 Steps Ex: 5 10 7 9 8 6 11 1. Arrange data in increasing order. (minimum value & 5 6 7 8 9 10 11 maximum value are the endpoints) 2. Find median of the entire list (median value) 5 6 7 8 9 10 11 a. If there is a number in the list that is the middle term, circle it and draw a line thru it. (median) b. If there is not a number that is in the middle, draw a line between the two numbers. (median is the number halfway between the two numbers) 3. Look at the bottom half of the numbers. Find the median of the bottom half of numbers (lower quartile) (same as #2 above) 4. Look at the top half of the numbers. Find the median of the bottom half of numbers (upper quartile) (same as #2 above) 5. Draw a number line that will cover the range of data (Evenly spaced marks) 6. Slightly above the number line place dots at the following points: minimum, lower quartile, median, upper quartile, and maximum 8 is the median 5 6 7 8 9 10 11 Does not apply to this problem 5 6 7 9 10 11 4 6 8 10 12 14 16 4 6 8 10 12 14 16 McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 4 of 9
7. Draw a box with side borders being the lower and upper quartiles. Draw two lines, one from each side of the pox connecting the minimum point on one side and the maximum point on the other side. Draw a vertical line at the median point from the top to the bottom of the box. 4 6 8 10 12 14 16 minimum Lower Quartile median Upper Quartile maximum 4 6 8 10 12 14 16 The data in the box represents the Interquartile Range IQR, the average, the middle 50%. The whisker on the left represents the bottom quartile, the bottom 25%; the whisker on the right represents the top 25%. The difference between the upper and lower quartiles is called the interquartile range (IQR). A statistic useful for identifying extremely large or small values of data is called an outlier. An outlier is commonly defined as any value of the data that lies more than 1.5 IQR units below the lower quartile or more than 1.5 IQR units above the upper quartile. In our example the lower quartile was at 6, the upper at 10. Using that the IQR =10 6 = 4. Multiplying that by 1.5, we have ( 1.5)( 4) = 6 Therefore, any score below 6 6 = 0 is an outlier, as is any score above 10 + 6 = 16. There are no points below 0, so we are OK on the left. There are no points greater than 16, so we are OK on the right. This would be an ideal place to use technology. Next are the instructions for drawing a box-and-whisker plot on the TI84. The examples will address outliers. McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 5 of 9
Entering Data and Drawing a Box-and-Whisker plot on the TI84 modified 1. STAT 2. EDIT regular 3. Enter the numbers in List 1 (L 1 ) or List 2 (L 2 ) 4. Return to the home screen 5. STAT PLOT 6. Turn Stat Plot 1 on and select the type of boxplot (modified or regular) 7. ZOOM 8. ZoomStat (9) 9. GRAPH Modified Regular The Ti 84 graphing calculator may indicate whether a box-and-whisker plot includes outliers. One setting on the graphing calculator gives the regular box-and-whisker plot which uses all numbers, so the furthest outliers are shown as being the endpoints of the whiskers Another calculator setting (modified) gives the box-and-whisker plot with the outliers specially marked (in this case, with a simulation of an open dot), and the whiskers going only as far as the highest and lowest values that aren't outliers: Find the outliers and extreme values, if any, for the following data set, and draw the box-and-whisker plot. Mark any outliers with an asterisk and any extreme values with an open dot. 20, 21, 21, 23, 23, 24, 25, 25, 26, 27, 29, 33, 40 To find the outliers and extreme values, I first have to find the IQR. Since there are thirteen values in the list, the median is the seventh value, so Q 2 = 25. The first half of the list is 20, 21, 21, 23, 23, 24, so Q 1 = 22; the second half is 25, 26, 27, 29, 33, 40 so Q 3 = 28. Then IQR = 28 22 = 6. The outliers will be any values below 22 1.5 6 = 22 9 = 13 or above 28 + 1.5 6 = 28 + 9 = 37. The extreme values will be those below 22 3 6 = 22 18 = 4 or above 28 + 3 6 = 28 + 18 = 46 Another example: L 2 =21, 23, 24, 25, 29, 33, 49 So I have an outlier at 49 but no extreme values, I won't have a top whisker because Q 3 is also the highest non-outlier, and my plot looks like this: McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 6 of 9
Measures of Central Tendency Objectives: (6.2) The student will apply measures of data distribution, including central tendency. 3 Measures of Central Tendency 1. Mean 2. Median 3. Mode The mean is the one you are probably most familiar with; it s the one often used in school for grades. To find the mean, you simply add all the scores and divide by the number of scores. In other words, if you had a 70, 80, and 90 on three tests, you d add those and divide by three. The mean is 80. Your average is 80. The median, often used in finance, is the middle score when the data is listed in either ascending or descending order. If there is no middle score, then you take the two middle scores, add them and divide by 2 (find the average of the two scores) Example: Find the median of 72, 65, 93, 85, and 55. Rewriting in order, I have 55, 65, 72, 85 and 93. The middle score is 72, so the median is 72. Piece of cake, right? The mode is the piece of information that appears most frequently. Let s look at some data: 55, 64, 64, 76, 78, 81, 81, 81, and 92. What scores appears most often? If you said 81, you just named the mode. You ve used the mode quite often before. If you have ever described the average weight of a particular population, the average height, shoe size, shirt size, the number of points scored in a particular type of game those are all examples of you using the mode. Interpreting Data Objectives: (6.4) The student will formulate inferences and predictions through interpolation and extrapolation of data to solve practical problems. (6.5) The student will evaluate statistical arguments that are based on data analysis for accuracy and validity. When looking at results, you must consider several questions like: Is the data display misleading? What is the margin of error? Are the conclusions supported by the data? Margin of error of a random sample defines an interval centered on the sample percent in which the population percent is most likely to lie. McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 7 of 9
Example: A sample percent of 27% has a margin of error of ±4%. Find the interval in which the population percent is most likely to lie. 27% 4% = 23% and 27% + 4% = 31%. The interval is between 23% and 31%. Predictions can be made about populations. If p % of a sample gives a particular response and the samples is representative of the population, then p% (# of people in population ) = predicted # of people in the population giving the response Example: A survey of 300 randomly selected cat owners finds that 120 cat owners prefer Brand C cat food. Predict how many owners in a town of 2000 cat owners prefer Brand C cat food. The percent of cat owners in the sample that preferred Brand C is 120 = 40% ; 40% of 2000 = 800 cat owners 300 Translating the objectives for interpreting data into instruction can also be done by looking at samples. McDougal Littell s Pre-Algebra Book, sections 11.3 to 11.5, contains many such examples and problems. Here are some other sample problems with which to begin: Example 1: Which of the following graphs most accurately depicts the hourly wages earned with respect to time worked? A. B. C. D. Earnings Earnings Earnings Earnings Hours Worked Hours Worked Hours Worked Hours Worked Discussion: Students should recognize that if one is paid by the hour, then only Graph C could illustrate it. A person would not earn money at no (zero) hours as shown in graph A. Looking at the other graphs, you could rule out B as a person would not earn the same amount for different amounts of hours worked. Graph D shows a person earning less money as the person works more hours. McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 8 of 9
Example 2: The graphs below show the numbers of baskets made by Player A and Player B during 5 basketball practices. Each player takes 100 practice shots during each practice. According to the graphs, who was more successful at making baskets? Player A Player B # of Baskets # of Baskets 80 78 76 74 72 A. Player A did much better. B. Player B did much better. C. Their scores appear to be about the same. D. More information is needed. Practice Session Practice Session Discussion: In this problem, there will be students who think Player B has scored the most baskets because of the steepness of the linear segments. Some students will think Player A scored the most baskets because his line segments looks consistently higher. They need to look carefully at the vertical scaling and note that Player A started about 70-71 and ended at 80, the same as Player B. (Answer is C.) Example 3: Why is this graph misleading? Season Attendance (in thousands) The heights of the balls are used to represent the number of spectators. # of spectators (thousands) However, the area of the balls distorts the comparison. The attendance for both sports doubled in the 10 year period; the size of the basketball makes it look like that increase may have been a lot more. 1990 2000 1990 2000 Year McDougal Littell, Chapter 11, Sections 1-5 Pre-Algebra, Unit 11B: Data Analysis Page 9 of 9