Math 6, Unit 6 Notes: Collecting and Displaying Data

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1 Math 6, Unit 6 Notes: Collecting and Displaying Data Objectives: (4.3) The student will design strategies for collecting data to solve problems. (4.4) The student will compare ways to represent collected data. (4.5) The student will interpret data from various formats including circle graphs and scatter plots. (4.6) The student will analyze the effect changes in formant will have on the interpretation of data. (4.7) The student will compute measures of central tendency including mean, median, and mode. (4.8) The student will draw conclusions from sets of data. (4.9) The student will make predictions from sets of data. (4.10) The student will describe the limitations of various graphical representations. The term statistics refers to both a set of data (information) and methods used to analyze the data. Imagine a little boy or girl running home from school all excited one day telling his mother he got eight right on a spelling test. The mother might wonder, what does that mean? Is that good or bad? She might decide she needs more information to make that decision. So she may asks, how many questions were on the test? When data is received and it must be analyzed, that process is known as statistics. Often times more information is needed to analyze the data, so more questions must be asked. The first statistic that many learn formally is a percent. If a child takes a quiz in school and scores 14 correct out of 17, the child needs to know what that means. The teacher usually converts that score to a percentage 82%, then states that a B was earned. The interpretation given to a grade of B in school is above average. Again, information was received, then using a statistical method (finding percentages), the information was analyzed. It s possible to look at a whole set of data and use a single number to describe it. Whether a child is a bowler, golfer, basketball player, or a student, a single number is used to describe their performance. However, when that is done, some information will be lost with such a simple description. There are three measures of central tendency, sometimes referred to as measures of averages. Math 6 Notes Unit 6: Collecting and Displaying Data Page 1 of 22

2 3 Measures of Central Tendency: 1. Mean 2. Median 3. Mode Mean The mean is the one that is probably most familiar, 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. For example, if a student scores 70, 80, and 90 on three tests, the mean is calculated as follows: Add the three scores, =240, then divide the sum by the number of data pieces: 240/3 = 80. The mean is 80. The average is 80. Example Find the mean of 72, 65, 93, 85, and 55. First add those 5 scores together; = 370 Second, divide that total by the number of scores, = 74 The mean is 74. Example Five kids just finished bowling one game. The average score of the five kids is 82. What is the total of all 5 scores? Having a mean of 82 does not mean each kid scored an 82, it means if the scores were distributed equally, they would each have 82. Their total score, is 82 x 5 = 410. When thinking of the mean, you need to think of the TOTAL points being distributed EQUALLY. CRT example: Math 6 Notes Unit 6: Collecting and Displaying Data Page 2 of 22

3 Median 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, take the two middle scores, add them and divide by 2. It s also referred to as the average of the two middle scores. Example Find the median of 72, 65, 93, 85, and st. Rewrite the data in ascending order: 55, 65, 72, 85 and nd. The middle score is 72, therefore the median is 72. Notice in the first two examples the data is the same, but the mean and median are not the same. Example Find the median of 72, 65, 93, 85, 74, and st. Rewrite the data in ascending order: 55, 65, 72, 74, 85, 93 2 nd. Notice there is no middle score. 3 rd. Add the two middle scores together and divide by = 146, = 73. The median is 73. When thinking about the median, you need to think of the MIDDLE score if they are listed in ORDER. Mode The mode is a piece of data that appears most frequently. Example The following are test scores: 55, 64, 64, 76, 78, 81, 81, 81, and 92. Find the mode. The score that appears most often is 81. When thinking about the mode, you need to think about the score that appears most frequently. Math 6 Notes Unit 6: Collecting and Displaying Data Page 3 of 22

4 These three measures of central tendency, most often referred to as averages, describe a set of data using a single number. By condensing the information like that, the whole picture may not be seen. Just like when the little kid ran home and indicated he had 8 right, more information was needed to determine what that meant. In math, that is called analyzing data. The following example shows how using an average or mean to describe a performance may be misleading. Let s say Abe, Ben and Carl each bowl 3 games. Three games later, they all found they had a mean average of 80. Here are the scores for each person: Abe s scores 80, 80, 80 Ben s scores 70, 80, 90 Carl s scores 65, 75, 100 In this case, the mean may not be a good indicator or each person s performance. Looking at Carl s scores, it appears he s a little erratic. It might be difficult to predict what he might score on the next game using the average. Abe, on the other hand, looks pretty stable as he ll probably score an 80 on the next game. Knowing Abe and Ben both have the same average, this example shows that one mean is a pretty good descriptor, which would allow to predict more comfortably what might happen next. In other words, the mean is doing a pretty good job of describing what s happening. Carl s mean is not as good of a descriptor as Abe s. Although his average is 80, like Abe s, the mean does not do a good job of describing what is happening. Abe s mean better describes what is occurring than Carl s mean. But, if the scores were not shown, it would not be evident how consistent Abe is and how erratic Carl is because their averages are both 80. Sometimes, more information is needed. One way to do this is to look at all the scores and try to determine consistency. In math, rather than looking at the whole set of data, only the high and low scores might be examined. It might be someone just had one super high or low score that really affected the mean. In other words, determine the spread of the scores. In statistics, that s referred to as dispersion. There are three ways to measure this spread or dispersion. 3 Measures of Dispersion 1. Range 2. Variance 3. Standard Deviation The significance of Range will be discussed in this unit but will leave Variance and Standard Deviation topics for another time. Math 6 Notes Unit 6: Collecting and Displaying Data Page 4 of 22

5 Range The range is just the difference between the top score and the bottom score. The larger the range, the less likely the mean can be depended upon as a good descriptor or predictor. In the last example, the range of Abe s scores was zero. The range of Carl s scores was 35 and Ben s range was 20. It would appear, the smaller the range, the mean is a more accurate descriptor. Graphing There are many ways to organize data. The most commonly used method of analyzing data is to use a percentage. Another good way to represent data is pictorially, using statistical graphs. Such graphs include line graphs, stem and leaf plots, frequency tables, histograms, bar graphs, pictographs, circle graphs and box and whisker plots to name a few. All these graphs allow us to look at a picture, rather than numbers, and draw some conclusions. LINE PLOTS A line plot consists of a line, on which each score is denoted by an X or a dot above the corresponding value on the number line. CRT example: Math 6 Notes Unit 6: Collecting and Displaying Data Page 5 of 22

6 LINE GRAPHS A line graph displays a set of data using line segments. This graph shows data that has changed over time. Example The following data represents the monthly class average over a 12 month period. 82, 97, 70, 72, 83, 75, 76, 84, 76, 88, 80, 81 The data can be arranged on a line graph: J F M A M J J A S O N D The advantage of a line graph is that it shows changes in data over a period of time. Another type of graph is called the Stem & Leaf Plot. STEM & LEAF PLOT The following test scores are used to construct a stem and leaf plot: 82, 97, 70, 72, 83, 75, 76, 84, 76, 88, 80 81, 81, 82 First determine how the stems will be defined. In this case, the stem will represent the tens column in the scores, the leaf will be represented by the ones column. When the information is presented, it will be in two parts, the stem and leaf. For instance, would be read as follows: The stem represents fifty, and the leaf has two scores, 7 and 4. Reading that information then gives 57 and a 54. Since the lowest score is in the 70 s and the highest is in the 90 s, the stem will consist of 7, 8, and 9. Usually, the smaller stems are placed on top, but are can be arranged from largest to smallest. Another decision to be made is whether or not to put the scores in order in the leaf portion. Math 6 Notes Unit 6: Collecting and Displaying Data Page 6 of 22

7 Notice that the 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. CRT example: If the stem and leaf plot were to be rotated 90 degrees (a quarter turn) the graph would resemble a bar graph which leads to the next type of graph to discuss. BAR GRAPH Using the same information above, let s construct a bar graph to show how many A, B, C, D, and F s there are. A s are defined as 90 and above, B s from 80 to 89, C s 70 to 79, etc A B C D F Math 6 Notes Unit 6: Collecting and Displaying Data Page 7 of 22

8 Another way to represent data is through the use of a histogram. HISTOGRAM A histogram is made up of adjoining vertical rectangles or bars. If the last stem and leaf were to be rotated 90 degrees and the rectangles were made as high as the left portion, we would have a histogram. A histogram looks just like bar graph except the rectangles are connected. Use the previous data for the following example: In a histogram the horizontal axis typically identifies the topic of the graph and the vertical axis describes the frequency of those observations. Two problems that might be encountered on a histogram; one is when data falls on the line that divides two rectangles. Which rectangle does the data belong to? Another problem is the width of the rectangles, how wide should they be? Both of these problems are easily overcome. To determine the width, find the range, then divide the range by the number of desired bars. 70, 72, 75, 76, 76, 82, 83, 84, 88, 80, 81, 81, 82, 82 and 97 Using the data from the above example the range is = 27. If four bars are needed, take the range and divide by 4. This means that each bar would be 6.75 units wide. Round up to make each bar 7 units wide. If the bars are 7 units wide, the intervals of the bars would be as follows: Where would 84 fit in this histogram? Would it fit in the second or third interval? To avoid this problem, change the endpoints of the intervals to the following: Math 6 Notes Unit 6: Collecting and Displaying Data Page 8 of 22

9 These intervals are still 7 units wide but now, 84 is clearly in the third interval. CIRCLE GRAPH A circle graph consists of a circular region partitioned into disjoint sections, each section representing a percentage of the whole. Example A family weekly income of $200 is budgeted in this manner; $60 food, $50 rent, $20 clothing, $20 books, $30 entertainment and $20 other. Construct a pie chart to illustrate that information. A circle has a total of 360º, therefore 360 represents the total amount of the budget or 100% of the expenses. To fill in the pie chart, we have to determine what percent is spent for each expense. To find that percent, I divide the expense by the total budgeted for the week. $60 out of $200 is budgeted for food. Converting that to a percent, we have 60/200 = 30%. So let s do percents. Food 60/200 or 30% Rent 50/200 or 25% Clothing 20/200 or 10% Books 20/200 or 10% Entert. 30/200 or 15% Other 20/200 or 10% The reason the dollar amounts are converted to percents is to determine what percentage of the circle will be dedicated to that category.since food represents 30% of the pie, find 30% of 360º, which equals 108º..30 x 360 =108 ( ). Doing the same for clothing, 25% of 360 º is 90 º, 10% is 360 º is 36 º, and 15% of 360 º is 54 º. Math 6 Notes Unit 6: Collecting and Displaying Data Page 9 of 22

10 Let s see what the pie chart looks like using those degree equivalents. Other Entert. Food Books Clothing Rent BOX AND WHISKER In a box and whisker graph, information is visually broken into four groups quartiles. To graph this information, first divide the data in half, actually, that is, find the median (the middle score). The median splits the information into two groups. Next, find the median of the top half, then find the median of the bottom half. Those three medians form a box. Example Create a box and whisker plot for the following data. 70, 72, 75, 76, 76, 82, 83, 84, 88, 80, 81, 81, 82, 82 and 97. First, find the median. Remember that the data must be either in ascending or descending order first. 70, 72, 75, 76, 76, 80, 81, 81, 82, 82, 82, 83, 84, 88 and 97. The median is 81. Next, find the median of the top half of the data and the median of the bottom half of the data. The top half consists of: 82, 82, 82, 83, 84, 88, and 97. The median of the top half is 83. The bottom half consists of: 70, 72, 75, 76, 76, 80, and 81. The median of the bottom half is 76. Math 6 Notes Unit 6: Collecting and Displaying Data Page 10 of 22

11 Finally, create the box and whiskers. The median of the top half is the right side of the box and the median of the bottom half is the left side of the box. The median of the entire data just puts a divider in the box. 81 is the median of the entire data 76 is the median of the bottom 83 is the median of the top half To make the whiskers, put a dot on the lowest score and on the highest and connect those dots to the box. We now have a box and whisker plot The data in the box represent the Inter Quartile 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 are called outliers. 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 this example the lower quartile is at 76 and the upper quartile is at 83. The IQR = = 7. Multiplying that by 1.5, we have ( 1.5) ( 7)= 10.5 Math 6 Notes Unit 6: Collecting and Displaying Data Page 11 of 22

12 Therefore, any score that is 10.5 below the lower quartile (below 65.5) or any score that is 10.5 above the upper quartile (above 93.5) is an outlier. There are no points below 65.5 but, the 97 is above 93.5, so that score is an outlier. Outliers are indicated by using an asterisk. When there are outliers, the whiskers end at the value farthest away from the box that is within 1.5 IQR from the end. Here is the final box and whisker graph Line graphs are normally used to demonstrate trends of a single variable. Bar graphs are used for more than one variable. Like bar graphs, histograms show relationships in more than one variable but are typically continuous like time or number grades. Bar graphs are discontinuous, like letter grades. A stem and leaf is very much like a frequency polygon turned vertically. It s great for comparing data in much the same way as bar graphs. The circle graph is best used for comparisons. One piece of data can be compared to other pieces of data or to the whole data and get a feeling for what s biggest, smallest. With a box and whisker graph the median can be seen very quickly and how the data is dispersed. It also divides the data into quartiles. The smaller the boxes or whiskers, the more closely the scores are to that median. The asterisks are often referred to as outliers. Math 6 Notes Unit 6: Collecting and Displaying Data Page 12 of 22

13 Graphing Problems 1. The Jones family has a budget. Each month it uses its income in the following manner: 30% for food, 25% for rent, 20% for transportation, 10% for savings, 5% for entertainment, and 10% for unexpected expenses. Construct a pie graph representing this information. 2. Each dollar that the government obtains in taxes is spent in the following manner: 25 cents goes to defense, 30 cents goes to social security, 10 cents goes to farm subsidies, 15 cents goes to government salaries, and 20 cents is spent on miscellaneous social programs. Construct a circle graph representing this information. 3. In 1988 UNLV received the indicated amount of revenue from the following sources: Federal Aid: $600,000 State Aid: 700,000 Private Donations: 100,000 Corporate Donations: 200,000 Student Tuition: 300,000 Other: 100,000 Construct a pie graph representing this information. 4. There are 20,000 students attending a certain college. The classes are distributed in the following manner: 4,000 seniors, 3,000 juniors, 5,000 sophomores, 6,000 freshmen, and 2,000 graduate students. Construct a circle graph representing this information. 5. A statistics experiment consists of tossing a group of 8 fair coins and recording the number of heads. Construct a histogram and a frequency polygon for the thirty tosses listed below. 6, 1, 8, 3, 6, 7, 5, 4, 5, 3, 3, 3, 7, 8, 2, 5, 2, 8, 4, 5, 4, 6, 5, 4, 1, 2, 2, 4, 6, 1. Math 6 Notes Unit 6: Collecting and Displaying Data Page 13 of 22

14 6. A student in a math class recorded the number of doughnuts purchased by the first 30 customers in Al s doughnut shop. Construct a histogram and a frequency polygon for this data. 2, 3, 10, 1, 4, 5, 6, 7, 9, 8, 3, 6, 3, 2, 4, 2, 5, 10, 2, 6, 2, 8, 1, 8, 8, 7, 7, 6, 5, The following were test scores for 33 students in a math 114 class. 58, 92, 85, 66, 72, 81, 60, 90, 70, 71, 77, 84, 75, 58, 89, 67, 98, 96, 70, 87, 74, 64, 64, 59, 87, 73, 91, 63, 86, 81, 72, 72, 73. a. Construct a grouped frequency distribution for these scores using the intervals 95-99, 90-94, 85-89, and so on. b. Use the frequency distribution from part (a) to construct a histogram, a frequency polygon, and a cumulative frequency graph. 8. A survey of 32 college students was made to determine the number of books purchased for their classes in the fall semester. Construct a frequency distribution, a frequency polygon, and a cumulative frequency graph using this data. 8, 7, 14, 7, 8, 10, 16, 8, 9, 15, 14, 16, 10, 14, 8, 14, 13, 8, 13, 8, 13, 8, 12, 11, 9, 12, 13, 12, 12, 7, 15, The scores on a math test of 40 grade school students are as follows: 62, 65, 94, 85, 90, 43, 73, 87, 74, 42, 62, 61, 83, 68, 84, 90, 66, 71, 63, 84, 84, 76, 96, 47, 53, 78, 53, 64, 68, 58, 46, 58 58, 86, 84, 53, 87, 77, 75, 62. a. Construct a grouped frequency distribution for these grades using the intervals 95-99, 90-94, 85-89, and so on. b. Using the frequency distribution from part (a), construct a frequency polygon and a cumulative frequency graph. Math 6 Notes Unit 6: Collecting and Displaying Data Page 14 of 22

15 10. The heights of 40 high school students (in inches) are given as follows: 62, 65, 54, 55, 50, 73, 73, 57, 64, 52, 62, 61, 53, 68, 64, 70, 66, 71, 63, 54, 64, 66, 56, 57, 63, 68, 53, 64, 68, 58, 66, 58, 58, 56, 64, 53, 67, 67, 70, 62. a. Construct a grouped frequency distribution for these heights using the intervals 72-75, 69-71, 66-68, and so on. b. Using the frequency distribution from part (a), construct a frequency polygon and a cumulative frequency graph. Measures of Central Tendency Problems Find the mean, mode, median, range 1. 1, 2, 3, 3, 4, 7, 9, , 14, 16, 13, , 2, 2, 3, 4, 5, 6, 7, 5, , 4, 4, 5, 8, 10, 12, 11, , 3, 8, 12, 10, , 14, 6, 2, 5, 7, 13, , 12, 3, 5, 7, 16, 13, 6, 7, The following table contains the number of traffic accidents in Nevada for the years Find the mean, mode, median, and range for the number of accidents. YEAR NUMBER YEAR NUMBER , Math 6 Notes Unit 6: Collecting and Displaying Data Page 15 of 22

16 9. Employees working at the Akron plant of the United Bug company have complained that they are discriminated against in the company s pay scale when compared to the pay scale at the Canton plant. The employees and their salaries are listed below. Akron Plant Canton Plant Mr. Jones $30,000 Mrs. Stein $20,000 Ms. Arthur $15,000 Mr. Patrick $20,000 Mr. Brady $15,000 Mr. Baron $20,000 a. What is the mean salary for all employees? b. What is the mean salary for the Akron workers? For Canton workers? c. What is the median salary for Akron workers? For Canton workers? d. What is the midrange salary for Akron workers? For Canton workers? e. If you were a lawyer acting for United Bug, which measure of central tendency would you use? f. If you were a lawyer for the Akron workers which measure of central tendency would you use 10. The mean score of a set of 12 tests is 68. What is the sum of the 12 tests scores? 11. The mean score on a set of 15 college entrance exams is 87. What is the sum of the 15 exam scores? 12. Two sets of data are given: the first set of data has 20 scores with a mean of 50, and the second set of data has 33 scores with a mean of 75. What is the mean if the two sets of data are combined? 13. In a math class Joe takes 10 tests with a mean score of 79. To get a B in the course students need a mean score of 80. What is wrong with the argument that Joe missed getting a B by 1 point. 14. The table below indicates the grades for fifty freshmen registered for MAT 114. Find the mean, mode, median, and range for the grades of the freshmen. Grade #of People A 4 B 18 C 10 D 12 F 6 Math 6 Notes Unit 6: Collecting and Displaying Data Page 16 of 22

17 15, The table below indicates the heights for a group of teenagers. Find the mean, mode, median, and range, for the teenagers heights. Height (inches) # of PeopleHeight (inches) # of People On a math test the following scores were made in a class of ten students: 81, 74, 87, 94, 71, 68, 72, 77, 81, 89. Find the mean, mode, median, and range for the set of data. 17. An experiment consists of tossing 8 coins and recording the number of heads that appear. The coins are tossed 10 times and the number of heads were 2, 3, 4, 5, 5, 6, 3, 2, 7, 3, respectively; find the mean, mode, median, and range, for the number of heads shown. Finding Measures of Central Tendency Problems Problem Variations in Finding the Mean 1. Find the mean of the following data: 78, 74, 81, 83, and In Ted s class of thirty students, the average on the math exam was 80. Andrew s class of forty students had an average of 90. What was the mean of the two classes combined? 3. Ted s bowling scores last week were 85, 89, and 101. What score would he have to make on his next game to have a mean of 105? 4. One of your students was absent on the day of the test. The class average for 24 students was 75%. After the other student took the test, the mean increased to 76%, what did the last student make on the test? Math 6 Notes Unit 6: Collecting and Displaying Data Page 17 of 22

18 5. Use the following graph to find the mean. Problem set STATISTICS Problems Find the mean, median, mode, and range given raw data Find a missing score given other scores and the mean Find the mean, median, mode and range using bar, line, and frequency graphs Find the measure of the central angle given data for a circle graph. Construct bar, line, frequency, and pie charts. 1. Bob bowled three games, his scores were 82, 85, and 88. Find his mean average. 2. Ted s scores on his tests were 62, 87, 75, 72, and 62. Find the mean, median, mode, and range. 3. A student has average score of 81 on three tests, if the student scored an 84 on the first two tests, what was the score on the third test? 4. Find the average rainfall per month if it rained 2.18 inches in June, 4.07 inches in July, 5.2 inches in August, and 1.07 inches in September. Math 6 Notes Unit 6: Collecting and Displaying Data Page 18 of 22

19 5. Bill scored 7, 12, 15, and 5 points in four basketball games. How many points must he score in the next game to have an average of 12 points per game? 6. The average income for 5 people is $150,000 per year. Four of the people earn $50,000, how much does the fifth person earn? 7. Find the median of the following list. 7, 12, 8, 9, 10, 4, 15, 17, John s average on his first four tests is 88. To earn an A, he must have an average of at least 90, what is the lowest grade he can make on his next test to earn an A? 9. Find the median and mode. Math 6 Notes Unit 6: Collecting and Displaying Data Page 19 of 22

20 10. Find the average temperature (mean). For what days was the temperature below average? What was the range of the temperatures? 11. Each month the Smith family uses its income in the following way: 30% for food, 25% for rent, 20% for transportation, 10% for savings, 5% for entertainment, and 10% for other expenses. Construct a pie chart representing this information. 12. Each dollar the government obtains in taxes is spent in the following manner: 25 for defense, 30 for social security, 10 for subsidies, 15 for salaries and 20 on social programs. Construct a circle graph representing this data. 13. There are 2000 students attending a certain high school. There are 400 seniors, 300 juniors, 500 sophomores, 600 freshmen, and th year students. Construct a circle graph showing this information. 14. The heights of 40 students in inches are given as follows: 62, 65, 54, 55, 50, 73, 73, 57, 64,52, 62, 61, 53, 68, 64, 70, 66, 71, 63, 54, 64, 66, 56, 57, 63, 68, 53, 64, 68, 58, 66, 58, 58, 56, 64, 53, 67, 67, 70, 62 Construct a grouped frequency distribution for the following intervals: 75-72, 71-69, 68-66, etc. Find the median, mode, and range of the heights. 15. The following table contains the number of accidents last week. Find the mean, median, mode, and range for the number of accidents. Monday 10 Tuesday 12 Wednesday 8 Thursday 15 Friday 8 Saturday 9 Sunday 8 Math 6 Notes Unit 6: Collecting and Displaying Data Page 20 of 22

21 16. Use the following table to find the mean, median, mode, and range for teenagers heights. Height # of people Height # of people How many students have above average height? 17. The grade distribution for the final exam in math is as follows: Find the median. Grade Frequency A 4 B 10 C 37 D 8 F In the data in the table were represented in a circle graph, find the measure of the central angle used to describe the tip. Lunch Cost Sandwich $5.00 Drink $1.00 Dessert $3.00 Tip $ Draw a line graph to show the relationship between the number of hours worked and the amount of money earned. $ Hours worked Math 6 Notes Unit 6: Collecting and Displaying Data Page 21 of 22

22 20. A merchant found that as the price of candy bars increased, the number of sales decreased. Sketch a line graph to show that relationship. Number Of candy Bars Price 21. You contract with your neighbor to cut and trim his lawn for a fixed fee, construct a line graph to show this relationship. $ Hours worked Math 6 Notes Unit 6: Collecting and Displaying Data Page 22 of 22