Statistical Graphs – section 3.2

II. Statistical Graphs – section 3.2 Histograms and Frequency Polygons are statistical graphs used to illustrate frequency distributions. Example: Construct a histogram for the frequency distribution below. # of Children 0 1 2 3 4 5 6 7 8 9 # of families 8 11 18 11 6 4 2 1 2 1 A histogram is just a bar graph. I will make my bars go vertically. It wouldn’t be wrong if I made my bars go horizontally. Start by creating a graph of the first quadrant of an “x” and “y” axis. Use the label of the left column for the x-axis, and the label on the top of the right column for the y-axis. Place the numbers in the left column beneath the horizontal line. Place the numbers in the right column to the left of the vertical line. You may have to increment the “y-axis” when it is convenient. I incremented this y-axis by 2 as I didn’t want to write 18 numbers down. It would have been hard to read the y-axis had I written down 18 numbers. Make a rectangle above each number on the x-axis the appropriate height. Number of Families This is an example of a histogram that models the data in the frequency distribution. 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 Number of children (It is certainly okay if the bars touch. I can’t figure out how to make that happen.) A frequency polygon is just a “line graph”. Let’s construct a frequency polygon to model the same data. I just plotted the points from the frequency distribution and then connected each point with a line segment. Example: Construct a frequency polygon for the data we used to create our frequency histogram. Number of families Start as if you were creating a histogram. Place a point above each point on the “x-axis”. Connect each point with a line segment. 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Number of Children Homework #1-4 1) Create a frequency histogram and a frequency polygon from the data in the frequency distribution below. Frequency distribution # of Children 0 1 2 3 4 5 6 7 8 9 Total # of families 8 11 18 11 6 4 2 1 2 1 64 2) Create a frequency histogram and a frequency polygon from the data in the frequency distribution below. Frequency distribution Year Unemployment rate 1920 5.2% 1930 4.2% 1940 8.7% 1950 5.3% 1960 5.5% 1970 4.9% 1980 7.1% 1990 5.6% 2000 4.0% 2010 9.6% 3) Create a frequency histogram and a frequency polygon from the data in the frequency distribution below. Put the entire class on the horizontal axis. Income (in thousands of dollars) 16-22 23-29 30-36 37-43 44-50 Total Number of families 2 5 4 3 1 15 4) Create a frequency histogram and a frequency polygon from the data in the frequency distribution below. Frequency Distribution Class 40-49 50-59 60-69 70-79 80-89 90-99 Total Frequency 7 5 3 2 7 1 25 Stem and Leaf Displays: A stem and leaf display is a tool that organizes and groups the data while allowing us to see the actual values that make up the data. To construct a stem and leaf display each value is represented by two different groups of digits. The left group is called the stem and the remaining group of digits is called the leaf. There are no rules for the number of digits to be included in the stem. Basically stem and leaf displays break numbers up. Here are a few ways numbers can be broken up using a stem and leaf display. 53 could get broken up 5 Stem 483 could be either 48 3 Stem Leaf 6 could get written 0 6 Stem Leaf 3 Leaf or 4 83 Stem Leaf Example: The following data represents the ages of 20 guests at 4 PM when happy hour starts at the Yard House. A stem and leaf display has been started. Finish the stem and leaf display. 27 39 47 60 Stem 2 3 4 5 6 7 28 43 50 62 Leaves 789 129 29 44 56 68 31 44 58 71 32 45 59 72 Notice the Stems are in ascending order. The leaves can be in any order BUT most people like them to ascend also. Also notice that a stem and leaf display looks like a sideways histogram. I will just add the end of each number next to the appropriate number. Stem 2 3 4 5 6 7 Leaves 789 129 34457 0689 028 12 Homework #5-8: Create a stem and leaf display for the given data. 5) The table below indicates the ages of each student in a recent MAT 142 class. Construct a stem and leaf display. 18 20 31 18 20 35 18 21 36 19 25 44 19 28 49 6) Twenty workers at a local mall were asked how many miles they commute to work one way. The data below gives the result of the survey. Construct a stem and leaf display. 1 8 13 18 3 9 14 20 3 9 14 21 7 10 15 21 7 11 17 23 7) Starting salaries of recent computer science majors (in thousands of dollars) are given in the table below. Construct a stem and leaf display. 42 55 62 45 57 64 48 58 65 49 59 66 50 60 67 51 61 73 8) Fifteen students were asked how many times a week they ate fast food. The results are given in the table below. Construct a stem and leaf display. 0 5 14 1 8 15 2 11 20 4 11 21 4 13 22 Circle graphs (also known as pie charts) are often used to compare parts of one or more components of the whole to the whole. It will be important to know the relative sizes of regions we are constructing when we are creating circle graphs. Here are some common percent graphed. 50% takes up half of the graph 25% takes up a quarter of the graph. 12.5% takes up 1/8 of the graph 12,50% 12,50% 25% 50,00% Here is a representation of 90% and 10% 10% 90% For the most part I just eyeball it when I create circle graphs. I generally know about how large I need to make a region and I just try to get close. I never measure. Example: 600 people exiting the AMC movie theatre at Arrowhead Mall were asked what type of movie they just watched. The results can be seen below… Movie Type Action Comedy Drama Science Fiction # of People 150 300 75 75 Use this information to create a circle graph To do this “by hand” we have to do a little math. We need to change each number to a percent, or if you like a degree. I won’t show how to use degrees as percent seems to be more common. Movie type Action # of people 150 Comedy 300 Drama 75 Science Fiction 75 Totals % of total to nearest tenth 150 = 0.25 = 25% 600 300 600 75 600 75 600 = 0.50 = 50% = 0.125 = 12.5% = 0.125 = 12.5% 600 100% Note: The percent column must add up to 100%. Lastly, create a reasonable picture. The Comedy sector must take up half of the graph. It can be the top of the graph, the bottom or like I showed the right side of the graph. The Action region needs to take up 25% or one quarter of the graph. Название диаграммы 12,50% 12,50% 50% 25% Action Comedy Drama Science Fiction Homework #9-11: Construct a Circle graph from the given data. 9) Type of cell phone IPhone Galaxy Flip Phone Other No cell phone Number of users 50 25 10 10 5 10) IQ 85 - 100 100 - 115 115 - 130 Higher than 130 Number of people 40 30 10 5 11) Height 5 feet – 5 feet 6 inches 5 feet 7 inches to 6 feet Over 6 feet 1 inch Number of people 100 80 20 Example: A survey was given to 200 people. The results are summarized in the circle graph below. a) How many peoples favorite sport is Tennis? Simply multiply the total number of people by the percent (in decimal form) that say their favorite sport is tennis. 200 * 0.18 = 36 Answer: 36 people b) How many more people favored soccer than basketball? Find out how many favored each. Number that favored soccer: 200*.29 = 58 Number that favored basketball: 200*.22 = 44 Then subtract the amounts. Answer 58 – 44 = 14 more people favored soccer than basketball. Homework: 12 - 19 12) Andrew spends 25 hours a week doing leisure activities. The results are summarized in the circle graph below. a) How many hours a week does Andrew spend watching television? (Round to 2 decimals.) b) How much more time does Andrew spend playing sports than he does reading? (Round to 2 decimals.) 13) A survey was given to 500 students. The results are summarized in the circle graph below. a) How many students favorite type of music is country? b) What is the most popular type of music and how many students say it is their favorite? 14) The current population of the USA is approximately 320 million people. The following circle graph summarizes the population of the USA. Ages of people living in the USA 10% 20% 30% 40% 0-17 18-44 45-64 65 and older a) How many people in the USA are 45 or older? b) How many are under 45? 15) 300 people attended a concert. The circle graph below summarizes the ages of the people who attended the concert. a) How many people at the concert were 30 or older? b) How many were under 50? 16) This frequency histogram represents the salary of employees in a certain company. You will have to estimate some of the numbers, and your answer could be slightly different than mine if your estimates are different. a) How many employees made at least 44 thousand dollars per year? b) How many made less than 33 thousand per year? 17) The ages of people attending a school assembly are summarized in the histogram below. You will have to estimate some of the numbers, and your answer could be slightly different than mine if your estimates are different. Название диаграммы number of students 90 80 70 60 50 40 30 20 10 0 8 9 10 11 12 13 age in years a) How many people were at least 15? b) How many were under 12? 14 15 16 17 18) 200 students took a final exam and the results are displayed in the histogram below. a) What percent of the students got an A on the final exam? b) How many students got an A on the final exam? c) How many students failed the final, if a grade of a D or F is considered failing? 19) 300 men were measured. The results (in percent) are summarized in the histogram below. 20 Number with the height 18 16 14 12 10 8 6 4 2 0 63 64 65 66 67 68 69 70 71 72 73 Height in inches a) How many men had heights between 68 and 70 inches inclusive? b) How many men had heights greater than 72 inches? 74 number of families Answers: 1) Histogram 20 15 10 5 0 0 1 2 3 4 5 6 7 8 Number of Children Number of families Frequency polygon 20 15 10 5 0 0 1 2 3 4 5 6 7 8 Number of children Number of families 3) Histogram 20 15 10 5 0 Income in thousands of dollars Frequency polygon 9 9 Number of families 20 15 10 5 0 16-22 23-29 30-36 37-43 44-50 Total Income in thousands of dollars 5) Stem 1 2 3 4 Leaves 88899 00158 156 49 7) Stem 4 5 6 7 Leaves 2589 015789 0124567 3 9) 5% 10% 10% 50% 25% IPhone Galaxy Flip Phone Other No cell phone 11) 10% 40% 5 feet – 5 feet 6 inches 50% 5 feet 7 inches to 6 feet Over 6 feet 1 inch 13a) 15 13b) pop, 320 15a) 198 15b) 270 17a) 70 17b) 130 19a) 37 men 19b) 7 men