Chapter Nine Graphing and Introduction to Statistics Learning Objectives: Ch 9 What is mean, medians, and mode? Tables, pictographs, and bar charts Line graphs and predications Creating bar graphs and pie charts Describing and summarizing data sets 2 Statistics Applications: Mean, Median, and Mode
Mean (Average) The most common measure of central tendency is the mean (sometimes called the arithmetic mean or the average ). The mean (average) of a set of number items is the sum of the items divided by the number of items. 4 Finding the Mean Find the mean of the following list of numbers. 25 5.1 9.5 6.8 Continued. 5 Finding the Mean The mean is the average of the numbers: 25 +5.1+9.5 +6.8 + 5.1 5 9.5 =5.28 6.8 6
Median You may have noticed that a very low number or a very high number can affect the mean of a list of numbers. Because of this, you may sometimes want to use another measure of central tendency, called the median. The median of an ordered set of numbers is the middle number. If the number of items is even, the median is the mean (average) of the two middle numbers. 7 Finding the Median Find the median of the following list of numbers. 25 5.1 9.5 6.8 Continued. 8 Finding the Median List the numbers in numerical order: 25 5.1 Median 6.8 9.5 9
Helpful Hint In order to compute the median, the numbers must first be placed in order. 10 Mode The mode of a set of numbers is the number that occurs most often. (It is possible for a set of numbers to have more than one mode or to have no mode.) 11 Finding the Mode Find the mode of the following list of numbers. 25 5.1 9.5 6.8 Continued. 12
Finding the Mode List the numbers in numerical order: 5.1 51 The mode is. 9.5 6.8 13 Helpful Hint Don t forget that it is possible for a list of numbers to have no mode. For example, the list 2, 4, 5, 6, 8, 9 has no mode. There is no number or numbers that occur more often than the others. 14 Reading Pictographs, Bar Graphs, Histograms, and Line Graphs
Pictographs A pictograph is a graph in which pictures or symbols are used. This type of graph contains a key that explains the meaning of the symbol used. An advantage of using a pictograph to display information is that comparisons can easily be made. A disadvantage of using a pictograph is that it is often hard to tell what fractional part of a symbol is shown. 16 The pictograph shows the approximate number of passengers traveling on the leading U.S. passenger airlines. Leading U.S. Passenger Airlines Delta United American Southwest US Airways Northwest = 25 million passengers Source: Air Transport Association of America 1998 17 Bar Graphs Bar graphs can appear with vertical bars or horizontal bars. An advantage to using bar graphs is that a scale is usually included for greater accuracy. 18
Millions Bar Graphs 28 26 24 22 20 18 16 14 10 12 8 6 4 2 0 Sao Paulo New York City Mexico City Tokyo The bar graph shows the population of the world s largest cities. Source: United Nations, Dept. for Economic and Social Information and Policy Analysis 19 Histograms A histogram is a special bar graph. The width of each bar represents a range of numbers called a class interval. The height of each bar corresponds to how many times a number in the class interval occurred and is called the class frequency. The bars in a histogram lie side by side with no space between them. 20 Histograms Student Scores Frequency (# of students) nts 14 12 10 40-49 1 8 50-59 3 60-69 2 6 70-79 10 4 80-89 12 2 90-99 8 0 The test scores of 36 students are summarized in the table. Number of Stude 40-49 50-59 60-69 70-79 80-89 90-99 Student Test Scores 21
Line Graphs Another common way to display information graphically is by using a line graph. An advantage of a line graph is that it can be used to visualize relationships between two quantities. A line graph can also be very useful in showing change over time. 22 Line Graphs 30 25 20 15 10 5 0 Tornado Deaths Jan. Feb. March April May June July August Sept. Oct. Nov. Dec. Average Number of U.S. Deaths (1966-1999) Source: Storm Prediction Center 23 Reading Tables 24
Line Graphs and Predications 25 Line Graphs and Predications 26 Reading Circle Graphs A circle graph also called pie chart is often used to show percents in different categories, with the whole circle representing 100%. 27
A telephone survey was taken to identify favorite sport activities. The results of the four most popular activities are shown in the form of a circle graph below. Bowling 17% Biking 21% Swimming 25% Walking 37% 28 Drawing Circle Graphs To draw a circle graph, we use the fact that a whole circle contains 360 (degrees). 360 29 The following table shows the percent of U.S. armed forces personnel that are in each branch of the service. Branch of Service Percent Army 33% Navy 27% Marine Corps 12% Air Force 25% Coast Guard 3% Source: U.S. Department of Defense 30
To draw a circle graph showing this data, we find the number of degrees in each sector representing each branch of service. Sector Degrees in Each Sector Army 33% x 360 119 Navy 27% x 360 97 Marine Corps 12% x 360 43 Air Force 25% x 360 = 90 Coast Guard 3% x 360 11 31 We draw a circle and mark its center. Then we draw a line from the center of the circle to the circle itself. We use a protractor to construct the sectors. We place the hole in the protractor over the center of the circle. Then we adjust the protractor so that 0 on the protractor is aligned with the line that we drew. 32 To construct the Army sector, we find 119 on the protractor and mark our circle. Then we remove the protractor and use this mark to draw a second line from the center to the circle itself. 33
To construct the Navy sector next, we follow the same procedure as before except that we line 0 up with the second line we drew and mark the protractor this time at 97. 34 We continue in this manner until the circle graph is complete. 35 The Rectangular Coordinate System and Paired Data
The Rectangular Coordinate System y-axis quadrant II 5 4 3 2 1 quadrant I origin (0,0) -5-4 -3-2 -1 1 2 3 4 5-2 -3 quadrant III -4 quadrant IV -5 x-axis 37 Plotting Points y 5 4 3 2 1 (4,3) -5-4 -3-2 -1-2 -3-4 -5 1 2 3 4 5 x 38 In general, to plot the ordered pair (x,y), start at the origin. Next, (x,y) move x units left or right and then move y units up or down. right if x is positive, left if x is negative up if y is positive, down if y is negative 39
Helpful Hint Since the first number, or x-coordinate,, of an ordered pair is associated with the x-axis,, it tells how many units to move left or right. Similarly, the second number, or y-coordinate, tells how many units to move up or down. 40 Plot (2,1), (-3,4), (5,0), (0,-2), (1,-3),, and (-4,-5). y (-3,4) 5 4 3 2 1-5 -4-3 -2-1 -2-3 (-4,-5) -4-5 (2,1) (5,0) x 1 2 3 4 5 (0,-2) (1,-3) 41 Helpful Hint Remember that each point in the rectangular coordinate system corresponds to exactly one ordered pair and that each ordered pair corresponds to exactly one point. 42
Helpful Hint If an ordered pair has a y-coordinate of 0, its graph lies on the x-axis. If an ordered pair has an x-coordinate of 0, its graph lies on the y-axis. Order is the key word in ordered pair. The first value always corresponds to the x-value and the second value always corresponds to the y-value. 43 Completing Ordered Pair Solutions An equation in two variables, such as 3x + y = 9, has solutions consisting of two values, one for x and one for y. For example, x = 1 and y = 6 is a solution of 3x + y = 9, because, if x is replaced with h1 and y is replaced with 6, we get a true statement. 3x + y = 9? 3(1) + 6 = 9 9 = 9 True The solution x = 1 and y = 6 can be written as (1,6), an ordered pair of numbers. 44 In general, an ordered pair is a solution of an equation in two variables if replacing the variables by the values of the ordered pair results in a true statement. 45
If the x-value of an ordered pair is known, then the y-value can be determined, and vice-versa. Complete each ordered pair so that it is a solution to the equation 2x - y = 6. (0, ) (, 4) Let x = 0 and solve for y. 2x - y = 6 2(0) - y = 6 0 - y = 6 y = - 6 Let y = 4 and solve for x. 2x - y = 6 2x - 4 = 6 2x = 10 x = 5 The ordered pair is (0,- 6). The ordered pair is (5,4). 46 Helpful Hint Have you noticed? Equations in two variables can have more than one solution. 47 Graphing Linear Equations in Two Variables
Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form ax + by = c where a, b,, and c are numbers, and a and b are not both 0. Examples 3x + 2y = 6 y = 8 3x = 4 Graphing Linear Equations by Plotting Points Every linear equation in two variables has infinitely many ordered-pair solutions. Since it is impossible to list every solution, we graph the solutions instead. Graphing Linear Equations by Plotting Points The pattern described by the solutions of a linear equation makes seeing the solutions possible by graphing because all the solutions of a linear equation in two variables correspond to points on a single straight line. If we plot a few of these points and draw the straight line connecting them, we have a complete graph of all the solutions.
To graph the equation x + y = 5, we plot a few ordered-pair solutions, say (2,3), (0,5),, and (-1, 6). Then we connect the points. y (-1,6) 5 4 3 2 1-5 -4-3 -2-1 (0,5) (2,3) 1 2 3 4 5 x To Graph a Linear Equation in Two Variables Find three ordered-pair solutions. Graph the solutions. Draw a line through h the plotted points. To Find an Ordered-Pair Solution of an Equation Choose either an x-value or y-value of the ordered pair. Complete the ordered pair by replacing the variable with the chosen value and solving for the unknown variable.
Helpful Hint All three points should fall on the same straight line. If not, check your ordered-pair solutions for a mistake, since every linear equation is a line. Horizontal Lines/Vertical Lines y y x x y = b x = a Counting and Introduction to Probability
Likelihood or Probability In our daily conversations, we often talk about the likelihood or the probability of a given result occurring for a chance happening. We call the chance happening an experiment. The possible results of an experiment are called outcomes. 58 Using a Tree Diagram Flipping a coin is an experiment and the possible outcomes are heads (H) or tails (T) and are equally likely to happen. One way to picture the outcomes of an experiment is to draw a tree diagram. Each outcome is shown on a separate branch. For example, the outcomes of flipping a coin are H T 59 A Tree Diagram for Tossing a Coin Twice There are 4 possible outcomes when tossing a coin twice. First Toss Second Toss Outcomes H T H T H T H,H H,T T,H T,T 60
The Probability of an Event probability of an event = number of ways that the event can occur number of possible outcomes To find the probability of an event, divide the number of ways that the event can occur by the number of possible outcomes. 61 Helpful Hint Note from the definition of probability that the probability of an event is always between 0 and 1, inclusive (i.e., including 0 and 1). A probability of 0 means an event won t occur, and a probability of 1 means that an event is certain to occur. 62 Box-and-Whisker Plots Statistics assumes that your data points (the numbers in your list) are clustered around some central value. The box in the box-and-whisker plot contains, and thereby highlights, the middle half of these data points. 63
Box-and-Whisker Plots 1. To create a box-and-whisker plot, you start by ordering your data (putting the values in numerical order), if they aren't ordered already. Then you find the median of your data. 2. The median divides the data into two halves. To divide the data into quarters, you then find the medians of these two halves. 3. Note: If you have an even number of values, so the first median was the average of the two middle values, then you include the middle values in your sub-median computations. 64