Organizing and Visualizing Data © 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 2 Populations and Samples Statistics is an area of mathematics in which we are interested in gathering, organizing, analyzing, and making predictions from numerical information called data. The set of all items under consideration is called the population. Often only a sample or subset of the population is considered. We will describe a sample as biased if it does not accurately reflect the population as a whole with regard to the data that we are gathering.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 3 Populations and Samples Bias could occur because of the way in which we decide how to choose the people to participate in the survey. This is called selection bias. Another issue that can affect the reliability of a survey is the way we ask the questions, which is called leading-question bias.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 4 Frequency Tables We show the percent of the time that each item occurs in a frequency distribution using a relative frequency distribution. We often present a frequency distribution as a frequency table where we list the values in one column and the frequencies of the values in another column.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 5 Example: 25 viewers evaluated the latest episode of CSI. The possible evaluations are (E)xcellent, (A)bove average, a(V)erage, (B)elow average, (P)oor. After the show, the 25 evaluations were as follows: A, V, V, B, P, E, A, E, V, V, A, E, P, B, V, V, A, A, A, E, B, V, A, B, V Construct a frequency table and a relative frequency table for this list of evaluations. Frequency Tables (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 6 Solution: We organize the data in the frequency table shown below. Frequency Tables (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 7 We construct a relative frequency distribution for these data by dividing each frequency in the table by 25. For example, the relative frequency Frequency Tables of E is.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 8 Example: Suppose 40 health care workers take an AIDS awareness test and earn the following scores: 79, 62, 87, 84, 53, 76, 67, 73, 82, 68, 82, 79, 61, 51, 66, 77, 78, 66, 86, 70, 76, 64, 87, 82, 61, 59, 77, 88, 80, 58, 56, 64, 83, 71, 74, 79, 67, 79, 84, 68 Construct a frequency table and a relative frequency table for these data. Frequency Tables (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 9 Solution: Because there are so many different scores in this list, the data is grouped in ranges. Frequency Tables (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 10 We divide each count in the frequency column by 40. For example, in the row labeled 55–59, we divide 3 by 40 to get in the third column. Frequency Tables

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 11 Representing Data Visually A bar graph is one way to visualize a frequency distribution. In drawing a bar graph, we specify the classes on the horizontal axis and the frequencies on the vertical axis. If we are graphing a relative frequency distribution, then the heights of the bars correspond to the size of the relative frequencies. Graphing the relative frequencies, rather than the actual values in data sets, allows us to compare the distributions.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 12 Representing Data Visually Example: Draw a bar graph of the frequency distribution of viewers’ responses to an episode of CSI. (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 13 Representing Data Visually Solution: The bar graph is shown below.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 14 Representing Data Visually Example: Draw a bar graph of the relative frequency distribution of viewers’ responses to an episode of CSI. (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 15 Representing Data Visually Solution: The bar graph for the relative frequency is shown below.

Warm up Suppose 40 health care workers take an AIDS awareness test and earn the following scores: 79, 62, 87, 84, 53, 76, 67, 73, 82, 68, 82, 79, 61, 51, 66, 77, 78, 66, 86, 70, 76, 64, 87, 82, 61, 59, 77, 88, 80, 58, 56, 64, 83, 71, 74, 79, 67, 79, 84, 68 1)Construct a frequency table 2)Construct a relative frequency table 3)Create a bar graph for each

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 17 Representing Data Visually A variable quantity that cannot take on arbitrary values is called discrete. Other quantities, called continuous variables, can take on arbitrary values. The number of children in a family is an example of a discrete variable. Weight is an example of a continuous variable. We use a special type of bar graph called a histogram to graph a frequency distribution when we are dealing with a continuous variable quantity or a variable quantity that is discrete, but has a very large number of different possible values.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 18 Representing Data Visually Example: A clinic has the following data regarding the weight lost by its clients over the past 6 months. Draw a histogram for the relative frequency distribution for these data. (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 19 Representing Data Visually Solution: We first find the relative frequency distribution. (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 20 Representing Data Visually Draw the histogram exactly like a bar graph except that we do not allow spaces between the bars.

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 21 Representing Data Visually Example: The bar graph shows the number of Atlantic hurricanes over a period of years. Answer the questions on the slides that follow. (continued on next slide)

Representing Data Visually What was the smallest number of hurricanes in a year during this period? What was the largest? a)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 23 Representing Data Visually (continued on next slide) What was the smallest number of hurricanes in a year during this period? What was the largest? Solution: The smallest number of hurricanes in any year during this time period was 4. The largest number was 19. a)

Representing Data Visually b) What number of hurricanes per year occurred most frequently?

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 25 Representing Data Visually b) (continued on next slide) What number of hurricanes per year occurred most frequently? Solution : We look for the tallest bar, which appears over the number 11. Therefore, 11 hurricanes occurred in 10 different years.

Representing Data Visually c) How many years were the hurricanes counted?

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 27 Representing Data Visually c) (continued on next slide) How many years were the hurricanes counted? Solution : We add the heights of all of the bars to get = 58 years.

Representing Data Visually d) In what percentage of the years were there more than 10 hurricanes?

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 29 Representing Data Visually d) values: = 25. There are 58 years of data, so. Solution: We count the number of years in which there were more than 10 hurricanes and add the heights of the bars above these In what percentage of the years were there more than 10 hurricanes?

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 30 Stem and Leaf Display Example: The following are the number of home runs hit by the home run champions in the National League for the years 1975 to 1989 and for 1993 to –1989: 38, 38, 52, 40, 48, 48, 31, 37, 40, 36, 37, 37, 49, 39, –2007: 46, 43, 40, 47, 49, 70, 65, 50, 73, 49, 47, 48, 51, 58, 50 Compare these home run records using a stem- and-leaf display. (continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 31 Stem and Leaf Display Solution: In constructing a stem-and- leaf display, we view each number as having two parts. The left digit is considered the stem and the right digit the leaf. For example, 38 has a stem of 3 and a leaf of 8. (continued on next slide) 1975 to to 2007

© 2010 Pearson Education, Inc. All rights reserved.Section 15.1, Slide 32 Stem and Leaf Display We can compare these data by placing these two displays side by side as shown below. Some call this display a back-to-back stem-and-leaf display. It is clear that the home run champions hit significantly more home runs from 1993 to 2007 than from 1975 to 1989.