Organizing and Visualizing Data

© 2010 Pearson Education, Inc. All rights reserved. Section 15.1, Slide 1

15.1

© 2010 Pearson Education, Inc. All rights reserved. Section 15.1, Slide 2

Populations and SamplesStatistics 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

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© 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

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We construct a relative frequency distributionfor these data by dividing each frequency in the table by 25. For example, the relative frequency

Frequency Tables

of E is .

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• 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

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• Solution: Because there are so many different scores in this list, the data is grouped in ranges.

Frequency Tables

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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 0.075 in the third column.

Frequency Tables

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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.

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© 2010 Pearson Education, Inc. All rights reserved. Section 15.1, Slide 13

Representing Data Visually• Solution: The bar graph is shown below.

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Representing Data Visually• Example: Draw a bar graph of the relative frequency distribution of viewers’ responses to an episode of CSI.

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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, 681)Construct a frequency table 2)Construct a relative frequency table3)Create a bar graph for each

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Representing Data VisuallyA 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.

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Representing Data Visually• Solution: We first find the relative frequency distribution.

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Representing Data VisuallyDraw the histogram exactly like a bar graph except that we do not allow spaces between the bars.

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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.

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

a)

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Representing Data Visually

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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 Visuallyb) What number of hurricanes per year occurred most frequently?

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Representing Data Visuallyb)

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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 Visuallyc) How many years were the hurricanes counted?

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Representing Data Visuallyc)

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How many years were the hurricanes counted?

• Solution: We add the heights of all of the bars to get1 + 1 + 6 + 6 + 9 + 4 + 6 + 10 + 5 + 5 + 3 + 1 + 1 = 58 years.

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

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Representing Data Visuallyd)

values: 10 + 5 + 5 + 3 + 1 + 1 = 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 2007. 1975–1989: 38, 38, 52, 40, 48, 48, 31, 37, 40,

36, 37, 37, 49, 39, 47 1993–2007: 46, 43, 40, 47, 49, 70, 65, 50, 73,

49, 47, 48, 51, 58, 50Compare these home run records using a stem-and-leaf display.

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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.

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1975 to 19891993 to 2007

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Stem and Leaf DisplayWe 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.