Slide 2-2 Copyright © 2008 Pearson Education, Inc. Chapter 2 Organizing Data.

Copyright © 2008 Pearson Education, Inc. Slide 2-2

Chapter 2

Organizing Data


Definition 2.1

Variables

Variable: A characteristic that varies from one person orthing to another.

Qualitative variable: A nonnumerically valued variable.

Quantitative variable: A numerically valued variable.

Discrete variable: A quantitative variable whosepossible values can be listed.

Continuous variable: A quantitative variable whosepossible values form some interval of numbers.


Figure 2.1


Definition 2.2

Data

Data: Values of a variable.

Qualitative data: Values of a qualitative variable.

Quantitative data: Values of a quantitative variable.

Discrete data: Values of a discrete variable.

Continuous data: Values of a continuous variable.


The table displays the number of days to maturity for 40 short-term investments. Getting a clear picture of these data is difficult, but is much easier if we group them into categories, or classes.

The first step is to decide on the classes. One convenient way to group these data is by 10s.

Table 2.1

Table 2.2


Table 2.3The percentage of a class, expressed as a decimal, is called the relative frequency of the class. A table that provides all classes and their relative frequencies is called a relative-frequency distribution. The table displays a relative-frequency distribution for the days-to-maturity data. Note that the relative frequencies sum to 1 (100%).


Definition 2.3Terms Used in Grouping

Classes: Categories for grouping data.

Frequency: The number of observations that fall in a class.

Frequency distribution: A listing of all classes and their frequencies.

Relative frequency: The ratio of the frequency of a class to the total number of observations.

Relative-frequency distribution: A listing of all classes and their relativefrequencies.

Lower cutpoint: The smallest value that could go in a class.

Upper cutpoint: The smallest value that could go in the next higher class(equivalent to the lower cutpoint of the next higher class).

Midpoint: The middle of a class, found by averaging its cutpoints.

Width: The difference between the cutpoints of a class.


Table 2.4

A table that provides the classes, frequencies, relative frequencies, and midpoints of a data set is called a grouped-data table. The table is a grouped-data table for the days-to-maturity data.


Example 2.7

TVs per Household Trends in Television, published by the Television Bureau of Advertising, provides information on television ownership. The table gives the number of TV sets per household for 50 randomly selected households. Use classes based on a single value to construct a grouped-data table for these data.

Table 2.8


Table 2.9

Solution Example 2.7 grouped-data table


Table 2.10

Example 2.8

Professor Weiss asked his introductory statistics students to state their political party affiliations as Democratic (D), Republican (R), or Other (O). The responses are given in the table. Determine the frequency and relative-frequency distributions for these data.


Table 2.11

Solution Example 2.8


Table 2.12

Example 2.10

The table shows frequency and relative-frequency distributions for the days-to-maturity data. Obtain graphical displays for these grouped data.


Solution Example 2.10One way to display these grouped data pictorially is to construct a graph, called a frequency histogram, that depicts the classes on the horizontal axis and the frequencies on the vertical axis.

Figure 2.2


Definition 2.4

Frequency and Relative-Frequency Histograms

Frequency histogram: A graph that displays the classeson the horizontal axis and the frequencies of the classeson the vertical axis. The frequency of each class is represented by a vertical bar whose height is equal to thefrequency of the class.

Relative-frequency histogram: A graph that displays theclasses on the horizontal axis and the relative frequenciesof the classes on the vertical axis. The relative frequencyof each class is represented by a vertical bar whoseheight is equal to the relative frequency of the class.


Table 2.14

Example 2.12

One of Professor Weiss’s sons wanted to add a new DVD player to his home theater system. He used the Internet to shop and went to pricewatch.com. There he found 16 quotes on different brands and styles of DVD players. Construct a dotplot for these data.



To construct a dotplot for the data, we begin by drawing a horizontal axis that displays the possible prices. Then we record each price by placing a dot over the appropriate value on the horizontal axis. For instance, the first price is $210, which calls for a dot over the “210” on the horizontal axis.

Figure 2.4


Table 2.15


For Table 2.15, let’s construct a stem-and-leaf diagram, which simultaneously groups the data and provides a graphical display similar to a histogram.



First, we list the leading digits of the numbers in the table (3, 4, . . . , 9) in a column, as shown to the left of the vertical rule. Next, we write the final digit of each number from the table to the right of the vertical rule in the row containing the appropriate leading digit.

Table 2.15

Figure 2.5


Table 2.16

Example 2.14

A pediatrician tested the cholesterol levels of several young patients and was alarmed to find that many had levels higher than 200 mg per 100 mL. Table 2.16 presents the readings of 20 patients with high levels. Construct a stem-and-leaf diagram for these data by using

a. one line per stem. b. two lines per stem.


Figure 2.6


The stem-and-leaf diagram in Fig. 2.6(a) is only moderately helpful because there are so few stems. Figure 2.6(b) is a better stem-and-leaf diagram for these data. It uses twolines for each stem, with the first line for the leaf digits 0-4 and the second line for the leaf digits 5-9


Example 2.15

Political Party Affiliations: The table shows the frequency and relative-frequency distributions for the political party affiliations of Professor Weiss’s introductory statistics students. Display the relative-frequency distribution of these qualitative data with a a. pie chart. b. bar graph.

Table 2.17



Figure 2.7


Definition 2.5

Distribution of a Data Set

The distribution of a data set is a table, graph, or formula that provides the values of the observations andhow often they occur.


The figure on the next slide displays a relative-frequency histogram for the heights of the 3264 female students who attend a midwestern college. Also included is a smooth curve that approximates the overall shape of the distribution. Both the histogram and the smooth curve show that this distribution of heights is bell shaped (or mound shaped), but the smooth curve makes seeing the shape a little easier.


Figure 2.8


Figure 2.9

Common distribution shapes


Example 2.20

The relative-frequency histogram for household size in the United States shown in the figure is based on data contained in Current Population Reports, a publication of the U.S. Census Bureau. Identify the distribution shape for sizes of U.S. households.

Figure 2.10

Solution - Skewed right


Definition 2.6

Population and Sample Data

Population data: The values of a variable for the entire population.

Sample data: The values of a variable for a sample of the population.


Definition 2.7

Population and Sample Distributions; Distribution of a Variable

The distribution of population data is called the population distribution, or the distribution of the variable.

The distribution of sample data is called a sample distribution.


Household Size Here the variable is household size, and the population consists of all U.S. households. It gives the population distribution or, equivalently, the distribution of the variable “household size.”

Figure 2.11


Key Fact 2.1

Population and Sample Distributions

For a simple random sample, the sample distributionapproximates the population distribution (i.e., thedistribution of the variable under consideration). The larger the sample size, the better the approximationtends to be.

Slide 2-2 Copyright © 2008 Pearson Education, Inc. Chapter 2 Organizing Data.

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