# © Copyright McGraw-Hill 2004 2-1 CHAPTER 2 Frequency Distributions and Graphs.

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© Copyright McGraw-Hill 2004 2-2 Objectives Organize data using frequency distributions. Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives. Represent data using Pareto charts, time series graphs, and pie graphs. Draw and interpret a stem and leaf plot.

© Copyright McGraw-Hill 2004 2-3 Introduction This chapter will show how to organize data and then construct appropriate graphs to represent the data in a concise, easy-to- understand form.

© Copyright McGraw-Hill 2004 2-4 Basic Vocabulary When data are collected in original form, they are called raw data. A frequency distribution is the organization of raw data in table form, using classes and frequencies. The two most common distributions are categorical frequency distribution and the grouped frequency distribution.

© Copyright McGraw-Hill 2004 2-5 Grouped Frequency Distributions When the range of the data is large, the data must be grouped into classes that are more than one unit in width. The lower class limit represents the smallest data value that can be included in the class.

© Copyright McGraw-Hill 2004 2-6 Grouped Frequency Distributions (cont’d.) The upper class limit represents the largest value that can be included in the class. The class boundaries are used to separate the classes so that there are no gaps in the frequency distribution.

© Copyright McGraw-Hill 2004 2-7 Class Boundaries Significant Figures Rule of Thumb: Class limits should have the same decimal place value as the data, but the class boundaries have one additional place value and end in a 5.

© Copyright McGraw-Hill 2004 2-8 Finding Class Boundaries The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the the lower (or upper) class limit of the next class. The class midpoint is found by adding the lower and upper boundaries (or limits) and dividing by 2.

© Copyright McGraw-Hill 2004 2-9 Class Rules There should be between 5 and 20 classes. The class width should be an odd number. The classes must be mutually exclusive. The classes must be continuous. The classes must be exhaustive. The classes must be equal width.

© Copyright McGraw-Hill 2004 2-10 Types of Frequency Distributions A categorical frequency distribution is used when the data is nominal. A grouped frequency distribution is used when the range is large and classes of several units in width are needed. An ungrouped frequency distribution is used for numerical data and when the range of data is small.

© Copyright McGraw-Hill 2004 2-11 Why Construct Frequency Distributions? To facilitate computational procedures for measures of average and spread. To enable the reader to determine the nature or shape of the distribution. To enable the reader to make comparisons among different data sets. To organize the data in a meaningful, intelligible way. To enable the researcher to draw charts and graphs for the presentation of data.

© Copyright McGraw-Hill 2004 2-12 Categorical Frequency 1. Make a table p. 36(List all categories) 2. Tally the data 3. Count the tallies 4. Find the percentage of values in each class by using the formula p. 36 5. Find the totals for the Frequency and Percent.

© Copyright McGraw-Hill 2004 2-13 Categorical Frequency Group Assignment Objective: To construct a Categorical Frequency Distribution Table 1. Data Bank p.745-746 2. Use the Marital Status Data ( 1- 25) 3. Material Needed: Chart Paper, Textbook, Markers, Notebook Paper, Ruler, Pencil

© Copyright McGraw-Hill 2004 2-14 Grouped Frequency 1. Determine the classes. Find the highest and lowest value. Find the range. Select the number of classes desired. Find the width by dividing the range by the number of classes and rounding up. Select a starting point; add the width to get the lower limit. Find the upper class limits. Find the boundaries.

© Copyright McGraw-Hill 2004 2-15 Grouped Frequency Distribution 2. Tally the data 3. Find the numerical frequencies from the tallies. 4.Find the cumulative frequencies.( Adding the frequency in each class to the total of the frequencies of the classes preceding that class)

© Copyright McGraw-Hill 2004 2-16 Grouped Frequency Distribution Class limitsClass BoundariesTallyFrequencyCum. Freq

© Copyright McGraw-Hill 2004 2-17 Ungrouped Frequency Distribution What is the difference between Grouped and Ungrouped? Range of Data is small. Class limits consisting of a single data value.

© Copyright McGraw-Hill 2004 2-18 The Role of Graphs The purpose of graphs in statistics is to convey the data to the viewer in pictorial form. Graphs are useful in getting the audience’s attention in a publication or a presentation.

© Copyright McGraw-Hill 2004 2-19 Three Most Common Graphs The histogram displays the data by using vertical bars of various heights to represent the frequencies.

© Copyright McGraw-Hill 2004 2-20 Three Most Common Graphs (cont’d.) The frequency polygon displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes.

© Copyright McGraw-Hill 2004 2-21 Three Most Common Graphs (cont’d.) The cumulative frequency or ogive represents the cumulative frequencies for the classes in a frequency distribution.

© Copyright McGraw-Hill 2004 2-22 Relative Frequency Graphs A relative frequency graph is a graph that uses proportions instead of frequencies. Relative frequencies are used when the proportion of data values that fall into a given class is more important than the frequency.

© Copyright McGraw-Hill 2004 2-23 Other Types of Graphs A Pareto chart is used to represent a frequency distribution for categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest.

© Copyright McGraw-Hill 2004 2-24 Other Types of Graphs (cont’d.) A time series graph represents data that occur over a specific period of time.

© Copyright McGraw-Hill 2004 2-25 Other Types of Graphs (cont’d.) A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

© Copyright McGraw-Hill 2004 2-26 Stem-and-Leaf Plots A stem-and-leaf plot is a data plot that uses part of a data value as the stem and part of the data value as the leaf to form groups or classes. It has the advantage over grouped frequency distribution of retaining the actual data while showing them in graphic form.

© Copyright McGraw-Hill 2004 2-27 2.4 Group Assignment 1. Create a graph according to the given data 2. Write-UP(on chart paper) Write the steps Key Points to Remember

© Copyright McGraw-Hill 2004 2-28 2.4 Group Assignment Data Pie Data Bank: X –AL, MS, GA, FL, TN Parento Data Bank: XII –Lighting, Tornadoes, Wind, Extreme Heat Time Data Bank: XII -Tornadoes 1993-1988 Stem and Leaf Data Bank: XIV- 1 st and 2 nd Column

© Copyright McGraw-Hill 2004 2-29 Summary of Graphs and Uses Histograms, frequency polygons, and ogives are used when the data are contained in a grouped frequency distribution. Pareto charts are used to show frequencies for nominal variables.

© Copyright McGraw-Hill 2004 2-30 Summary of Graphs and Uses (cont’d.) Time series graphs are used to show a pattern or trend that occurs over time. Pie graphs are used to show the relationship between the parts and the whole.

© Copyright McGraw-Hill 2004 2-31 Conclusions Data can be organized in some meaningful way using frequency distributions. Once the frequency distribution is constructed, the representation of the data by graphs is a simple task.