Chapter 2 Frequency Distributions and Graphs 1 Copyright © 2012 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

C H A P T E R Outline 2 Frequency Distributions and Graphs 2-1Organizing Data 2-2Histograms, Frequency Polygons, and Ogives 2-3Other Types of Graphs

C H A P T E R Objectives 2 Frequency Distributions and Graphs 1Organize data using a frequency distribution. 2Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives. 3Represent data using bar graphs, Pareto charts, time series graphs, and pie graphs. 4Draw and interpret a stem and leaf plot.

Introduction to Chapter 2 Now that we know what data is, we can begin organizing it to help us (and others) understand what the data has to say. The organization of data is what Chapter two is all about. Bluman, Chapter 24

2-1 Organizing Data raw data Data collected in original form is called raw data. frequency distribution A frequency distribution is the organization of raw data in table form, using classes and frequencies. categorical frequency distributions Nominal- or ordinal-level data that can be placed in categories is organized in categorical frequency distributions. 5 Bluman Chapter 2

Categorical Freq. Distributions… …have the following three characteristics: The categories (classes) The frequency (count) of each class The percentage of each class Bluman, Chapter 26

Chapter 2 Frequency Distributions and Graphs Section 2-1 Example 2-1 Page #38 7 Bluman Chapter 2

Categorical Frequency Distribution Twenty-five army inductees were given a blood test to determine their blood type. Raw Data: A,B,B,AB,O O,O,B,AB,B B,B,O,A,O A,O,O,O,AB AB,A,O,B,A Construct a frequency distribution for the data. 8 Bluman Chapter 2

Categorical Frequency Distribution Twenty-five army inductees were given a blood test to determine their blood type. Raw Data: A,B,B,AB,O O,O,B,AB,B B,B,O,A,O A,O,O,O,AB AB,A,O,B,A ClassTallyFrequencyPercent A B O AB IIII IIII II IIII Bluman Chapter 2

Grouped Frequency Distribution Grouped frequency distributions Grouped frequency distributions are used when the range of the data is large. lower upper class limitsClass boundaries The smallest and largest possible data values in a class are the lower and upper class limits. Class boundaries separate the classes. 10 Bluman Chapter 2

Grouped Frequency Distribution Columns of the table:  Class Limits  Class Boundaries  Class Midpoints  Tally  Frequency (count)  Cumulative Frequency Procedure Chart, Page 44. Bluman, Chapter 211 Sometimes used and sometimes not. I want us to be in the habit of always finding them when we need them.

Rules for Classes in Grouped Frequency Distributions 1. There should be 5-20 classes. 2. The class width should be an odd number (ends with prettier midpoint). 3. The classes must be mutually exclusive. 4. The classes must be continuous. 5. The classes must be exhaustive. 6. The classes must be equal in width. 12 Bluman Chapter 2

Grouped Frequency Distribution class width The class width can be calculated by subtracting  successive lower class limits (or boundaries)  successive upper class limits (or boundaries)  upper and lower class boundaries class midpoint X m The class midpoint X m can be calculated by averaging  upper and lower class limits (or boundaries) 13 Bluman Chapter 2

Chapter 2 Frequency Distributions and Graphs Section 2-1 Example 2-2 Page #41 14 Bluman Chapter 2

Constructing a Grouped Frequency Distribution The following data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes Bluman Chapter 2

Constructing a Grouped Frequency Distribution STEP 1 Determine the classes. Find the class width by dividing the range by the number of classes 7. Range = High – Low = 134 – 100 = 34 Width = Range/7 = 34/7 = 5 Rounding Rule: Always round up if a remainder. 16 Bluman Chapter 2

Constructing a Grouped Frequency Distribution For convenience sake, we will choose the lowest data value, 100, for the first lower class limit. The subsequent lower class limits are found by adding the width to the previous lower class limits. Class Limits The first upper class limit is one less than the next lower class limit. The subsequent upper class limits are found by adding the width to the previous upper class limits. 17 Bluman Chapter 2

Constructing a Grouped Frequency Distribution The class boundary is midway between an upper class limit and a subsequent lower class limit. 104,104.5, Bluman Chapter 2 Class Limits Class Boundaries Class midp. Freq. Cum. Freq

Constructing a Grouped Frequency Distribution STEP 2 Find the class midpoints. STEP 3 Find the frequencies and cumulative frequencies. Class Limits Class Boundaries Class midp. Freq. Cum. Freq Bluman Chapter 2

Grup Work Break up into groups of 3 and do #9 on page 46. Bluman, Chapter 220

2-2 Histograms, Frequency Polygons, and Ogives 3 Most Common Graphs in Research 1. Histogram 2. Frequency Polygon 3. Cumulative Frequency Polygon (Ogive) 21 Bluman Chapter 2

Histogram histogram The histogram is a graph that displays the data by using adjacent vertical bars of various heights to represent the frequencies of the classes. Horizontal Values: Class Boundaries Vertical Values: Frequencies Label x-axis and y-axis accordingly 22 Bluman Chapter 2

Chapter 2 Frequency Distributions and Graphs Section 2-2 Example 2-4 Page #51 23 Bluman Chapter 2

Histograms Construct a histogram to represent the data for the record high temperatures for each of the 50 states (see Example 2–2 for the data). 24 Bluman Chapter 2

Histograms Class Limits Class Boundaries Frequency Histograms use class boundaries and frequencies of the classes. 25 Bluman Chapter 2

Histograms Histograms use class boundaries and frequencies of the classes. 26 Bluman Chapter 2

Frequency Polygons frequency polygon The frequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the class midpoints. Horizontal Values: Class Midpoints Vertical Values: Frequencies Label x-axis and y-axis accordingly Anchor graph to x-axis at beginning and end 27 Bluman Chapter 2

Reminder: Class Midpoints Class Midpoints are found by adding the lower and upper limits (or boundaries) of a class and dividing by two. Find the midpoints first, and then begin to construct the graph. Bluman, Chapter 228

Chapter 2 Frequency Distributions and Graphs Section 2-2 Example 2-5 Page #53 29 Bluman Chapter 2

Frequency Polygons Construct a frequency polygon to represent the data for the record high temperatures for each of the 50 states (see Example 2–2 for the data). 30 Bluman Chapter 2

Frequency Polygons Class Limits Class Midpoints Frequency Frequency polygons use class midpoints and frequencies of the classes. 31 Bluman Chapter 2

Frequency Polygons Frequency polygons use class midpoints and frequencies of the classes. A frequency polygon is anchored on the x-axis before the first class and after the last class. 32 Bluman Chapter 2

Ogives ogive The ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution. Horizontal Values: Class Boundaries Vertical Values: Cumulative Frequencies Label x-axis and y-axis accordingly Anchor graph to x-axis at 1 st lower class boundary (no point drawn) 33 Bluman Chapter 2

Chapter 2 Frequency Distributions and Graphs Section 2-2 Example 2-6 Page #54 34 Bluman Chapter 2

Ogives Construct an ogive to represent the data for the record high temperatures for each of the 50 states (see Example 2–2 for the data). 35 Bluman Chapter 2

Ogives Ogives use class boundaries and cumulative frequencies of the classes. Class Limits Class Boundaries Frequency Cumulative Frequency Bluman Chapter 2

Ogives Ogives use class boundaries and cumulative frequencies of the classes. Class Boundaries Cumulative Frequency Less than 99.5 Less than Less than Less than Less than Less than Less than Less than Bluman Chapter 2

Ogives Ogives use class boundaries and cumulative frequencies of the classes. 38 Bluman Chapter 2

Procedure Table Constructing Statistical Graphs Step 1 Draw and label the x and y axes. Step 2 Choose a suitable scale for the frequencies or cumulative frequencies, and label it on the y axis. Step 3 Represent the class boundaries for the histogram or ogive, or the midpoint for the frequency polygon, on the x axis. Step 4 Plot the points and then draw the bars or lines.

Shapes of Distributions 40 Bluman Chapter 2

Shapes of Distributions 41 Bluman Chapter 2

2.3 Other Types of Graphs 42 Bluman Chapter 2 Bar Graphs Pareto Chart Time Series Graph Pie Graph Stem and Leaf Plot

Bar Graphs Used with qualitative or categorical data. Has horizontal or vertical bars:  Each bar represents a category of the data  The height (length) of each bar is the frequency of the data in that category Bars are not adjacent. Label graph. Bluman, Chapter 243

Bar Graphs Bluman, Chapter 244

Pareto Charts Used with qualitative or categorical data. Categories on x-axis, frequencies on the y. Categories are arranged from highest frequency to the lowest. Bars are adjacent to each other. Label graph. Bluman, Chapter 245

Pareto Charts 46 Bluman Chapter 2

Time Series Graphs Used when data is collected over a period of time. X-axis is the time (unit depends on data). Y-axis represents the frequencies / amount/ percent/ etc. Plot points that correspond to the data. Draw straight line segments that connect each pair of points. Do not anchor to axis. Label graph. Can be used to compare multiple sets of data. Bluman, Chapter 247

Time Series Graphs 48 Bluman Chapter 2

Pie Graphs Used extensively in statistics. Shows the relationship of the parts of the data (categories) to the whole. Visually compares the size (%) of the sections. Used with categorical data. Find the % of each category and construct pie graph. Be sure to label each category and %. Bluman, Chapter 249

Pie Graphs 50 Bluman Chapter 2

Misleading Graphs Be careful of misleading graphs that may appear to skew the data at hand. This is why we need to be careful of the scales we use to construct the graphs. Page 76. Bluman, Chapter 251

Stem and Leaf Plots stem and leaf plot 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. It sorts and graphs. 52 Bluman Chapter 2

Stem and Leaf Plots How to construct:  Arrange the data in order.  Separate the data according to the first digit.  Use the first digit of every data value as the stem and the remaining digit at the leaf.  If more than two digits, then first two digits are the stem, last one is the leaf, etc. Bluman, Chapter 253

Chapter 2 Frequency Distributions and Graphs Section 2-3 Example 2-13 Page #80 54 Bluman Chapter 2

At an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a stem and leaf plot for the data Bluman Chapter 2

Stem and Leaf Plot 56 Bluman Chapter 2