STATISTICS for the Utterly 1-1 Confused, 2nd ed. SLIDES PREPARED By Lloyd R. Jaisingh Ph.D. Morehead State University Morehead KY 1-2 Part 1 DESCRIPTIVE STATISTICS Chapter 1 Graphical Displays of Univariate Data 1-3 Outline Do I Need to Read This Chapter? 1-1 Introduction 1-2 Frequency Distributions 1-3 Dot Plots 1-4 Bar Charts or Bar Graphs 1-5 Histograms 1-4 Outline 1-6 Frequency Polygons 1-7 Stem-and-Leaf Displays or Plots 1-8 Time Series Graphs 1-9 Pie Graphs or Pie Charts 1-10 Pareto Charts It’s a Wrap 1-5 Objectives Introduction of some basic statistical terms. Introduction of some graphical displays. 1-6 Introduction What is statistics? Statistics is the science of collecting, organizing, summarizing, analyzing, and making inferences from data. The subject of statistics is divided into two broad areas—descriptive statistics and inferential statistics. Breakdown of the subject of statistics 1-7 Statistics Descriptive Inferential Statistics Statistics Includes Includes Collecting Making inferences Organizing Hypothesis testing Summarizing Determining Presenting relationships data Making predictions 1-8 Introduction (contd.) Explanation of the term data: Data are the values or measurements that variables describing an event can assume. Variables whose values are determined by chance are called random variables. Types of variables – there are two types: qualitative and quantitative. 1-9 Introduction (contd.) What are qualitative variables: These are variables that are nonnumeric in nature. What are quantitative variables: These are variables that can assume numeric values. Quantitative variables can be classified into two groups – discrete variables and continuous variables. Breakdown of the types of variables 1-10 Variables Quantitative Qualitative Includes Discrete Continuous variables 1-11 Introduction (contd.) • What are quantitative data: These are data values that are numeric. • Example: the heights of female basketball players. • What are qualitative data: These are data values that can be placed into distinct categories according to some characteristic or attribute. • Example: the eye color of female basketball players. 1-12 Introduction (contd.) • What are discrete variables: These are variables that can assume values that can be counted. • Example: the number of days it rained in your neighborhood for the month of March. • What are continuous variables: These are variables that can assume all values between any two values. • Example: the time it takes to complete a quiz. 1-13 Introduction (contd.) • In order for statisticians to do any analysis, data must be collected or sampled. • We can sample the entire population or just a portion of the population. • What is a population? A population consists of all elements that are being studied. • What is a sample?: A sample is a subset of the population. 1-14 Introduction (contd.) • Example: If we are interested in studying the distribution of ACT math scores of freshmen at a college, then the population of ACT math scores will be the ACT math scores of all freshmen at that particular college. • Example: If we selected every tenth ACT math scores of freshmen at a college, then this selected set will represent a sample of ACT math scores for the freshmen at that particular college. 1-15 Introduction (contd.) Population – all freshmen ACT math scores Sample – every 10th ACT math score 1-16 Introduction (contd.) • What is a census? A census is a sample of the entire population. • Example: Every 10 years the U.S. government gathers information from the entire population. Since the entire population is sampled, this is referred to as a census. 1-17 Introduction (contd.) • Both populations and samples have characteristics that are associated with them. • These are called parameters and statistics respectively. • A parameter is a characteristic of or a fact about a population. • Example: The average age for the entire student population on a campus is an example of a parameter. 1-18 Introduction (contd.) • A statistic is a characteristic of or a fact about a sample. • Example: The average ACT math score for a sample of students on a campus is an example of a statistic. 1-19 Introduction (contd.) Population – Described by Parameters Sample – Described by Statistics 1-20 1-2 Frequency Distributions • What is a frequency distribution? A frequency distribution is an organization of raw data in tabular form, using classes (or intervals) and frequencies. • What is a frequency count? The frequency or the frequency count for a data value is the number of times the value occurs in the data set. 1-21 Categorical or Qualitative Frequency Distributions • NOTE: We will consider categorical, ungrouped, and grouped frequency distributions. What is a categorical frequency distribution? A categorical frequency distribution represents data that can be placed in specific categories, such as gender, hair color, political affiliation etc. 1-22 Categorical or Qualitative Frequency Distributions -- Example • Example: The blood types of 25 blood donors are given below. Summarize the data using a frequency distribution. AB B A O B O B O A O B O B B B A O AB AB O A B AB O A 1-23 Categorical Frequency Distribution for the Blood Types -- Example Continued Note: The classes for the distribution are the blood types. 1-24 Quantitative Frequency Distributions -- Ungrouped • What is an ungrouped frequency distribution? An ungrouped frequency distribution simply lists the data values with the corresponding frequency counts with which each value occurs. 1-25 Quantitative Frequency Distributions – Ungrouped -- Example • Example: The at-rest pulse rate for 16 athletes at a meet were 57, 57, 56, 57, 58, 56, 54, 64, 53, 54, 54, 55, 57, 55, 60, and 58. Summarize the information with an ungrouped frequency distribution. 1-26 Quantitative Frequency Distributions – Ungrouped -- Example Continued Note: The (ungrouped) classes are the observed values themselves. 1-27 Relative Frequency • NOTE: Sometimes frequency distributions are displayed with relative frequencies as well. • What is a relative frequency for a class? The relative frequency of any class is obtained dividing the frequency (f) for the class by the total number of observations (n). 1-28 Relative Frequency Relative for Frequency aclass for frequency theclass number observatiointhedistributi total of ns on Example: The relative frequency for the ungrouped class of 57 will be 4/16 = 0.25. 1-29 Relative Frequency Distribution Note: The relative frequency for a class is obtained by computing f/n. 1-30 Cumulative Frequency and Cumulative Relative Frequency • NOTE: Sometimes frequency distributions are displayed with cumulative frequencies and cumulative relative frequencies as well. 1-31 Cumulative Frequency and Cumulative Relative Frequency • What is a cumulative frequency for a class? The cumulative frequency for a specific class in a frequency table is the sum of the frequencies for all values at or below the given class. 1-32 Cumulative Frequency and Cumulative Relative Frequency • What is a cumulative relative frequency for a class? The cumulative relative frequency for a specific class in a frequency table is the sum of the relative frequencies for all values at or below the given class. 1-33 Cumulative Frequency and Cumulative Relative Frequency Note: Table with relative and cumulative relative frequencies. 1-34 Quantitative Frequency Distributions -- Grouped • What is a grouped frequency distribution? A grouped frequency distribution is obtained by constructing classes (or intervals) for the data, and then listing the corresponding number of values (frequency counts) in each interval. 1-35 Quantitative Frequency Distributions -- Grouped • There are several procedures that one can use to construct a grouped frequency distribution. • However, because of the many statistical software packages (MINITAB, SPSS etc.) and graphing calculators (TI-83 etc.) available today, it is not necessary to try to construct such distributions using pencil and paper. 1-36 Quantitative Frequency Distributions -- Grouped • Later, we will encounter a graphical display called the histogram. We will see that one can directly construct grouped frequency distributions from these displays. 1-37 Quantitative Frequency Distributions – Grouped -- Quick Tip • A frequency distribution should have a minimum of 5 classes and a maximum of 20 classes. • For small data sets, one can use between 5 and 10 classes. • For large data sets, one can use up to 20 classes. 1-38 Quantitative Frequency Distributions – Grouped -- Example • Example: The weights of 30 female students majoring in Physical Education on a college campus are as follows: 143, 113, 107, 151, 90, 139, 136, 126, 122, 127, 123, 137, 132, 121, 112, 132, 133, 121, 126, 104, 140, 138, 99, 134, 119, 112, 133, 104, 129, and 123. Summarize the data with a frequency distribution using seven classes. 1-39 Quantitative Frequency Distributions – Grouped -- Example Continued • NOTE: We will introduce the histogram here to help us construct a grouped frequency distribution. 1-40 Quantitative Frequency Distributions – Grouped -- Example Continued • What is a histogram? A histogram is a graphical display of a frequency or a relative frequency distribution that uses classes and vertical (horizontal) bars (rectangles) of various heights to represent the frequencies. 1-41 Quantitative Frequency Distributions – Grouped -- Example Continued • The MINITAB statistical software was used to generate the histogram. • The histogram has seven classes. • Classes for the weights are along the x-axis and frequencies are along the y-axis. • The number at the top of each rectangular box, represents the frequency for the class. 1-42 Quantitative Frequency Distributions – Grouped -- Example Continued Histogram with 7 classes for the weights. 1-43 Quantitative Frequency Distributions – Grouped -- Example Continued • Observations • From the histogram, the classes (intervals) are 85 – 95, 95 – 105, 105 – 115 etc. with corresponding frequencies of 1, 3, 4, etc. • We will use this information to construct the group frequency distribution. 1-44 Quantitative Frequency Distributions – Grouped -- Example Continued • Observations (continued) • Observe that the upper class limit of 95 for the class 85 – 95 is listed as the lower class limit for the class 95 – 105. • Since the value of 95 cannot be included in both classes, we will use the convention that the upper class limit is not included in the class. 1-45 Quantitative Frequency Distributions – Grouped -- Example Continued • Observations (continued) • That is, the class 85 – 95 should be interpreted as having the values 85 and up to 95 but not including the value of 95. • Using these observations, the grouped frequency distribution is constructed from the histogram and is given on the next slide. 1-46 Quantitative Frequency Distributions – Grouped -- Example Continued 1-47 Quantitative Frequency Distributions – Grouped -- Example Continued • Observations (continued) • In the previous slide with the grouped frequency distribution, the sum of the relative frequencies did not add up to 1. This is due to rounding to four decimal places. • The same observation should be noted for the cumulative relative frequency column. 1-48 Dot Plots • What is a dot plot? A dot plot is a plot that displays a dot for each value in a data set along a number line. If there are multiple occurrences of a specific value, then the dots will be stacked vertically. • Example: The following frequency distribution shows the number of defectives observed by a quality control officer over a 30 day period. Construct a dot plot for the data. 1-49 Dot Plots – Example Continued The next slide shows the dot plot for the number of defectives. 1-50 Dot Plots – Example Continued 1-51 Bar Charts or Bar Graphs • What is a bar chart (graph)? A bar chart or a bar graph is a graph that uses vertical or horizontal bars to represent the frequencies of the categories in a data set. • Example: A sample of 300 college students was asked to indicate their favorite soft drink. The results of the survey are shown on the next slide. Display the information using a bar chart. 1-52 Bar Charts – Example Continued The next slide shows the bar chart for the soft drink preferences of the students. 1-53 Bar Chart – Example Continued 1-54 Bar Charts -- Quick Tip • Bar charts are effective at reinforcing differences in magnitude. • Bar charts are useful when the data set has categories (for example, hair color, gender, etc.). • Bar charts are useful when the data are qualitative in nature. • Note: The bars are equally separated. 1-55 Histograms Revisited Histogram with 7 classes for the weights. 1-56 Histograms -- Quick Tip • Histograms are useful when the data values are quantitative. • A histogram gives an estimate of the shape of the distribution of the population from which the sample was taken. • If the relative frequencies were plotted along the vertical axis to produce the histogram, the shape will be the same as when the frequencies are used. 1-57 Frequency Polygons • What is a frequency polygon? A frequency polygon is a graph that displays the data using lines to connect points plotted for the frequencies. • Note: The frequencies represent the heights of the vertical bars in the histogram. • Example: Display a frequency polygon for the weights of the 30 female students (presented previously). 1-58 Frequency Polygons -- Example Continued Frequency Polygon 1-59 Frequency Polygons – Observations • The frequency polygon is superimposed on the histogram. • The polygon is mound-shaped. • This indicates that the shape of the population from which the sample was taken is mound shaped. • The line segments pass through the mid points at the top of the rectangles. • The polygon is tied down at both ends. 1-60 Stem-and-Leaf Displays or Plots • What is a stem-and-leaf plot? A stem-and- leaf plot is a data plot that uses part of a data value as the stem to form groups or classes and part of the data value as the leaf. • Note: A stem-and-leaf plot has an advantage over a grouped frequency distribution, since a stem-and-leaf plot retains the actual data by showing them in graphic form. 1-61 Stem-and-Leaf Displays or Plots -- Example • Example: Consider the following values – 96, 98, 107, 110, and 112. Construct a stem-and-leaf plot by using the units digits as the leaves. 1-62 Stem-and-Leaf Plot – Example Continued Stems and leaves for the Stem-and-leaf plot for the data values. data values. Stem Leaf 09 6 8 10 7 11 0 2 1-63 Stem-and-Leaf Displays or Plots -- Example • Example: A sample of the number of admissions to a psychiatric ward at a local hospital during the full phases of the moon is as follows: 22, 30, 21, 27, 31, 36, 20, 28, 25, 33, 21, 38, 32, 35, 26, 19, 43, 30, 30, 34, 27, and 41. • Display the data in a stem-and-leaf plot with the leaves represented by the unit digits. 1-64 Stem-and-Leaf Plot – Example Continued Stem Leaf 1 9 2 0 1 1 2 5 6 7 7 8 3 0 0 0 1 2 3 4 5 6 8 4 1 3 1-65 Time Series Graphs • What is a time series graph? A time series graph is a plot which displays data that are observed over a given period of time. • Note: From a time series graph, one can observe and analyze the behavior of the data over time. 1-66 Time Series Graphs -- Example • Example: The following table gives the number of hurricanes for the years 1981 to 1990. • Display the data with a time series graph. 1-67 Time Series Graphs – Example Continued Graph seem to display an upward trend over the years. Highest number was in 1990. Time Series Graph for the Dow Jones Industrial Average 1-68 from October 1999 to October 2000 – Example 1-69 Pie Graphs or Pie Charts • What is a pie graph (chart)? A pie graph is a circular display that is divided into sectors (classes) according to the percentage of data values in each class. • Note: A pie chart allows us to observe the proportions of the classes relative to the entire data set. • Pie charts are readily used to display qualitative data. 1-70 Pie Graphs or Pie Charts -- Example • Example: present a pie chart for the blood type data given earlier. • The pie chart is presented on the next slide. • Note: Each sector (slice) is proportional to the frequency count or percentage relative to the total sample size. 1-71 Pie Graphs or Pie Charts – Example Continued 1-72 Pareto Charts • What is a Pareto chart? A Pareto chart is a type of bar chart in which the horizontal axis represents the categories of interest. The bars are ordered from largest to smallest in terms of the frequency counts for the categories. • Note: A Pareto chart can help you determine which of the categories make up the critical few and which are the insignificant many. 1-73 Pareto Charts -- Example • A cumulative percentage line helps you judge the added contribution of each category. • Example: The following information over a six month period, relating to the number of defects in a manufacturing process in a company, was obtained by the quality control team. • The data is given on the next slide. • Present a Pareto chart for the data. 1-74 Pareto Charts – Example Continued Note: The Pareto chart, on the next slide, combined the information for the scrap, unconnected wire, and missing studs as an Others category. 1-75 Pareto Charts – Example Continued 1-76 It’s a Wrap • All of the graphs that were presented in this chapter can be generated or constructed with many of the statistical software packages which are on the market today. • All of the graphs presented were constructed using the MINITAB for Windows software. • Other packages which can be used are SPSS, SAS, and EXCEL.