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SLIDES PREPARED By Lloyd R. Jaisingh Ph.D. Morehead State University Morehead KY

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## SLIDES PREPARED By Lloyd R. Jaisingh Ph.D. Morehead State University Morehead KY

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**STATISTICS for theUtterly Confused, 2nd ed.**SLIDES PREPARED By Lloyd R. Jaisingh Ph.D. Morehead State University Morehead KY**Part 1 DESCRIPTIVE STATISTICS**Chapter 1 Graphical Displays of Univariate Data**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**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**Objectives**• Introduction of some basic statistical terms. • Introduction of some graphical displays.**Introduction**• What is statistics?Statistics is the science of collecting, organizing, summarizing, analyzing, and makinginferences from data. • The subject of statistics is divided into two broad areas—descriptive statistics and inferential statistics.**Breakdown of the subject of statistics**Statistics Descriptive Statistics Inferential Statistics • Includes • Making inferences • Hypothesis testing • Determining relationships • Making predictions • Includes • Collecting • Organizing • Summarizing • Presenting data**Introduction (contd.)**• Explanation of the termdata: 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.**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**Variables Quantitative Qualitative • Includes • Discrete • Continuous variables**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 aredata values that can be placed into distinct categories according to some characteristic or attribute. • Example: the eye color of female basketball players.**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.**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.**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.**Introduction (contd.)**Population – all freshmen ACT math scores Sample – every 10th ACT math score**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.**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.**Introduction (contd.)**• A statistic is a characteristic of or a fact about a sample. • Example: The average ACT math score for a sampleof students on a campus is an example of a statistic.**Introduction (contd.)**Population – Described by Parameters Sample – Described by Statistics**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.**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.**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**Categorical Frequency Distribution for the Blood Types --**Example Continued Note: The classes for the distribution are the blood types.**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.**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.**Quantitative Frequency Distributions – Ungrouped --**Example Continued Note:The (ungrouped) classes are the observed values themselves.**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).**Relative Frequency**Example: The relative frequency for the ungrouped class of 57 will be 4/16 = 0.25.**Relative Frequency Distribution**Note:The relative frequency for a class is obtained by computing f/n.**Cumulative Frequency and Cumulative Relative Frequency**• NOTE: Sometimes frequency distributions are displayed with cumulative frequencies and cumulative relative frequencies as well.**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.**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.**Cumulative Frequency and Cumulative Relative Frequency**Note:Table with relative and cumulative relative frequencies.**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.**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.**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.**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.**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.**Quantitative Frequency Distributions – Grouped -- Example**Continued • NOTE: We will introduce the histogram here to help us construct a grouped frequency distribution.**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.**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.**Quantitative Frequency Distributions – Grouped -- Example**Continued Histogram with 7 classes for the weights.**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.**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.**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.**Quantitative Frequency Distributions – Grouped -- Example**Continued**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.**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.**Dot Plots – Example Continued**The next slide shows the dot plot for the number of defectives.