Chapter 0-1 Graphs, Charts, and Tables – Describing Your Data

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STAT 280: Elementary Applied Statistics. Chapter 0-1 Graphs, Charts, and Tables – Describing Your Data. Chapter Goals. After completing this chapter, you should be able to: Construct a frequency distribution both manually and with a computer Construct and interpret a histogram
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STAT 280: Elementary Applied StatisticsChapter 0-1Graphs, Charts, and Tables – Describing Your DataChapter GoalsAfter completing this chapter, you should be able to:
  • Construct a frequency distribution both manually and with a computer
  • Construct and interpret a histogram
  • Create and interpret bar charts, pie charts, and stem-and-leaf diagrams
  • Present and interpret data in line charts and scatter diagrams
  • Frequency DistributionsWhat is a Frequency Distribution?
  • A frequency distribution is a list or a table …
  • containing the values of a variable (or a set of ranges within which the data falls) ...
  • and the corresponding frequencies with which each value occurs (or frequencies with which data falls within each range)
  • Why Use Frequency Distributions?
  • A frequency distribution is a way to summarize data
  • The distribution condenses the raw data into a more useful form...
  • and allows for a quick visual interpretation of the data
  • Frequency Distribution: Discrete Data
  • Discrete data: possible values are countable
  • Example: An advertiser asks 200 customers how many days per week they read the daily newspaper.Relative FrequencyRelative Frequency: What proportion is in each category?22% of the people in the sample report that they read the newspaper 0 days per weekFrequency Distribution: Continuous Data
  • Continuous Data: may take on any value in some interval
  • Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27(Temperature is a continuous variable because it could be measured to any degree of precision desired)Grouping Data by ClassesSort raw data in ascending order:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
  • Find range: 58 - 12 = 46
  • Select number of classes: 5(usually between 5 and 20)
  • Compute class width: 10(46/5 then round off)
  • Determine class boundaries:10, 20, 30, 40, 50
  • Compute class midpoints: 15, 25, 35, 45, 55
  • Count observations & assign to classes
  • Frequency Distribution ExampleData in ordered array:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58Frequency DistributionClassFrequency RelativeFrequency10 but under 20 3 .15 20 but under 30 6 .30 30 but under 40 5 .25 40 but under 50 4 .20 50 but under 60 2 .10 Total 20 1.00Histograms
  • The classesorintervals are shown on the horizontal axis
  • frequency is measured on the vertical axis
  • Bars of the appropriate heights can be used to represent the number of observations within each class
  • Such a graph is called a histogram
  • Histogram ExampleData in ordered array:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58No gaps between bars, since continuous dataClass MidpointsQuestions for Grouping Data into Classes
  • 1. How wide should each interval be?(How many classes should be used?)
  • 2. How should the endpoints of the intervals be determined?
  • Often answered by trial and error, subject to user judgment
  • The goal is to create a distribution that is neither too "jagged" nor too "blocky”
  • Goal is to appropriately show the pattern of variation in the data
  • How Many Class Intervals?
  • Many (Narrow class intervals)
  • may yield a very jagged distribution with gaps from empty classes
  • Can give a poor indication of how frequency varies across classes
  • Few (Wide class intervals)
  • may compress variation too much and yield a blocky distribution
  • can obscure important patterns of variation.
  • (X axis labels are upper class endpoints)General Guidelines
  • Number of Data Points Number of Classes
  • under 50 5 - 7 50 – 100 6 - 10 100 – 250 7 - 12 over 250 10 - 20
  • Class widths can typically be reduced as the number of observations increases
  • Distributions with numerous observations are more likely to be smooth and have gaps filled since data are plentiful
  • Class Width
  • The class width is the distance between the lowest possible value and the highest possible value for a frequency class
  • The minimum class width is
  • Largest Value  Smallest ValueW =Number of ClassesHistograms in ExcelSelectTools/Data Analysis1Histograms in Excel(continued)Choose Histogram23Input data and bin rangesSelect Chart OutputStem and Leaf Diagram
  • A simple way to see distribution details in a data set
  • METHOD: Separate the sorted data series into leading digits (the stem) and the trailing digits (theleaves)Example: Data in ordered array:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
  • Here, use the 10’s digit for the stem unit:
  • Stem Leaf 1 2 3 5
  • 12 is shown as
  • 35 is shown as
  • Example: Data in ordered array:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
  • Completed Stem-and-leaf diagram:
  • Using other stem units
  • Using the 100’s digit as the stem:
  • Round off the 10’s digit to form the leaves
  • 613 would become 6 1
  • 776 would become 7 8
  • . . .
  • 1224 becomes 12 2
  • Stem LeafGraphing Categorical DataCategorical DataPie ChartsBar ChartsPareto DiagramBar and Pie Charts
  • Bar charts and Pie charts are often used for qualitative (category) data
  • Height of bar or size of pie slice shows the frequency or percentage for each category
  • Pie Chart ExampleCurrent Investment Portfolio Investment Amount PercentageType(in thousands $)Stocks 46.5 42.27Bonds 32.0 29.09CD 15.5 14.09Savings 16.0 14.55Total110 100Savings 15%Stocks42%CD 14%Bonds 29%Percentages are rounded to the nearest percent(Variables are Qualitative)Bar Chart ExamplePareto Diagram Example% invested in each category (bar graph)cumulative % invested (line graph)Bar Chart ExampleTabulating and Graphing Multivariate Categorical Data
  • Investment in thousands of dollars
  • Investment Investor A Investor BInvestor CTotal CategoryStocks 46.5 55 27.5 129Bonds 32.0 44 19.0 95CD 15.5 20 13.5 49Savings 16.0 28 7.0 51 Total 110.0 147 67.0 324Tabulating and Graphing Multivariate Categorical Data(continued)
  • Side by side charts
  • Side-by-Side Chart Example
  • Sales by quarter for three sales territories:
  • Line Charts and Scatter Diagrams
  • Line charts show values of one variable vs. time
  • Time is traditionally shown on the horizontal axis
  • Scatter Diagrams show points for bivariate data
  • one variable is measured on the vertical axis and the other variable is measured on the horizontal axis
  • Line Chart ExampleScatter Diagram ExampleTypes of Relationships
  • Linear Relationships
  • Types of Relationships(continued)
  • Curvilinear Relationships
  • Types of Relationships(continued)
  • No Relationship
  • Chapter Summary
  • Data in raw form are usually not easy to use for decision making -- Some type of organization is needed:
  • Table Graph
  • Techniques reviewed in this chapter:
  • Frequency Distributions and Histograms
  • Bar Charts and Pie Charts
  • Stem and Leaf Diagrams
  • Line Charts and Scatter Diagrams
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