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1 Chapter Two Organizing and Summarizing Data 2.2 Organizing Quantitative Data I

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2 The first step in summarizing quantitative data is to determine whether the data is discrete or continuous. If the data is discrete, the categories of data will be the observations (as in qualitative data), if the data is continuous, the categories of data (called classes) must be created using intervals of numbers.

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3 (i) Discret Data EXAMPLE1 Constructing Frequency and Relative Frequency Distribution from Discrete Data

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4 The following data represent the number of available cars in a household based on a random sample of 50 households. Construct a frequency and relative frequency distribution. 3012111202422212202411324121223321220322232122113530121112024222122024113241212233212203222321221135 Data based on results reported by the United States Bureau of the Census. EXAMPLE1( cont’)

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5 Frequency and Relative Frequency Table EXAMPLE 1 (cont’)

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6 A histogram is constructed by drawing rectangles for each class of data whose height is the frequency or relative frequency of the class. The width of each rectangle should be the same and they should touch each other.

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7 EXAMPLE 2 Drawing a Histogram for Discrete Data Draw a frequency and relative frequency histogram for the “number of cars per household” data.

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10 (II) Continuous Data Categories of data are created for continuous data using intervals of numbers called classes.

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11 Group the data into intervals ( or classes), construct the frequency distribution table or relative frequency distribution table. i) Group data, Constructing tables

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12 The following data represents the number of persons aged 25 - 64 who are currently work disabled. The lower class limit of a class is the smallest value within the class while the upper class limit of a class is the largest value within the class. The lower class limit of first class is 25. The lower class limit of the second class is 35. The upper class limit of the first class is 34. The class width is the difference between consecutive lower class limits. The class width of the data given above is 35 - 25 = 10.

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13 EXAMPLE Organizing Continuous Data into a Frequency and Relative Frequency Distribution The following data represent the time between eruptions (in seconds) for a random sample of 45 eruptions at the Old Faithful Geyser in California. Construct a frequency and relative frequency distribution of the data. Source: Ladonna Hansen, Park Curator

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14 The smallest data value is 672 and the largest data value is 738. We will create the classes so that the lower class limit of the first class is 670 and the class width is 10 and obtain the following classes:

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15 The smallest data value is 672 and the largest data value is 738. We will create the classes so that the lower class limit of the first class is 670 and the class width is 10 and obtain the following classes: 670 - 679 680 - 689 690 - 699 700 - 709 710 - 719 720 - 729 730 - 739

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16 Frequency Table Using class width of 10

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17 Frequency Table Using class width of 5

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18 ii) Frequency or Relative Frequency Histogram

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19 EXAMPLE Constructing a Frequency and Relative Frequency Histogram for Continuous Data Using class width of 10: Frequency Histogram

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20 Using class width of 10: Relative Frequency Histogram

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21 Using class width of 5: Frequency Histogram

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22 Stem-and-Leaf Plot for Continuous data

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23 Construction of a Stem-and-Leaf Plot Step 1: The stem of the graph will consist of the leading digits The leaf of the graph will be the rightmost digit. The choice of the stem depends upon the class width desired. Step 2: Write the stems in a vertical column in increasing order. Draw a vertical line to the right of the stems. Step 3: Write each leaf corresponding to the stems to the right of the vertical line. The leafs must be written in ascending order.

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24 EXAMPLE Constructing a Stem-and-Leaf Diagram The employment ratio is the number of employed to population ratio. It is found by dividing the number of employed individuals in a population by the size of the population. The following data represent the employment ratio by state in the United States for 1999. Construct a stem-and-leaf diagram.

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26 We let the stem represent the integer portion of the number and the leaf will be the decimal portion. For example, the stem of Alabama will be 60 and the leaf will be 3.

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27 52 7 53 54 55 56 57 58 0 5 59 4 7 60 3 0 1 61 5 6 62 8 9 7 63 3 4 2 6 64 7 1 0 3 4 65 4 1 6 1 6 6 66 8 1 7 2 5 0 4 0 9 67 3 8 68 8 1 69 6 9 70 1 0 3 1 71 4 0 1 72 73 0

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28 52 7 53 54 55 56 57 58 0 5 59 4 7 60 0 1 3 61 5 6 62 7 8 9 63 2 3 4 6 64 0 1 3 4 7 65 1 1 4 6 6 6 66 0 0 1 2 4 5 7 8 9 67 3 8 68 1 8 69 6 9 70 0 1 1 3 71 0 1 4 72 73 0

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29 Once a frequency distribution or histogram of continuous data is created, the raw data is lost (unless reported with the frequency distribution), however, the raw data can be retrieved from the stem-and-leaf plot. Advantage of Stem-and-Leaf Diagrams over Histograms

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30 If the value of a variable is measured at different points in time, the data is referred to as time series data. A time series plot is obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Lines are then drawn connecting the points.

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31 The following data represent the closing value of the Dow Jones Industrial Average for the years 1980 - 2001.

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32 Time Series Plot

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33 Distribution Shapes Symmetric Uniform Normal Nonsymmetric Skewed to the left Skewed to the right

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34 Distribution Shapes---Uniform

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35 Distribution Shapes---Bell-Shaped

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36 Distribution Shapes-Skewed Right

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37 Distribution Shapes-Skewed Left

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38 EXAMPLEIdentifying the Shape of the Distribution Identify the shape of the following histogram which represents the time between eruptions at Old Faithful.

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39 Answer: Slightly skewed left

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40 Misleading Graphs

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41 Distorted Vertical Scale

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42 Characteristics of Good Graphics Clearly label the graphic and provide explanations, if needed.

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43 Characteristics of Good Graphics Clearly label the graphic and provide explanations, if needed. Avoid distortion. Don’t lie about the data.

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44 Characteristics of Good Graphics Clearly label the graphic and provide explanations, if needed. Avoid distortion. Don’t lie about the data. Avoid three dimensions. Three dimensional pie charts may look nice, but they distract the reader and often result in misinterpretation of the graphic.

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45 Characteristics of Good Graphics Clearly label the graphic and provide explanations, if needed. Avoid distortion. Don’t lie about the data. Avoid three dimensions. Three dimensional pie charts may look nice, but they distract the reader and often result in misinterpretation of the graphic. Do not use more than one design in the same graphic. Sometimes, graphs use a different design in a portion of the graphic in order to draw attention to this area. Don’t use this technique. Let the numbers speak for themselves.

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