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

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. Tatal

(i) Discret Data EXAMPLE 1 Constructing Frequency and Relative Frequency Distribution from Discrete Data

EXAMPLE1( cont’) 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. 3 0 1 2 1 1 1 2 0 2 4 2 2 2 1 2 2 0 2 4 1 1 3 2 4 1 2 1 2 2 3 3 2 1 2 2 0 3 2 2 2 3 2 1 2 2 1 1 3 5 Data based on results reported by the United States Bureau of the Census. Tatal

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

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. Tatal

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

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

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

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. Tatal

EXAMPLE. Organizing Continuous Data into a 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 Tatal

The smallest data value is 672 and the largest data value is 738 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: Tatal

The smallest data value is 672 and the largest data value is 738 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 Tatal

Frequency Table Using class width of 10 Tatal

Frequency Table Using class width of 5 Tatal

ii) Frequency or Relative Frequency Histogram

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

Using class width of 10: Relative Frequency Histogram Tatal

Using class width of 5: Frequency Histogram Tatal

Stem-and-Leaf Plot for Continuous data

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. Tatal

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. Tatal

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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. Tatal

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 Tatal

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 Tatal

Advantage of Stem-and-Leaf Diagrams over Histograms 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. Tatal

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.

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

Time Series Plot

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

Distribution Shapes---Uniform Tatal

Distribution Shapes---Bell-Shaped Tatal

Distribution Shapes-Skewed Right Tatal

Distribution Shapes-Skewed Left Tatal

EXAMPLE Identifying the Shape of the Distribution Identify the shape of the following histogram which represents the time between eruptions at Old Faithful. Tatal

Answer: Slightly skewed left Tatal

Misleading Graphs

Distorted Vertical Scale

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

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

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.

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.