ORGANIZING AND GRAPHING DATA CHAPTER 2 ORGANIZING AND GRAPHING DATA

2.1 ORGANIZING AND GRAPHING QUALITATIVE DATA Definition Data recorded in the sequence in which they are collected and before they are processed or ranked are called raw data.

Table 2.1 Ages of 50 Students (See Pg. 29)

Table 2.2 Status of 50 Students (See Pg. 29) Definition Ungroup data set is a data set containing information on each member of a sample or population individually. Tables 2.1 and 2.2 are examples of ungroup data set.

ORGANIZING AND GRAPHING QUANTITATIVE DATA Qualitative data sets can be: Organized into tables and Displayed using graphs. To organize and display qualitative data set: Frequency Distributions table Relative Frequency and Percentage Distributions table Graphical Presentation of Qualitative Data

TABLE 2.3 Types of Employment Students Intend to Engage In Definition Frequency of a category is the number of element or member of the data set that belong to a specific category.

Frequency Distributions Definition A frequency distribution table for qualitative data lists all categories and the number of elements that belong to each of the categories.

Example 2 A sample of 30 employees from large companies was selected, and these employees were asked how stressful their jobs were. The responses of these employees are recorded below, where very represents very stressful, somewhat means somewhat stressful, and none stands for not stressful at all. somewhat none very Construct a frequency distribution table for these data.

Example 2: Solution Frequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions Calculating Relative Frequency of a Category Relative frequency distribution lists the relative distribution for all categories. Calculating Percentage Percentage = (Relative frequency) · 100 Percentage distribution lists the percentages for all categories.

Example 2 Example 2: Solution Determine the relative frequency and percentage for the data in Table 2.4. Example 2: Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights represent the frequencies of respective categories. Bar graph for the frequency distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories. Calculating Angle Sizes for the Pie Chart Pie chart for the percentage distribution

Case Study: In or Out in 30 Minutes

2.2 ORGANIZING AND GRAPHING QUANTITATIVE DATA Quantitative data sets can be: grouped into tables and Displayed using graphs. To group and display quantitative data set: Frequency Distributions table Constructing Frequency Distribution tables Relative and Percentage Distributions table Graphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that falls between two numbers, lower and upper limits. Note that classes are mutually exclusive, meaning that no one value in the data set belong in more than one class. A frequency is the number of values in the data set that belong to a specific class. A frequency distribution for quantitative data lists all the classes and the number of values that belong to each class. Data presented in the form of a frequency distribution are called grouped data.

Table 2.7 Weekly Earnings of 100 Employees of a Company – Frequency Distribution of Quantitative Data

Frequency Distribution Table – Conversion of Class Limits to Class Boundary Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class. A class width or size is define as: A class midpoint or mark is define as:

Table 2.8 Class Boundaries, Class Widths, and Class Midpoints for Table 2.7

Constructing Frequency Distribution Tables Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set. It can also be estimated by using: c = 1 + 3.3 log n; c is the number of classes and n is the number of observations A class width or size is define as: The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set.

Example 2-3 – Frequency Distribution Table The following data give the total number of iPods® sold by a mail order company on each of 30 days. Construct a frequency distribution table. 8 25 11 15 29 22 10 5 17 21 13 26 16 18 12 9 20 23 14 19 27

Example 2-3: Solution Number of Classes Suppose we decide to use 5 classes of equal width. Class Width Lower limit of the first class Suppose we take 5 as the lower limit of the first class. Then our classes will be 5 – 9, 10 – 14, 15 – 19, 20 – 24, and 25 – 29

Relative Frequency and Percentage Distributions Calculating Relative Frequency and Percentage

Example 2-4 Example 2-4: Solution Calculate the relative frequencies and percentages for Table 2.9. Example 2-4: Solution Table 2.10 Relative Frequency and Percentage Distributions for Table 2.9

Graphing Grouped Data Definition A histogram graph is used to display frequency, relative frequency, and percentage distributions for quantitative data. In a histogram graph classes are marked on the horizontal axis and the frequencies, relative frequencies, or percentages are marked on the vertical axis. The frequencies, relative frequencies, or percentages are represented by the heights of the bars. In a histogram, the bars are drawn adjacent to each other.

Graphs for Displaying Quantitative Data Figure 2.4 Relative frequency histogram for Table 2.10. Figure 2.3 Frequency histogram for Table 2.9.

Graphing Grouped Data Definition A polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines. Figure 2.6 Frequency distribution curve. Figure 2.5 Frequency polygon for Table 2.9.

Method to Use for Quantitative Data Set Use class limit for data sets without fractional numbers or values Use “less than method for data sets with fractional numbers or values

Example 2-5 On April 1, 2009, the federal tax on a pack of cigarettes was increased from 39¢ to $1.0066, a move that not only was expected to help increase federal revenue, but was also expected to save about 900,000 lives (Time Magazine, April 2009). Table 2.11 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1, 2009. Construct a frequency distribution table. Calculate the relative frequencies and percentages for all classes.

Example 2-5: Solution Number of Classes: Suppose we decide to use 6 classes of equal width. Class Width Lower limit of the first class Suppose we take 1.0 as the lower limit of the first class. Then our classes will be written as 1.00 to less than 1.5, 1.50 to less than 2.00, and so on.

Example 2-6 – Single-Value Classes The administration in a large city wanted to know the distribution of vehicles owned by households in that city. A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned: 5 1 1 2 0 1 1 2 1 1 1 3 3 0 2 5 1 2 3 4 2 1 2 2 1 2 2 1 1 1 4 2 1 1 2 1 1 4 1 3 Construct a frequency distribution table for these data, and draw a bar graph.

Example 2-6: Solution Table 2 Example 2-6: Solution Table 2.13 Frequency Distribution of Vehicles Owned The observations assume only six distinct values: 0, 1, 2, 3, 4, and 5. Each of these six values is used as a class in the frequency distribution in Table 2.13.

SHAPES OF HISTOGRAMS Symmetric Skewed Uniform or Rectangular

Figure 2.8 Symmetric and Skewed histograms. Symmetric histogram has the same shape on both sides of its central points. Skewed histogram is not symmetric, and the tail of one side is longer than the tail of the other side.

Uniform Distribution Histogram, Symmetric frequency curves, and Skewed Frequency curves

2.3 CUMULATIVE FREQUENCY DISTRIBUTIONS Definition A cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class. To construct a cumulative frequency distribution table: All the classes have the same lower limit but different upper limit. The cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es).

Example 2-7 Using the frequency distribution of Table 2.9, reproduced here, prepare a cumulative frequency distribution for the number of iPods sold by that company.

CUMULATIVE FREQUENCY DISTRIBUTIONS Calculating Cumulative Relative Frequency and Cumulative Percentage

Table 2.15 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS Definition An ogive is a curve for displaying cumulative frequency, cumulative relative frequency, and cumulative percentage distributions. To Draw an ogive: Label the vertical and horizontal axes. Mark a dot at the upper boundary of the corresponding class at a height equal to the: Cumulative frequencies, or Cumulative relative frequencies, or Cumulative percentages of respective classes. Join the dot with straight line.

Figure 2.12 Ogive for the cumulative frequency distribution of Table 2.14. Find the number of days for which 17 or fewer iPods were sold.

2.4 STEM-AND-LEAF DISPLAYS Definition Stem-and-leaf display is a method for displaying quantitative raw data in a concise manner. It divides quantitative data into two portions, which are separated by a vertical line. The left of the vertical line is the stem, which represents the first digit(s) of the values in the data set and arranged in increasing order. The right of the line is the leaf, which represents the remaining digit(s) of the values in the data set, and aligned with the row of the corresponding stem

Example 2-8 The following are the scores of 30 college students on a statistics test: Construct a stem-and-leaf display. 75 69 83 52 72 84 80 81 77 96 61 64 65 76 71 79 86 87 68 92 93 50 57 95 98

Figure 2.13 Stem-and-leaf display.

Example 2-9 The following data are monthly rents paid by a sample of 30 households selected from a small city. Construct a stem-and-leaf display for these data. 880 1210 1151 1081 985 630 721 1231 1175 1075 932 952 1023 850 1100 775 825 1140 1235 1000 750 915 965 1191 1370 960 1035 1280

Example 2-9: Solution Figure 2.16 Stem-and-leaf display of rents.

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month. Prepare a new stem-and-leaf display by grouping the stems.

Example 2-10: Solution

2.5 DOTPLOTS Definition Values that are very small or very large relative to the majority of the values in a data set are called outliers or extreme values.

Example 2-11 Table 2.16 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season. Create a dotplot for these data.

Table 2.16 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11: Solution Step1 Step 2

Example 2-12 Refer to Table 2.16 in Example 2-11, which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season. Table 2.17 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season. Make dotplots for both sets of data and compare these two dotplots.

Table 2.17 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12: Solution