Sta220 - Statistics Mr. Smith Room 310 Class #3

Section

1-3 Lesson Objectives You will be able to: 1.Describe Qualitative Data with Graphs 2.Describe Quantitative Data with Graphs

1-4 Objective 1 Describe Qualitative Data with Graphs

Describing Data Two methods for describing data are presented in this chapter, one graphical and the other numerical.

Qualitative data are nonnumerical in nature; thus, the value of a qualitative variable can only be classified into categories called classes.

A qualitative variable with three classes (X, Y, and Z) is measured for each of 20 units randomly sampled from a target population. The data (observed class for each unit are as follows: ClassesFrequencyRelative Frequency Class Percentage X 0.4 Y Z3 Total20

TYPES OF GRAPHS Three of the most widely used graphical methods for describing qualitative data: bar graphs, pie chart, and Pareto Diagram.

When using a bar graph, the categories (classes) of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency, or class percentage.

Bar Graph ClassesFrequency Relative Frequency Percentage X80.440% Y % Z % Total201100%

Bar Chart Restaurant Complaints ComplaintCount Overpriced789 Small portions621 Wait time109 Food is tasteless65 No atmosphere45 Not clean30 Too noisy27 Food is too salty15 Unfriendly staff12 Food not fresh9 Frequency Bar Chart

Pie Chart Classes Relative Frequency X0.4 Y0.45 Z0.15 Total1

Pie Chart Restaurant Complaints ComplaintCount Percent Overpriced Small portions Wait time Food is tasteless No atmosphere Not clean Too noisy Food is too salty Unfriendly staff Food not fresh

Pareto diagram is a bar graph with the categories (classes) of the qualitative variable (i.e., the bars) arranged by height in descending order from left to right. The goal of the Pareto diagram is to make it easy to locate the “most important” categories – those with the largest frequencies.

ClassesCount Cumulative % X945 Y885 Z3100

Pareto Diagram ComplaintCountCumulative % Overpriced Small portions Wait time Food is tasteless No atmosphere Not clean Too noisy Food is too salty Unfriendly staff1299.5

Some Bad/Misleading Graphs

Misleading Graphs A survey was conducted to determine what food would be served at the French Club Party. Explain how the following graph is misleading.

The ratio of the heights of bars within each category does not reflect the actual ratio. There is an implied precision that is unrealistic. (To the penny really?) The percentages are computed incorrectly. A doubling of costs is only 100% increase.

Sta220 - Statistics Mr. Smith Room 310 Class #4

Section

1-29 Lesson Objectives You will be able to: 1.Describe Quantitative Data with Graphs 2.Use Summation Notation 3.Understanding Central Tendency

Lesson Objective # 2: Describe Quantitative Data with Graphs

Recall that quantitative data sets consist of data that are recorded on a meaningful numerical scale. To describe, summarize, and detect patterns in such data, we can use three graphical methods: dot plots, stem-and-leaf displays and histograms.

Test Scores

Dot Plot When using a dot plot, the numerical value of each quantitative measurement in the data set is represented by a dot on a horizontal scale. When data values repeat, the dots are placed above one another vertically. The dot plot condenses the data by grouping all value that are the same.

Dot Plot

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 2.8 MINITAB dot plot for 100 EPA mileage ratings

Stem-and-Leaf The stem-and-leaf display condenses the data by grouping all data with the same stem. The possible stems are listed in order in a column. The leaf for each quantitative measurement in the data set is placed in the corresponding stem row. Leaves for observations with the same stem value are listed in increasing order horizontally.

Decimal point is 1 digit(s) to the right of the colon. 5 : 69 6 : 4 6 : 59 7 : : : : 56 9 : : 589

Histogram When using a histogram, the possible numerical values of the quantitative variable are partitioned into class intervals, each of which has the same width. These intervals form the scale of the horizontal axis. A vertical bar is placed over each class interval, with the height of the bar equal to either the class frequency or class relative frequency. When constructing histograms, use more classes as the number of values in the data set gets larger.

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 2.10 SPSS histogram for 100 EPA gas mileage ratings

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Table 2.3

Determining the Number of Classes in a Histogram Number of Observations in Data Set Number of Classes Fewer than More than

Lesson Objective # 3: Use Summation Notation

= = 12

Lesson Objective # 4: Understanding Central Tendency

The central tendency of the set of measurements – that is, the tendency of the data to cluster, or center, about certain numerical values The variability of the set of measurements – that is, the spread of the data.

There are three measures of central tendency: mean, median and mode.

MEAN

StatCrunch

MEDIAN The median of a quantitative data is the middle number when the measurements are arranged in ascending (or descending) order.

Calculating a Sample Median Μ Arrange the n measurements from the smallest to the largest. 1.If n is odd, Μ is the middle number. 2.If n is even, Μ is the mean of the middle two numbers. NOTE: Remember to order the data before calculating a value for the median

MODE The mode is the measurement that occurs most frequently in the data set. Mode is the only measure of center that has to be an actual data value in the samples. NOTE: For some quantitative data sets, the mode may not be very meaningful.

The modal class is the measurement class containing the largest relative frequency. (Ex. Relative frequency histogram for quantitative data.)

SKEWED A data set is said to be skewed if one tail of the distribution has more extreme observations than the other tail.

Center of a Distribution – Mean The mean is the center of gravity because it is the point where the histogram balances:

Center of a Distribution – Mean The median is the value with exactly half the data values below it and half above it.

With rightward skewed data, the right tail (high end) of the distribution has more extreme observations. Conversely, with leftward skewed data, the left tail (low end) of the distribution has more extreme observations.