STA1610 - Introduction To Statistics Module 2.

STA1610 - Introduction To Statistics Module 2. How to Summarize Qualitative Data Using Tables & Graphs Note: For access to statistical tables and equations, please click "Help Files" on the upper right of any module/exam page. Chapter Two: Descriptive Statistics: Tabular, Graphical & Numerical Methods Section 2.0: Introduction In this module, we will consider ways to describe and represent data. We will consider frequency distributions, measures of central tendency, variances, standard deviations, percentiles, and quartiles. Section 2.1: How to Summarize Qualitative Data Using Tables & Graphs We will begin this section by looking at a frequency distribution. A frequency distribution is a tabular representation of the summary data which shows the numerical count of items in each class in the data set. Look at the following data set in Table One: NursingABC - Module 7/25/19, 11)15 PM Table 2.1 is a list of cars sold at The Ajax Used Car Emporium for the month of July 2014. The sales data displayed in this manner is not particularly useful to Ajax. Display the data in the form of a frequency distribution. Solution. In order to create a frequency distribution, we count the number of each model of car sold then display this information in a table. The frequency distribution for the data is shown in Table 2.2. The frequency distribution shows the number sold of each model of car. By looking at this data, Ajax can learn which brands of cars sold better than the others. The frequency distribution tells us that the bestselling brand is Ford followed closely by Nissan. On the other hand, the brand least sold was Jeep. NOTE: For any frequency distribution, the classes must not be allowed to overlap. An overlap will result in double counting of an item and will lead to erroneous results. For some types of data, such as the car brands in example 2.1, there is usually not much chance of creating classes that overlap. However, when working with certain types of data, particularly numerical data, care must be taken to ensure that the classes do not overlap. Next, we will consider relative frequency. Relative frequency is a calculated value that represents the proportion of the items in each class. The following formula calculates the relative frequency: where n is the total count of all the classes being compared. The relative frequency can be converted to the relative percentage by multiplying the relative frequency by 100. NursingABC - Module 7/25/19, 11)15 PM Relative Percentage of a Category = Relative Frequency of a Category *100 Example 2.2 Consider the data in Table 2.1. Make a relative frequency and relative percentage distribution for this data. Solution. We have already made a frequency distribution for this data in Table 2.2. Note that Ajax sold a total of fifty cars in July 2014. Eight of the fifty sold were Chrysler In order to find the relative frequency for Chrysler, divide eight by fifty to get 8/50 = .16. The relative frequency for Ford is found by divide the of Ford’s sold, fifteen by fifty to get 15/50 = .30 and so on and so on. In order to get the relative percentages, multiply each of the relative frequencies by 100. All of the calculations are shown Table 2.3. This table shows the calculated relative frequencies and percentages for Ajax's July 2014 car sales. In conclusion, frequency distribution tables are a way to help us understand data numerically. While frequency distributions help us to organize and understand data numerically, charting is a means to represent frequencies visually. While there are many different types of charts available, we will focus on the following the three types of charts: the Column Chart, the Bar Chart and the Pie Chart. The basic column chart uses rectangular shapes of varying sizes to represent the different classes being evaluated. The rectangular bars for each class are lined up vertically and labeled on the x-axis of the chart. The height of each rectangle is scaled to correspond with the y-axis of the chart. Graph 2.1, which is a column chart, presents the frequency distribution of Table 2.2, in a visual format. NursingABC - Module 7/25/19, 11)15 PM Graph 2.1 listed the classes in alphabetical order. While the information presented in that chart is valid, it is often more helpful to arrange the data so that the class with the largest count value is listed first and the remaining classes listed in descending order. Graph 2.2 shows the data from Table 2.2 in descending order. Listing the classes in descending order is particularly helpful to those who must look at many, many graphs. (Such a person, wants to just scan the graphs, to see which class has the highest frequency, which has the second highest, etc.) The Bar Chart is the same as the column chart except the axis are transposed - the information that was represented on the x-axis is moved to the y-axis and the y-axis is then moved to the x-axis. Graph 2.3 is the NursingABC - Module 7/25/19, 11)15 PM bar chart of the sorted data in Graph 2.1. Once again, if your personal preference is to have the greatest values shown on top, simply rearrange the data in the original table. Graph 2.4 shows this minor change. While the information used in the column and bar charts were based on the initial frequency distribution table, both chart types could also have been used to display the information found in the relative frequency and percentage frequency data tables. The last chart type we will discuss here is the pie chart. While the pie chart can also be used for any of the distribution summaries, it is best suited for the distributions focused on the proportions. Since the "pie" is NursingABC - Module 7/25/19, 11)15 PM representative of the "whole", the wedges can easily be seen as representing the individual classes of the pie along with their respective proportions. Graph 2.5 is a pie chart representing the relative frequency proportion values from Table 2.3. Graph 2.6 is a variation on the pie chart called the exploded pie chart and represents the relative percentage values found in Table 2.3. NursingABC - Module 7/25/19, 11)15 PM Summarizing Quantitative Data Using Tables & Graphs Section 2.3: Summarizing Quantitative Data Using Tables & Graphs In this section, we summarize quantitative data. We begin by considering frequency distributions of numerical, or quantitative, data. The most important difference between summarizing numerical data and non-numerical data is in the defining of the classes. In example 2.1 of the previous section, each class was clearly defined because each class (i.e. each car model), was independent of every other car model. That is, when one car was sold it was either a Ford or a Chrysler or a Nissan… In other words, a car cannot be a Ford and a Nissan. A car cannot be a Nissan and a Chrysler. Therefore, in example 2.1, there was no possible overlap of the data. Each car will fit into one and only one class. However, defining classes for numerical data is not as clear-cut. In fact, there are usually many ways to define classes for numerical data. Consider the data in Table 2.4: Table 2.4 shows the hourly wages of workers at a certain factory. How would one define the classes for a frequency distribution for this data? There are many possibilities. For example, one could make each class $ 5.00 wide such as in Table 2.5: One could make each class $ .50 wide such as in Table 2.6: NursingABC - Module 7/25/19, 11)15 PM One could make each class $ 1.00 wide such as in Table 2.7: One could make each class $ 2.50 wide such as in Table 2.8: NursingABC - Module 7/25/19, 11)15 PM Tables 2.5, 2.6, 2.7, and 2.8 are all correct frequency distributions. Therefore, this illustrates the fact that there are many ways to create classes for this type of data. Of course, some of these distributions may be more helpful than the others. For example, Table 2.5 probably has too few classes to provide useful information. Table 2.6 probably has too many distributions to be helpful. On the other hand, Tables 2.7 and 2.8 are very useful. Which is best? Which table should we use to represent our data? That is a matter of opinion and depends upon the point that we are trying to make with the frequency distribution. Three step method. There are “systematic” ways to define the classes for quantitative data. One can use three steps to produce an effective frequency distribution table. Step One: Determine the number of classes to be evaluated. Step Two: Determine the width of the classes. Step Three: Determine each class's limits. Step one Use the fewest number of classes possible to effectively explain your data. You should set the number of classes in order to best represent the data. There are no definite rules for setting the number of classes - two people may define the classes differently. However, it should be recognized that the more classes used, the more likely the results in each class will be diluted (see Table 2.6 above). Two criteria to use in determining the number of classes are: 1) The number of data in the data set 2) The range of the data (i.e. the largest minus the smallest data value.) Step two. Evaluate the range of the data by locating the smallest data value and the largest data value. Then, calculate the approximate width of the classes using the following formula: Step three Set the limits for each class. It is imperative that one sets the upper and lower limits for each class such that each value in the data set is counted in one and only one class. NursingABC - Module 7/25/19, 11)15 PM Example 2.3. Consider the following customer service times: The twenty numbers in Table 2.9 represent a sample of customer waiting times in a bank. A time of 1 indicates that the customer waited between 0 to 1 minutes before the next teller was available to assist him or her. A value of 3 means the customer waited longer than 2 minutes but no longer than 3 minutes before receiving assistance, etc. Use the three step method listed above to make a frequency distribution. Also include the relative frequency and the percent frequency in your table. Solution. Step one The data set contains twenty values and those values range from 1 to 12 minutes. Since this data set is somewhat small, let's plan on using five classes. Step two We see that the smallest data value is 1 and the largest value is 12. Now use the formula. Step three In step two, we calculated a class width of 2.2. However, since all the values in our data set are integers, our class width should be an integer. Therefore, we will use standard rounding rules for arithmetic and round this the approximate width of 2.2 down to an integer value of 2. So, according to the calculations, the class width should be two. However, one must be flexible using a certain amount of judgment and discretion in selecting the class width. You might find that a class width of two produces too many classes for your application. In that case, you might try a class width of three, or more. We will use two as the first try for the class width. Using 2, we find that the first class will range from 1 - 2, class two ranges from 3 - 4, the third class ranges from 5 - 6 and the fourth class ranges from 7 - 8. For the fifth class, there is an anomaly. If we use the class width 2, then the class would range from 9 - 10. However, there is a 12 in the data set … where do we put it? To fully understand the problem, we need to understand that the issue is a direct result to fact we rounded down a calculated value when making the class width. NursingABC - Module 7/25/19, 11)15 PM Issues like this will occur anytime a value is rounded up or rounded down! If we had rounded the approximate width up to 3 from the 2.2, then the classes would have been 1 - 3, 4 - 6, 7 - 9 and 10 - 12. Since our data range has a maximum value of 12, we would have only had 4 classes in our analysis. For our purposes in this example, we can easily set the fifth class to "9+" so that the class includes all values greater than 9. Table 2.10 shows the frequency distribution with the class widths we set. We need to show the relative frequency and percent frequency. Using the same formula used in the previous section, the values are easily calculated and are shown in Table 2.11. Just as with qualitative data, quantitative data can also be visually displayed in the form of a histogram. NursingABC - Module 7/25/19, 11)15 PM Problem Set 2.1 Problem Set 2.1 1. The students in a statistics class have the following majors: NursingABC - Module 7/25/19, 11)15 PM a) Make a frequency distribution for this data. b) Make a relative frequency distribution for this data. Include relative percentages on this table. Using Crosstabulations and Scatter Diagrams Section 2.4: Using Crosstabulations and Scatter Diagrams Most real-world applications involve not just one but several variables. For most applications, one is usually interested in studying the interaction among these variables. Cross-tabulations are useful for such studies. The use of cross-tabulations is an effective method for comparing qualitative-to-qualitative data, qualitativeto-quantitative data and quantitative-to-quantitative data. Consider the following example: Example 2.3. In a recent survey taken our fictitious Analysis State University, the students are asked to fill out an anonymous survey on each course they take each semester and on each instructor they have each semester. The following Table 2.12 is a summary of this year's results for the instructor ratings for the Intro to Statistics classes. In Table 2.12 we see that each student rated each of the for instructors (Smith, Wilson, Jones, and Baker) as Poor, Good, Very Good, Excellent. NursingABC - Module 7/25/19, 11)15 PM The data, as presented in Table 2.12, is not particularly useful for evaluating the performance of the instructors. It would be helpful to organize the data into a form that is easier to analyze and evaluate. We will begin organizing this data by setting up a cross-tabulation. We will place the classes (Poor, Good, Very Good, Excellent) in the left column and place the professors (Smith, Wilson, Jones, and Baker) along the top horizontal row. Under Smith, we will list the number of this instructor’s Poor, Good, Very Good, and Excellent evaluations. Then do the same for Wilson, Jones, Baker. This is done in Table 2.13. Notice that there are two sets of totals in the table, one total is in the far right column and another total is NursingABC - Module 7/25/19, 11)15 PM along the bottom row. The totals in the bottom row represent the total student responses for each professor, in this example there were 25 students in each class submitting responses. The last column on the right is the totaling of the number of response in each rating class. Notice that Table 2.13 is very useful. This table shows the number of Poor, Good, Very Good, and Excellent evaluations for each instructor. It will be even more helpful to see the percentage of each evaluation that each instructor received. For this, we create a percentage tabulation. There are different ways to set up percentage tabulations, depending upon what we are trying to express or evaluate. Suppose that we wish to know what percentage of the total Poor evaluations that each instructor received, what percentage of the total Good evaluations that each instructor received, etc. To do this, we first consider the Poor evaluations. Smith has 2 Poor evaluations out of a total of 9 Poor evaluations (find 9 in the far right column). So, for Smith, 2/9 =.2222 or 22% of the Poor Evaluations. Wilson has 1 Poor evaluation out of a total of 9 Poor evaluations (we find 9 in the far right column). So for Wilson, 1/9 =.1111 or 11% of the Poor Evaluations. We continue this until the Poor row is complete. See Table 2.14. Next do the Good evaluations. Smith has 7 Good evaluations out of a total of 34 Good evaluations (we find 34 in the far right column). For Smith 7/34 = .2058 or 21% of the Good evaluations. Wilson has 8 Good evaluations out of a total of 34 Good evaluations (we find 34 in the far right column). For Wilson 8/34 = .235 or 24% of the Good evaluations. We continue to do these calculations until we complete Table 2.14. We have transformed the raw data presented in Table 2.12 into the very useful for presented in Table 2.14. Next, we will consider a way for presenting data for two variables called scatter diagrams. A scatter diagram is a two dimensional plot, or graphical display, of data. A scatter diagram helps to determine whether or not there is relationship between the two variables.