What kind of graph consist of tabular frequencies shown as adjacent rectangles erected over intervals?

Example 2.2.a

Table 2.2 is a relative frequency table for the data of Table 2.1. The relative frequencies are obtained by dividing the corresponding frequencies of Table 2.1 by 42, the size of the data set. ■

Table 2.2.

Example 2.2.b

The following data relate to the different types of cancers affecting the 200 most recent patients to enroll at a clinic specializing in cancer. These data are represented in the pie chart presented in Figure 2.4. ■

2.2.1 Frequency Tables and Graphs

A data set having a relatively small number of distinct values can be conveniently presented in a frequency table. For instance, Table 2.1 is a frequency table for a data set consisting of the starting yearly salaries (to the nearest thousand dollars) of 42 recently graduated students with B.S. degrees in electrical engineering. Table 2.1 tells us, among other things, that the lowest starting salary of $57,000 was received by four of the graduates, whereas the highest salary of $70,000 was received by a single student. The most common starting salary was $62,000, and was received by 10 of the students.

Table 2.1. Starting Yearly Salaries

Data from a frequency table can be graphically represented by a line graph that plots the distinct data values on the horizontal axis and indicates their frequencies by the heights of vertical lines. A line graph of the data presented in Table 2.1 is shown in Figure 2.1.

When the lines in a line graph are given added thickness, the graph is called a bar graph. Figure 2.2 presents a bar graph.

Another type of graph used to represent a frequency table is the frequency polygon, which plots the frequencies of the different data values on the vertical axis, and then connects the plotted points with straight lines. Figure 2.3 presents a frequency polygon for the data of Table 2.1.

2.2.2 Relative Frequency Tables and Graphs

Consider a data set consisting of n values. If f is the frequency of a particular value, then the ratio f/n is called its relative frequency. That is, the relative frequency of a data value is the proportion of the data that have that value. The relative frequencies can be represented graphically by a relative frequency line or bar graph or by a relative frequency polygon. Indeed, these relative frequency graphs will look like the corresponding graphs of the absolute frequencies except that the labels on the vertical axis are now the old labels (that gave the frequencies) divided by the total number of data points.

A pie chart is often used to indicate relative frequencies when the data are not numerical in nature. A circle is constructed and then sliced into different sectors; one for each distinct type of data value. The relative frequency of a data value is indicated by the area of its sector, this area being equal to the total area of the circle multiplied by the relative frequency of the data value.

Example 2.2b

The following data relate to the different types of cancers affecting the 200 most recent patients to enroll at a clinic specializing in cancer. These data are represented in the pie chart presented in Figure 2.4. ■

Figure 2.4.

2.2.3 Grouped Data, Histograms, Ogives, and Stem and Leaf Plots

As seen in Subsection 2.2.2, using a line or a bar graph to plot the frequencies of data values is often an effective way of portraying a data set. However, for some data sets the number of distinct values is too large to utilize this approach. Instead, in such cases, it is useful to divide the values into groupings, or class intervals, and then plot the number of data values falling in each class interval. The number of class intervals chosen should be a trade-off between (1) choosing too few classes at a cost of losing too much information about the actual data values in a class and (2) choosing too many classes, which will result in the frequencies of each class being too small for a pattern to be discernible. Although 5 to 10 class intervals are typical, the appropriate number is a subjective choice, and of course, you can try different numbers of class intervals to see which of the resulting charts appears to be most revealing about the data. It is common, although not essential, to choose class intervals of equal length.

The endpoints of a class interval are called the class boundaries. We will adopt the left- end inclusion convention, which stipulates that a class interval contains its left-end but not its right-end boundary point. Thus, for instance, the class interval 20–30 contains all values that are both greater than or equal to 20 and less than 30.

Table 2.3 presents the lifetimes of 200 incandescent lamps. A class frequency table for the data of Table 2.3 is presented in Table 2.4. The class intervals are of length 100, with the first one starting at 500.

Table 2.3. Life in Hours of 200 Incandescent Lamps

Table 2.4. A Class Frequency Table

A bar graph plot of class data, with the bars placed adjacent to each other, is called a histogram. The vertical axis of a histogram can represent either the class frequency or the relative class frequency; in the former case the graph is called a frequency histogram and in the latter a relative frequency histogram. Figure 2.5 presents a frequency histogram of the data in Table 2.4.

We are sometimes interested in plotting a cumulative frequency (or cumulative relative frequency) graph. A point on the horizontal axis of such a graph represents a possible data value; its corresponding vertical plot gives the number (or proportion) of the data whose values are less than or equal to it. A cumulative relative frequency plot of the data of Table 2.3 is given in Figure 2.6. We can conclude from this figure that 100 percent of the data values are less than 1,500, approximately 40 percent are less than or equal to 900, approximately 80 percent are less than or equal to 1,100, and so on. A cumulative frequency plot is called an ogive.

An efficient way of organizing a small- to moderate-sized data set is to utilize a stem and leaf plot. Such a plot is obtained by first dividing each data value into two parts—its stem and its leaf. For instance, if the data are all two-digit numbers, then we could let the stem part of a data value be its tens digit and let the leaf be its ones digit. Thus, for instance, the value 62 is expressed as

and the two data values 62 and 67 can be represented as

8.5.1 Dot Plots

A dot plot, also called a dot chart, is used for relatively small data sets. The plot groups the data as little as possible and the identity of an individual observation is not lost. A dot plot uses dots to show where the data values (or scores) in a distribution are. The dots are plotted against their actual data values that are on the horizontal scale. If there are identical data values, the dots are “piled” on top of each other. Thus, to draw a dot plot, count the number of data points falling in each data value and draw a stack of dots that corresponds to the number of items in each data value. The plot makes it easy to see gaps and clusters in a data set as well as how the data spreads along the axis. For example, consider the following data set:

35,48,50,50,50,54,56,65,65,70,75,80

Since the value 50 occurs three times in the data set, there are three dots above 50. Similarly, since the value 65 occurs twice, there are two dots above 65. Thus, each dot in the plot represents a data item; there are as many dots on the plot as there are data items in the observation set. The dot plot for the data set is shown in Figure 8.4.

8.5.2 Frequency Distribution

A frequency distribution is a table that lists the set of values in a data set and their frequencies (i.e., the number of times each occurs in the data set). For example, consider the following data set X:

35,48,50,50,50,54, 56,65,65,70,75,80

The frequency distribution of the set is shown in Table 8.2.

Table 8.2. Example of Frequency Distribution

Sometimes a set of data covers such a wide range of values that a list of all the X values would be too long to be a “simple” presentation of the data. In this case we use a grouped frequency distribution table in which the X column lists groups of data values, called class intervals, rather than individual data values. The width of each class can be determined by dividing the range of observations by the number of classes. It is advisable to have equal class widths, and the class intervals should be mutually exclusive and non-overlapping.

Class limits separate one class from another. The class width is the difference between the lower limits of two consecutive classes or the upper limits of two consecutive classes. A class mark (midpoint) is the number in the middle of the class. It is found by adding the upper and lower limits and dividing the sum by two. Table 8.3 shows how we can define the group frequency distribution of the following data set:

Table 8.3. Example of Group Frequency Distribution

35,48,50,50,50,54,56,65,65,70,75 ,80

From the table we find that the class width is 41 − 34 = 47 − 40 = 7. The table also shows the class marks (or the midpoints of the different classes). Observe that the class marks are also separated by the class width.

8.5.3 Histograms

A frequency histogram (or simply histogram) is used to graphically display the grouped frequency distribution. It consists of vertical bars drawn above the classes so that the height of a bar corresponds to the frequency of the class that it represents and the width of the bar extends to the real limits of the score class. Thus, the columns are of equal width, and there are no spaces between columns. For example, the histogram for the Table 8.3 is shown in Figure 8.5.

8.5.4 Frequency Polygons

A frequency polygon is a graph that is obtained by joining the class marks of a histogram with the two end points lying on the horizontal axis. It gives an idea of the shape of the distribution. It can be superimposed on the histogram by placing the dots on the class marks of the histogram, as shown in Figure 8.6, which is the frequency polygon of Figure 8.5.

8.5.5 Bar Graphs

A bar graph (or bar chart) is a type of graph in which each column (plotted either vertically or horizontally) represents a categorical variable. (A categorical variable is a variable that has two or more categories with no intrinsic ordering to the categories. For example, gender is a categorical variable with two categories: male and female.) A bar graph is used to compare the frequency of a category or characteristic with that of another category or characteristic. The bar height (if vertical) or length (if horizontal) shows the frequency for each category or characteristic.

For example, assume that data has been collected from a survey of 100 ECE students to determine how many of them indicated that Probability, Electronics, Electromechanics, Logic Design, Electromagnetics, or Signals and Systems is their best subject. Let the data show that 30 students indicated that Probability is their best subject, 20 students indicated that Electronics is their best subject, 15 students indicated that Electromechanics is their best subject, 15 students indicated that Logic Design is their best subject, 10 students indicated that Electromagnetics is their best subject, and 10 students indicated that Signals and Systems is their best subject. This result can be displayed in a bar graph as shown in Figure 8.7.

Because each column represents an individual category rather than intervals for a continuous measurement, gaps are included between the bars. Also, the bars can be arranged in any order without affecting the data.

Bar charts have a similar appearance as histograms. However, bar charts are used for categorical or qualitative data while histograms are used for quantitative data. Also, in histograms, classes (or bars) are of equal width and touch each other, while in bar charts the bars do not touch each other.

8.5.6 Pie Chart

A pie chart is a special chart that uses “pie slices” to show relative sizes of data. For example, consider the survey discussed earlier of ECE students to find out their favorite subjects. We noted that the survey result is as follows:

Probability 30%

Electronics 20%

Electromechanics 15%

Logic Design 15%

Electromagnetics 10%

Signals and Systems 10%

The result can be displayed in the pie chart shown in Figure 8.8. The size of each slice is proportional to the probability of the event that the slice represents.

Sometimes we are given a raw score of a survey and are required to display the result in a pie chart. For example, consider a survey of 25 customers who bought a particular brand of television. They were required to rate the TV as good, fair or bad. Assume that the following are their responses:

Good, good, fair, fair, fair, bad, fair, bad, bad, fair, good, bad, fair, good, fair, bad, fair, fair, good, bad, fair, good, fair, bad, and bad.

To construct the pie chart we first create a list of the categories and tally each occurrence. Next, we add up the number of tallies to determine the frequency of each category. Finally, we obtain the relative frequency as the ratio of the frequency to the sum of the frequencies, where the sum is 25. This is illustrated in Table 8.4.

Table 8.4. Construction of Relative Frequencies

With this table we can then construct the pie chart as shown in Figure 8.9.

8.5.7 Box and Whiskers Plot

The box and whisker diagram (or box plot) is a way to visually organize data into fourths or quartiles. The diagram is made up of a “box,” which lies between the first and third quartiles, and “whiskers” that are straight lines extending from the ends of the box to the maximum and minimum data values. Thus, the middle two-fourths are enclosed in a “box” and lower and upper fourths are drawn as whiskers. The procedure for drawing the diagram is as follows:

Arrange the data in increasing order

Find the median

Find the first quartile Q1, which is the median of the lower half of the data set; and the third quartile, Q3, which is the median of the upper half of the data set.

On a line, mark points at the median, Q1, Q3, the minimum value of the data set and the maximum value of the data set.

Draw a box that lies between the first and third quartiles and thus represents the middle 50% of the data.

Draw a line from the first quartile to the minimum data value, and another line from the third quartile to the maximum data value. These lines are the whiskers of the plot.

Thus the box plot identifies the middle 50% of the data, the median, and the extreme points. The plot is illustrated in Figure 8.10 for the rank-order data set that we have used in previous sections:

12,34,48,50,50,54,56,65,66,80,88,90

As discussed earlier, the median is M = Q2 = 55, the first quartile is Q1 = 49, and the third quartile is Q3 = 73. The minimum data value is 12, and the maximum data value is 90. These values are all indicated in the figure.

When collecting data, sometimes a result is collected that seems “wrong” because it is much higher or much lower than all of the other values. Such points are known as “outliers.” These outliers are usually excluded from the whisker portion of the box and whiskers diagram. They are plotted individually and labeled as outliers.

As discussed earlier, the interquartile range, IQR, is the difference between the third quartile and the first quartile. That is, IQR = Q3 − Q1, which is the width of the box in the box and whiskers diagram. The IQR is one of the measures of dispersion, and statistics assumes that data values are clustered around some central value. The IQR can be used to tell when some of the other values are “too far” from the central value. In the box-and-whiskers diagram, an outlier is any data value that lies more than one and a half times the length of the box from either end of the box. That is, if a data point is below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR, it is viewed as being too far from the central values to be reasonable. Thus, the values for Q1 − 1.5 × IQR and Q3 + 1.5 × IQR are the “fences” that mark off the “reasonable” values from the outlier values. That is, outliers are data values that lie outside the fences. Thus, we define

Lower fence = Q1 − 1.5 × IQR

Upper fence = Q3 + 1.5 × IQR

For the example in Figure 8.10, IQR = Q3 − Q1 = 24 ⇒ IQR × 1.5 = 36. From this we have that the lower fence is at Q1 − 36 = 49 − 36 = 13, and the upper fence is at Q3 + 36 = 73 + 36 = 109. The only data value that is outside the fences is 12; all other data values are within the two fences. Thus, 12 is the only outliner in the data set.

The following example illustrates how to draw the box and whiskers plot with outliers. Supposed that we are given a new data set, which is 10, 12, 8, 1, 10, 13, 24, 15, 15, 24. First, we rank-order the data in an increasing order of magnitude to obtain:

1,8,10,10,12,13,15,15,24,24

Since there are 10 entries, the median is the average of the fifth and sixth numbers. That is, M = (12 + 13)/2 = 12.5. The lower half data set is 1, 8, 10, 10, 12 whose median is Q1 = 10. Similarly, the upper half data set is 13, 15, 15, 24, 24 whose median is Q3 = 15. Thus, IQR = 15 − 10 = 5, the lower fence is at 10 − (1.5)(5) = 2.5 and the upper fence is at 15 + (1.5)(5) = 22.5. Because the data values 1, 24 and 24 are outside the fences, they are outliers. The two values of 24 are stacked on top of each other. This is illustrated in Figure 8.11. The outliers are explicitly indicated in the diagram.