2.3: Graphical Displays

Statistical graphs are useful in getting the audience’s attention in a publication or presentation. Data presented graphically is easier to summarize at a glance compared to frequency distributions or numerical summaries. Graphs are useful to reinforce a critical point, summarize a data set, or discover patterns or trends over a period of time. Florence Nightingale (1820-1910) was one of the first people to use graphical representations to present data. Nightingale was a nurse in the Crimean War and used a type of graph that she called polar area diagram, or coxcombs to display mortality figures for contagious diseases such as cholera and typhus.

Nightingale

Nightingale-mortality.jpg. (2021, May 18). Wikimedia Commons, the free media repository. Retrieved July 2021 from https://commons.wikimedia.org/w/index.php?title=File:Nightingale-mortality.jpg&oldid=561529217.

It is hard to provide a complete overview of the most recent developments in data visualization with the onset of technology. The development of a variety of highly interactive software has accelerated the pace and variety of graphical displays across a wide range of disciplines.

2.3.1 Stem-and-Leaf Plot

Stem-and-leaf plots (or stemplots) are a useful way of getting a quick picture of the shape of a distribution by hand. Turn the graph sideways and you can see the shape of your data. You can now easily identify outliers. Each observation is divided into two pieces; the stem and the leaf. If the number is just two digits then the stem would be the tens digit and the leaf would be the ones digit. When a number is more than two digits then the cut point should split the data into enough classes that is useful to see the shape of the data.

To create a stem-and-leaf plot:

1. Separate each observation into a stem and a leaf.
2. Write the stems in a vertical column in ascending order (from smallest to largest). Fill in missing numbers even if there are gaps in the data. Draw a vertical line to the right of this column.
3. Write each leaf in the row to the right of its stem, in increasing order.

Create a stem-and-leaf plot for the sample of 35 ages.

46 47 49 25 46 22 42 24 46 40 39 27 25 30 33 27 46 21 29 20 26 25 25 26 35 49 33 26 32 31 39 30 39 29 26

Solution

Divide each number so that the tens digit is the stem and the ones digit is the leaf. The smallest observation is 20. The stem = 2 and the leaf = 0. The next value is 21 and the stem = 2 and the leaf = 1, up to the last value of 49 which would have a stem = 4 and a leaf = 9. If we use the tens categories we have the stems 2, 3 and 4. Line up the stems without skipping a number even if there are no values in that stem. In other words, the stems should have equal spacing (for example, count by ones, tens, hundreds, thousands, etc.). Then place a vertical line to the right of the stems. In each row put the leaves with a space between each leaf. Sort each row from smallest to largest. In Figure 2- 6 the 2 | 0 = 20.

\begin{array}{l|llllllllllllllll}
2 & 0 & 1 & 2 & 4 & 5 & 5 & 5 & 5 & 6 & 6 & 6 & 6 & 7 & 7 & 9 & 9 \\
3 & 0 & 0 & 1 & 2 & 3 & 3 & 5 & 9 & 9 & 9 \\
4 & 0 & 2 & 6 & 6 & 6 & 6 & 7 & 9 & 9
\end{array}

Figure 2-6

It is hard to see the shape with so few classes and so many leaves in each class.

We can break each stem in half, putting leaves 0-4 in the first row and 5-9 in the second row, as in Figure 2-7.

\begin{array}{l|llllllllllll}
2 & 0 & 1 & 2 & 4 \\
2 & 5 & 5 & 5 & 5 & 6 & 6 & 6 & 6 & 7 & 7 & 9 & 9 \\
3 & 0 & 0 & 1 & 2 & 3 & 3 \\
3 & 5 & 9 & 9 & 9 \\
4 & 0 & 2 \\
4 & 6 & 6 & 6 & 6 & 7 & 9 & 9
\end{array}

Figure 2-7

Now, add labels and make sure the leaves are in ascending order. Be careful to line the leaves up in columns. You need to be able to compare the lengths of the rows when you interpret the graph.

Imagine lines around the leaves and turn the graph 90 degrees to the left. You can now see in Figure 2-8 the shape of the distribution. Note that Excel uses the upper class limit for the axis label.

Figure 2-8

If a leaf takes on more than the ones category then supply a footnote at the bottom of the plot with the units.

A small sample of house prices in thousands of dollars was collected: 375, 189, 432, 225, 305, 275. Make a stem-and-leaf plot.

Solution

If we were to split the stem and leaf between the ones and tens place, then we would need stems going from 18 up to 43. Twenty-six stems for only six data points is too many. The next break then for a stem would be between the tens and hundreds. This would give stems from 1 to 4. Then each leaf will be the ones and tens. For example, then number 375 would have a stem = 3 and a leaf = 75.

\begin{array}{l|ll}
1 & 89 \\
2 & 25 & 75 \\
3 & 05 & 75 \\
4 & 32
\end{array}

Leaf = $1000

A small sample of coffee prices: 3.75, 1.89, 4.32, 2.25, 3.05, 2.75 was collected. Make a stem-and-leaf plot.

Solution

\begin{array}{l|ll}
1 & 89 \\
2 & 25 & 75 \\
3 & 05 & 75 \\
4 & 32
\end{array}

Leaf = $0.01

Note that the last two stem-and-leaf plots look identical except for the footnote. It is important to include units to tell people what the stems and leaves mean by inserting a legend.

Back-to-back stem-and-leaf plots let us compare two data sets on the same number line. The two samples share the same set of stems. The sample on the right is written backward from largest leaf to smallest leaf, and the sample on the left has leaves from smallest to largest.

Use the following back-to-back stem-and-leaf plot to compare pulse rates before and after exercise.

Solution

The group on the left has leaves going in descending order and represent the pulse rates before exercise. The stems are in the middle column. The group on the right has leaves going in ascending order and represent the pulse rates after exercise. The first row has pulse rates of 62, 65, 66, 67, 68, 68 and 69. The last row of pulse rates are 124, 125, and 128.

2.3.2 Histogram

A histogram is a graph for quantitative data (we call these bar graphs for qualitative data). The data is divided into a number of classes. The class limits become the horizontal axis demarcated with a number line and the vertical axis is either the frequency or the relative frequency of each class. Figure 2-9 is an example of a histogram.

The histogram for quantitative data looks similar to a bar graph, except there are some major differences.

First, in a bar graph the categories can be put in any order on the horizontal axis. There is no set order for these nominal data. You cannot say how the data is distributed based on the shape, since the shape can change just by putting the categories in different orders. With quantitative data, the data are in a specific order, since you are dealing with numbers. With quantitative data, you can talk about a distribution shape.

This leads to the second difference from bar graphs. In a bar graph, the categories that you made in the frequency table were the words used for the category name. In quantitative data, the categories are numerical categories, and the numbers are determined by how many classes you choose. If two people have the same number of categories, then they will have the same frequency distribution. Whereas in qualitative data, there can be many different categories depending on the point of view of the author.

The third difference is that the bars touch with quantitative data, and there will be no gaps in the graph. The reason that bar graphs have gaps is to show that the categories do not continue on, as they do in quantitative data. Since the graph for quantitative data is different from qualitative data, it is given a different name of histogram.

Some key features of a histogram:

• Equal spacing on each axis
• Bars are the same width
• Label each axis and title the graph
• Show the scale on the frequency axis
• Label the categories on the category axis
• The bars should touch at the class boundaries

Figure 2-9

To create a histogram, you must first create a frequency distribution. Software and calculators can create histograms easily when a large amount of sample data is being analyzed.

Excel

To create a histogram in Excel you will need to first install the Data Analysis tool.

If your Data Analysis is not showing in the Data tab, follow the directions for installing the free add-in here: https://support.office.com/en-us/article/Load-the-Analysis-ToolPak-in-Excel-6a63e598-cd6d-42e3-9317- 6b40ba1a66b4.

Type in the data into one blank column in any order. If you want to have class widths other than Excel’s default setting, type in a new column the endpoints of each class found in your frequency distribution, these are called the bins in Excel.

Using the sample of 35 ages, make a histogram using Excel.

46 47 49 25 46 22 42 24 46 40 39 27 25 30 33 27 46 21 29 20 26 25 25 26 35 49 33 26 32 31 39 30 39 29 26

Solution

Type the data in any order into column A and the bins in order in column B as shown below. Then select the Data tab, select Data Analysis, select Histogram, then select OK.

In the dialogue box, click into the Input Range box, then use your mouse and highlight the ages including the label.

Then click into the Bin Range box and use your mouse to highlight the bins including the label.

Select the box for Labels only if you included the labels in your ranges. You can have your output default to a new worksheet, or select the circle to the left of Output Range, click into the box to the right of Output Range and then select one blank cell on your spreadsheet where you want the top left-hand corner of your table and graph to start. Then check the boxes next to Cumulative Percentage and Chart Output. Then select OK, and see below.

A histogram needs to have bars that touch, which is not the default in Excel. To get the bars to touch, right-click on one of the blue bars and select Format Data Series and slide the Gap Width to 0%.

Excel produces both a frequency table and a histogram. The table has the frequencies and the cumulative relative frequencies.

Bin	Frequency	Cumulative %
24	4	11.43%
29	12	45.71%
34	6	62.86%
39	4	74.29%
44	2	80.00%
49	7	100.00%
More	0	100.00%

The histogram has bars for the height of each frequency and then makes a line graph of the cumulative relative frequencies over the bars. This red line is a line graph of the cumulative relative frequencies, also called an ogive and is discussed in a later section.

It is important to note that the number of classes that are used and the value of the first class boundary will change the shape of the histogram.

A relative frequency histogram is when the relative frequencies are used for the vertical axis instead of the frequencies and the y-axis will represent a percent instead of the number of people.

In Excel, after you create your histogram, you can manually change the frequency column to the relative frequency values by dividing each number by the sample size. Here is a screen shot just as the last number was changed, note as soon as you hit enter the bars will shrink and adjust.

After the last value =7/35 was entered and the label changed to Relative Frequency you get the following graph.

The shape of the histogram will be the same for the relative frequency distribution and the frequency distribution; the height, though, is the proportion instead of frequency.