2.3: Other Graphical Representations of Data

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Toros Berberyan, Tracy Nguyen, and Alfie Swan
Citrus College

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Learning Objectives

Introduction of additional statistical graphs to illustrate data structure and relationships.
Utilize stem-and-leaf plots, scatter plots, dot plots, and time series graphs for data visualization
Reveal patterns, trends, and distributions across different types of data

There are many other types of graphs. Some of the more common ones are the dot plot, the stem-and-leaf plot, the scatter plot, and the time-series plot. Many different graphs have emerged lately for qualitative data. Many are found in publications and websites. The following is a description of the stem-and-leaf plot, the scatter plot, and the time-series plot.

Stem-and-Leaf Plots

Stem-and-leaf plots are a quick and easy way to look at small samples of numerical data. You can look for any patterns or any strange data values. It is easy to compare two samples using stem-and-leaf plots.

The first step is to divide each number into 2 parts: the stem (such as the leftmost digit) and the leaf (such as the rightmost digit). There are no set rules, you just have to look at the data and see what makes sense.

Example $\PageIndex{1}$ stem plot for grade distribution

The following are the percentage grades of 25 students from a statistics course. Draw a stem-and-leaf plot of the data.

Table $\PageIndex{1}$: Data of Test Grades
62	87	81	69	87	62	45	95	76	76
62	71	65	67	72	80	40	77	87	58
84	73	93	64	89

Solution

Divide each number so that the tens digit is the stem and the ones digit is the leaf. 62 becomes 6|2.

Make a vertical chart with the stems on the left of a vertical bar. Be sure to fill in any missing stems. In other words, the stems should have equal spacing (for example, count by ones or count by tens). Graph 2.3.1 shows the stems for this example.

Stem values from 4 to 9. — Figure $\PageIndex{1}$: Stem Plot for Test Grades Step 1

Now go through the list of data and add the leaves. Put each leaf next to its corresponding stem. Don’t worry about the order yet just get all the leaves down.

When the data value 62 is placed on the plot, it looks like the plot in Graph 2.3.2.

Stem values from 4 to 9. The stem of 6 has a leaf of 2. — Figure $\PageIndex{2}$: Stem Plot for Test Grades Step 2

When the data value 87 is placed on the plot, it looks like the plot in Graph 2.3.3.

Stem values from 4 to 9. The stem of 6 has a leaf of 2. Stem of 8 has a leaf of 7. — Figure $\PageIndex{3}$: Stem Plot for Test Grades Step 3

Filling in the rest of the leaves to obtain the plot in Graph 2.3.4.

Stem values from 4 to 9. Next to it are the leaves of each stem. — Figure $\PageIndex{4}$: Stem Plot for Test Grades Step 4

Now you have to add labels and make the graph look pretty. You need to add a label and sort the leaves into increasing order. You also need to tell people what the stems and leaves mean by inserting a legend. 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. The final stem plot for the test grade data is in Graph 2.3.5.

Stem values from 4 to 9. Next to it are the leaves of each stem. The title is test scores. — Figure $\PageIndex{5}$: Stem Plot for Test Grades

Now you can interpret the stem-and-leaf display. The data is bimodal and somewhat symmetric. There are no gaps in the data. The center of the distribution is around 70.

Scatter Plot

Sometimes you have two different variables and you want to see if they are related in any way. A scatter plot helps you to see what the relationship would look like. A scatter plot is just a plotting of the ordered pairs.

Example $\PageIndex{2}$ scatter plot

Is there any relationship between elevation and high temperature on a given day? The following data are the high temperatures at various cities on a single day and the elevation of the city.

Table $\PageIndex{2}$: Data of Temperature versus Elevation
Elevation (in feet)	7000	4000	6000	3000	7000	4500	5000
Temperature (°F)	50	60	48	70	55	55	60

Solution

Preliminary: State the random variables

Let x = altitude

y = high temperature

Now plot the x values on the horizontal axis, and the y values on the vertical axis. Then set up a scale that fits the data on each axis. Once that is done, then just plot the x and y values as an ordered pair.

Scatter plot of temperature versus elevation. — Figure $\PageIndex{6}$: Scatter Plot of Temperature versus Elevation

Looking at the graph, it appears that there is a linear relationship between temperature and elevation. It also appears to be a negative relationship; thus, as elevation increases, the temperature decreases.

Time-Series Graph

A time-series plot is a graph showing the data measurements in chronological order, the data being quantitative. For example, a time-series plot is used to show profits over the last 5 years. To create a time-series plot, the time always goes on the horizontal axis, and the other variable goes on the vertical axis. Then plot the ordered pairs and connect the dots. The purpose of a time-series graph is to look for trends over time. Caution, you must realize that the trend may not continue. Just because you see an increase, doesn’t mean the increase will continue forever. As an example, before 2007, many people noticed that housing prices were increasing. The belief at the time was that housing prices would continue to increase. However, the housing bubble burst in 2007, and many houses lost value and haven’t recovered.

Example $\PageIndex{3}$ Time-series plot

The following table tracks the weight of a dieter, where the time in months is measuring how long since the person started the diet

Table $\PageIndex{3}$: Data of Weights versus Time
Time (months)	0	1	2	3	4	5
Weight (pounds)	200	195	192	193	190	187

Make a time-series plot of this data

Solution

Guidelines for creating a time-series graph

Put the title.
Create and label the x and y-axes.
Plot the points using time as the x-coordinate, and weight as the y-coordinate.
Connect the points with straight lines.

Time series graph weights versus time. — Figure $\PageIndex{7}$: Time Series Graph Weights Versus Time

Notice, that over the 5 months, the weight appears to be decreasing. However, it doesn’t look like there is a large decrease.

Be careful when making a graph. If you don’t start the vertical axis at 0, then the change can look much more dramatic than it really is. As an example, Graph 2.3.8 shows Graph 2.3.7 with a different scaling on the vertical axis. Notice the decrease in weight looks much larger than it really is.

Time series graph weights versus time with error. — Figure $\PageIndex{8}$: Time Series Graph Weights Versus Time With Error

Dot Plot

A dot plot is used to analyze the frequencies of a data set in a simplified manner. Each data value is plotted on the number line, and dots are stacked above the value to represent the frequencies.

Example $\PageIndex{1}$

Use the data below to construct a dot plot.

Number of Pets for Each Family
5	4	2	0	4	1	2
2	0	3	0	1	2	1
2	1	1	3	4	4	5
3	2	5	2	1	1	0
4	5	3	5	5	3	4
1	1	3	2	3	4	1
3	2	5	5	2	2	2

Table $\PageIndex{4}$: Number of Pets

Solution

Guidelines for Creating a Dot Plot

Create a title.
Draw a number line.
List the numbers from lowest to highest on the number line.
For each data value above, place a dot above its number on the number line. If the numbers repeat, stack them with more dots.

Dot plot of the data listed above. — Figure $\PageIndex{9}$: Dot Plot of the Number of Pets for Each Family

Authors

"2.3: Other Graphical Representations of Data" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY-SA 4.0

Attributions

"2.3: Other Graphical Representations of Data" by Kathryn Kozak is licensed CC BY-SA 4.0

Exercises

Students in a statistics class took their first test. Below are the scores they earned. Create a stem-and-leaf plot.

80, 79, 89, 74, 73, 79, 93, 70, 70, 76, 88, 83, 73, 81, 79, 80, 85, 79, 80, 79, 58, 93, 94, 74

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Students in a statistics class took their first test. Below are the scores they earned. Create a dot plot with the scores.

67, 67, 67, 67, 85, 70, 87, 80, 80, 72, 84, 67, 84, 67, 80, 80, 81, 81, 88, 74, 70, 80, 80, 84

When an anthropologist finds skeletal remains, they need to figure out the height of the remains. The height of the remains (in cm) and the length of one of their metacarpal bone (in cm) were collected in the table below. Create a scatter plot and state if there is a relationship between the height of the remains and the length of their metacarpal. Let x represent the length of the metacarpal and y represent the height of the remains.

Table $2.1. 4$ : Data of Metacarpal Versus Height
Length of Metacarpal	Height of the Remains
45	171
51	178
39	157
41	163
48	172
49	183
46	173
43	175
47	173

The table below presents the average 30-year fixed mortgage rates in the U.S. from 2016 to 2025. Create a time-series plot of the data below.

Table $2.3. 11$ : Year versus Mortgage Rates
Year	Average 30-Year Fixed Mortgage Rate
2016	3.65%
2017	3.99%
2018	4.54%
2019	3.94%
2020	3.10%
2021	2.96%
2022	5.34%
2023	6.81%
2024	6.72%
2025	6.89%

Answers

If you are an instructor and want the solutions to all the exercise questions for each section, please email Toros Berberyan.

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Example \(\PageIndex{1}\) stem plot for grade distribution

Solution

Example \(\PageIndex{2}\) scatter plot

Solution

Example \(\PageIndex{3}\) Time-series plot

Solution

Example \(\PageIndex{1}\)

Solution