Section 6.5: Bar Graphs
- Page ID
- 216505
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Draw bar graphs
A bar graph (also called a bar chart) is a graphical display that uses rectangular bars to represent data. Each bar corresponds to a category or group, and the length or height of the bar is proportional to the value or frequency it represents. Bar graphs are one of the most versatile and widely used methods for displaying categorical data, making comparisons between groups clear and straightforward.
When to Use Bar Graphs
Bar graphs are most effective when you want to:
- Compare values across different categories or groups
- Display categorical or discrete data
- Show frequencies, counts, or measurements for distinct groups
- Make it easy to compare the sizes of different categories
- Present data where categories have no natural order or connection
Key Components of a Bar Graph
- Horizontal Axis (x-axis): Typically shows the categories or groups being compared
- Vertical Axis (y-axis): Represents the scale of measurement (frequency, count, percentage, amount, etc.)
- Bars: Rectangular shapes whose heights (or lengths) represent the data values
- Bars are separated by spaces to emphasize that categories are distinct
- All bars should have the same width
- Labels: Clear identification of categories and the measurement scale
- Title: Describes what the bar graph represents
- Scale: Consistent intervals on the measurement axis, typically starting at zero
Each category is assigned a position along one axis (usually horizontal), and a bar is drawn to a height (or length) that corresponds to its value on the other axis. The bars are separated by spaces to show that the categories are distinct and unrelated. By comparing the heights of the bars, viewers can quickly see which categories have larger or smaller values.
Types of Bar Graphs
- Vertical Bar Graph: Bars extend upward from the horizontal axis
- Most common type
- Categories on x-axis, values on y-axis
- Also called a column chart
- Horizontal Bar Graph: Bars extend horizontally from the vertical axis
- Categories on y-axis, values on x-axis
- Useful when category names are long
- Easier to read with many categories
- Side-by-Side Bar Graph: Multiple bars for each category, placed side-by-side
- Compares subcategories within each main category
- Example: Comparing male and female enrollment across different majors
- Emphasizes comparison, not trends over time
Although bar graphs come in many forms, this section focuses only on these three most commonly used types. These examples provide the foundation students need to recognize how categorical data can be compared, grouped, or displayed over time. While other variations of bar graphs exist, such as segmented bars, paired bars, and population bars, they tend to be used in more specialized contexts or rely on the same underlying principles already demonstrated. By concentrating on the most widely used forms, this keeps the explanations clear and accessible without overwhelming learners with unnecessary variations. Students will still be well‑prepared to interpret other bar graph formats they may encounter, since the core ideas remain the same.
Advantages of Bar Graphs
- Easy to read and understand: Simple, intuitive format that most people recognize
- Excellent for comparisons: Bar heights make it obvious which values are larger or smaller
- Works with many categories: Can display numerous categories without becoming cluttered
- Precise values: Bar heights allow for accurate reading of exact values
- Flexible: Can be vertical or horizontal, grouped or stacked
- Shows individual values clearly: Each category stands alone
- No misleading connections: Spaces between bars prevent implying relationships that don't exist
Limitations of Bar Graphs
- Not ideal for trends over time: Line graphs are better for showing continuous change
- Can become cluttered: Too many categories or groups can make the graph hard to read
- Limited in showing relationships: Shows comparisons but not correlations between variables
- Scale manipulation: Starting the y-axis above zero can exaggerate differences
When NOT to use Bar Graphs
Avoid bar graphs when:
- You want to show trends or changes over continuous time (use line graphs)
- You need to display parts of a whole (use pie graphs for small numbers of categories)
- You're showing the relationship between two quantitative variables (use scatter plots)
- Data is continuous rather than categorical
- Step 1 — Organize your data in a table with categories and their corresponding values
- Step 2 —Draw and label the axes
- Horizontal axis (x-axis): List the categories with equal spacing
- Vertical axis (y-axis): Show the measurement scale (frequency, count, percentage, etc.)
- Include units where appropriate
- Step 3 — Determine the scale for the vertical axis (y-axis):
- Find the maximum value in your data
- Choose an upper limit that accommodates all values
- Use consistent intervals (e.g., 0, 10, 20, 30, 40...)
- Generally start at zero to avoid distorting comparisons
- Step 4 — Draw the bars
- For each category, draw a rectangular bar
- Height (or length) should accurately represent the value
- Keep all bars the same width
- Leave spaces between bars to show distinct categories
- Step 5 — Label Clearly
- Mark categories on the horizontal axis (x-axis)
- Ensure vertical axis (y-axis) scale is clear
- Consider adding value labels on top of bars for precision
- Step 6 — Add a descriptive title, if necessary
- Example: "Favorite Sports Among 100 High School Students"
A college surveyed 200 students about their class standing:
| Classes | Frequency |
|---|---|
| Freshman | 60 |
| Sophomore | 50 |
| Junior | 45 |
| Senior | 45 |
Create a bar graph to display this data.
✅ Solution:
- Step 1 — Organize your data in a table with categories and their corresponding values. This step is done, since data was given in a frequency distribution.
- Step 2 — Draw and label the axes. The horizontal axis is labeled "Classes" and the vertical axis is labeled "Frequency".
- Step 3 — Determine the scale for the vertical axis (y-axis). Since the maximum frequency is 60, then increments of 10 can be used for the tick marks on the axis.
- Step 4 — Draw the bars. Draw rectangular bars of equal width up to the frequency of each class.
- Step 5 — Label clearly. Label each of the rectangular bars: Freshman, Sophomore, Junior, and Senior.
- Step 6 — Add a descriptive title, if necessary. Add the title, "Class Standings" or some other similar title.
A teacher recorded final grades for 40 students:
A, B, F, C, B, D, B, C, B, C, A, C, B, C, B, C, A, D, C, B, B, C, C, A, C, C, B, A, A, B, A, D, B, D, F, B, B, A, B, C
Create a bar graph to display this data.
✅ Solution:
- Step 1 — Organize your data in a table with categories and their corresponding values. Create a frequency distribution for the data.
| Grades | Frequency |
|---|---|
| A | 8 |
| B | 14 |
| C | 12 |
| D | 4 |
| F | 2 |
- Step 2 — Draw and label the axes. The horizontal axis is labeled "Grades" and the vertical axis is labeled "Frequency".
- Step 3 — Determine the scale for the vertical axis (y-axis). Since the maximum frequency is 12, then increments of 2 can be used for the tick marks on the axis.
- Step 4 — Draw the bars. Draw rectangular bars of equal width up to the frequency of each letter grade.
- Step 5 — Label clearly. Label each of the rectangular bars: A, B, C, D, and F.
- Step 6 — Add a descriptive title, if necessary. Add the title, "Final Grades" or some other similar title.
A survey was conducted and asked adults their primary source of news:
| News Source | Respondents |
|---|---|
| Television | 57 |
| Internet/Social Media | 149 |
| Newspapers | 6 |
| Radio | 38 |
Create a bar graph to display this data.
✅ Solution:
- Step 1 — Organize your data in a table with categories and their corresponding values. This step is done, since data was given in a frequency distribution.
- Step 2 — Draw and label the axes. The horizontal axis is labeled "Grades" and the vertical axis is labeled "Frequency".
- Step 3 — Determine the scale for the vertical axis (y-axis). Since the maximum frequency is 149, then increments of 20 can be used for the tick marks on the axis.
- Step 4 — Draw the bars. Draw rectangular bars of equal width up to the frequency of each letter grade.
- Step 5 — Label clearly. Label each of the rectangular bars: Television, Internet/Social Media, Newspapers, and Radio.
- Step 6 — Add a descriptive title, if necessary. Add the title, "Primary Source of News" or some other similar title.
A survey of 500 smartphone users were asked which type of phone that they own/prefer is given below:
| Type of Phone | Users |
|---|---|
| iOS (iPhone) | 214 |
| Android | 258 |
| Other | 28 |
Create a bar graph to display this data.
✅ Solution:
- Step 1 — Organize your data in a table with categories and their corresponding values. This step is done, since data was given in a frequency distribution.
- Step 2 — Draw and label the axes. The horizontal axis is labeled "Grades" and the vertical axis is labeled "Frequency".
- Step 3 — Determine the scale for the vertical axis (y-axis). Since the maximum frequency is 258, then increments of 50 can be used for the tick marks on the axis.
- Step 4 — Draw the bars. Draw rectangular bars of equal width up to the frequency of each letter grade.
- Step 5 — Label clearly. Label each of the rectangular bars: iOS (iPhone), Android, and Other.
- Step 6 — Add a descriptive title, if necessary. Add the title, "Phone Device Preference" or some other similar title.
A travel agency surveyed their clients about preferred vacation types:
| Vacation Type | Clients |
|---|---|
| Beach | 95 |
| Mountains | 64 |
| City/Urban | 48 |
| Adventure/Safari | 44 |
| Cruise | 28 |
| National Park/Wildlife | 40 |
| Lake/Countryside | 30 |
Create a bar graph to display this data.
✅ Solution:
- Step 1 — Organize your data in a table with categories and their corresponding values. This step is done, since data was given in a frequency distribution.
- Step 2 — Draw and label the axes. The horizontal axis is labeled "Grades" and the vertical axis is labeled "Frequency".
- Step 3 — Determine the scale for the vertical axis (y-axis). Since the maximum frequency is 95, then increments of 10 can be used for the tick marks on the axis.
- Step 4 — Draw the bars. Draw rectangular bars of equal width up to the frequency of each letter grade.
- Step 5 — Label clearly. Label each of the rectangular bars: Beach, Mountains, City/Urnam, Adventure/Safari, Cruise, National Park/Wildfire, and Lake/Countryside.
- Step 6 — Add a descriptive title, if necessary. Add the title, "Vacation Destinations" or some other similar title.
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A bag of peanut M&M’s were opened and the following colors with their respective quantities were present: 6 Blue, 4 Brown, 1 Green, 5 Orange, 3 Red, 2 Yellow. Construct a bar graph for the data.
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A survey of 100 8th-graders was conducted. They were asked what their favorite sport is.The results are given in the frequency table.
Sport Number of 8th Graders Baseball 17 Basketball 26 Football 22 Hockey 15 Track 6 Volleyball 9 Wrestling 5 Create a bar graph to display this data.
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The table below provides the total electoral votes for the last eight U.S. Presidential elections.
Year Democrat Republican 1996 379 159 2000 266 271 2004 251 286 2008 365 173 2012 332 206 2016 227 304 2020 306 232 2024 226 312 Construct a bar graph to display this data.
- Answers
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