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Build your own math worksheets from 35,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Graphs for one categorical variable

Learning goals

  • Constructs bar charts or pie charts from categorical counts or proportions.
  • Reads category counts or proportions from a bar chart or pie chart.
  • Chooses between a bar chart and pie chart for categorical data.
  • Compares category counts or proportions.

Click problems to add them to your worksheet.

Graphs for one categorical variable 1
The bar chart shows the number of books selected in four genres. Which genre was selected most often, and by how many more selections than History?
Figure for problem 539660

Hints

- Check the category labels and the y-axis increments before reading any value. - Read the two bar heights from the scale, then subtract in the requested order.

Solution

1. Fantasy has the tallest bar at \(45\). 2. History has \(20\) selections. 3. The difference is \(45-20=25\).

Answer

Fantasy, by \(25\) selections.
Graphs for one categorical variable 2
What percentage of responses fall in the “Not reported” category?
Figure for problem 539669

Hints

- A full pie chart represents all responses. - Add the percentages printed on the other slices before finding what remains.

Solution

1. The labeled slices total \(37\%+29\%+18\%=84\%\). 2. The full pie is \(100\%\), so “Not reported” represents \(100\%-84\%=16\%\).

Answer

\(16\%\).
Graphs for one categorical variable 3
A categorical variable has labels “Less than once a month,” “One to three times a month,” “Once a week,” and “Several times a week.” Explain why a horizontal bar chart would usually be more readable than a vertical bar chart for these data. State what mathematical information is unchanged by changing orientation.

Hints

- Consider how much space each category label needs. - Separate a formatting decision from a data transformation. - Ask which axis carries the category names after rotation.

Solution

1. The category labels are long, so placing them beside horizontal bars allows the text to remain readable without rotation or crowding. 2. Orientation changes only the layout of the graph. 3. The category frequencies, relative frequencies, ordering, and comparisons among bar lengths remain unchanged.

Answer

A horizontal bar chart gives the long labels more space. Changing orientation does not change any frequency, relative frequency, order, or comparison.
Graphs for one categorical variable 4
A community center asks members to choose one weekend workshop. How many members chose Music?
Figure for problem 563391

Hints

- Locate the bar labeled Music. - Follow its top across to the numbered vertical scale.

Solution

1. Find the Music bar on the horizontal category axis. 2. Read its height from the vertical frequency scale: \(7\).

Answer

\(7\) members.
Graphs for one categorical variable 5
An arts center asks members which music class they prefer. Does any slice cover more than half the circle? Name the class and explain how you can tell.
Figure for problem 563392

Hints

- Find the largest part of the circle. - Compare its size with a semicircle, not a printed percentage.

Solution

1. The largest slice is labeled Choir. 2. The Choir slice is larger than a semicircle, so it represents more than half of the members.

Answer

Yes. Choir takes more than half of the circle.
Graphs for one categorical variable 6
The bar chart displays relative frequencies for the disposal method used for sampled cafeteria items. What percentage of items were recycled, and is that more than the other two categories combined?
Figure for problem 539661

Hints

- Translate the bar height into a percentage and compare it with the remainder. - The complement of the recycled share is the combined share of the other categories.

Solution

1. The Recycle bar has relative frequency \(0.45=45\%\). 2. The other categories total \(0.30+0.25=0.55=55\%\). 3. Recycling is not more than the other two combined.

Answer

\(45\%\) were recycled; no, the other two categories total \(55\%\).
Graphs for one categorical variable 7
The pie chart shows how \(360\) students usually travel to school. How many students usually walk?
Figure for problem 539662

Hints

- Use the slice’s share of the full sample. - Convert the Walk slice from a percent to a decimal before applying it to the total.

Solution

1. The Walk slice represents \(25\%\). 2. The count is \(0.25\cdot 360=90\).

Answer

\(90\) students.
Graphs for one categorical variable 8
A film club claims that more than half of members chose either Drama or Comedy. Does the pie chart support the claim?
Figure for problem 539663

Hints

- Combine every slice named in the claim before comparing with the benchmark. - A majority means a combined share strictly greater than \(50\%\).

Solution

1. Drama and Comedy account for \(32\%+28\%=60\%\). 2. Since \(60\%>50\%\), the claim is supported.

Answer

Yes. Together, Drama and Comedy account for \(60\%\).
Graphs for one categorical variable 9
The horizontal bar chart shows the region selected by \(80\) survey respondents. What fraction selected either South or West? Give the fraction in simplest form.
Figure for problem 539665

Hints

- Combine the relevant bar lengths and compare with the total sample size. - Add the South and West frequencies, place that sum over the chart total, and simplify.

Solution

1. South and West total \(30+20=50\). 2. The fraction is \(\frac{50}{80}=\frac{5}{8}\).

Answer

\(\frac{5}{8}\).
Graphs for one categorical variable 10
The pie chart summarizes responses from \(500\) people. How many more people answered Yes than No?
Figure for problem 539666

Hints

- Find the difference between the two slices before applying it to the total. - A percentage-point gap can be converted to a count gap because both slices refer to the same total.

Solution

1. The percentage-point difference is \(45\%-35\%=10\%\). 2. The count difference is \(0.10\cdot 500=50\).

Answer

\(50\) more people answered Yes.
Graphs for one categorical variable 11
The chart shows responses to a frequency question. Which statement is supported? a) A majority selected Often or Always. b) Exactly one-third selected Sometimes. c) More selected Sometimes than any other single category.
Figure for problem 539670

Hints

- Check each statement against the exact bar heights. - Test each choice separately rather than stopping after the first false statement.

Solution

1. Often or Always totals \(28+12=40\), not a majority of \(100\). 2. Sometimes is \(36\), not exactly one-third of \(100\). 3. The Sometimes bar is the tallest at \(36\).

Answer

c) More selected Sometimes than any other single category.
Graphs for one categorical variable 12
The bar chart represents a sample of \(80\) observations. Which relative frequency belongs to category \(C\)?
Figure for problem 539672

Hints

- Read the category count and compare it with the stated total. - Use \(\frac{\text{category frequency}}{\text{sample size}}\) for the requested bar.

Solution

1. Category \(C\) has frequency \(24\). 2. Its relative frequency is \(\frac{24}{80}=0.30\).

Answer

\(0.30\), or \(30\%\).
Graphs for one categorical variable 13
Does the pie chart support the statement, “A majority gave a favorable response,” if favorable means Strongly agree or Agree?
Figure for problem 539675

Hints

- Identify every slice included in the definition of a favorable response. - Compare their combined share with the strict \(50\%\) majority benchmark.

Solution

1. Favorable responses total \(26\%+31\%=57\%\). 2. Since \(57\%>50\%\), a majority gave a favorable response.

Answer

Yes. Favorable responses account for \(57\%\).
Graphs for one categorical variable 14
The graph shows relative frequencies rather than counts for \(250\) arrivals. Find the frequency represented by the Late bar.
Figure for problem 539676

Hints

- Read the Late bar as a relative frequency on the decimal scale shown. - Convert a relative frequency to a count by multiplying by the total number of arrivals.

Solution

1. The Late bar has relative frequency \(0.20\). 2. The frequency is \(0.20\cdot250=50\).

Answer

The Late frequency is \(50\).
Graphs for one categorical variable 15
Use the graph to express the frequency of category \(B\) as a ratio to category \(C\), in simplest form.
Figure for problem 539677

Hints

- Read both bar heights and simplify their comparison. - Preserve the order \(B:C\) and divide both counts by a common factor.

Solution

1. The ratio is \(40:16\). 2. Divide both terms by \(8\) to obtain \(5:2\).

Answer

\(5:2\).
Graphs for one categorical variable 16
The bar chart shows the primary neighborhood improvement selected by \(100\) survey respondents. A planner wants to rearrange the same bars into a Pareto-style display, with categories ordered from greatest to least frequency. List the categories in the new order. Then explain which features of the categorical distribution change and which remain unchanged when the bars are reordered.
Figure for problem 548666

Hints

- Read each bar height before deciding the order. - A rearrangement does not add, remove, or recode observations. - Separate visual position from the numerical information represented by a bar.

Solution

1. The bar frequencies are Transit \(35\), Parks \(30\), Housing \(20\), and Safety \(15\). 2. In descending order, the categories are Transit, Parks, Housing, and Safety. 3. Reordering changes only the horizontal placement of the bars. Each category's frequency, relative frequency, modal category, and the total sample size remain unchanged.

Answer

The order is Transit, Parks, Housing, Safety. Only the positions of the bars change; the frequencies, relative frequencies, mode, and total remain the same.
Graphs for one categorical variable 17
In a survey of \(120\) students, each student could select every after-school activity that interested them. The results were Theater \(68\), Coding \(54\), Music \(47\), and Community Service \(39\). Thierry proposes displaying the results in a pie chart. Evaluate the proposal and name a more appropriate graph.

Hints

- Check whether one respondent can belong to more than one category. - Consider what the full circle of a pie chart is supposed to represent. - Choose a display that compares category amounts without requiring them to total \(100\%\).

Solution

1. The category counts sum to \(68+54+47+39=208\), which is greater than the \(120\) respondents because students could select more than one category. 2. Pie-chart sectors represent mutually exclusive parts of one whole and must total \(100\%\). These overlapping selections do not meet that condition. 3. A bar chart can display each activity's count or selection rate without implying that the categories partition the students.

Answer

A pie chart is not appropriate because the categories overlap and do not form parts of one whole. A bar chart is appropriate.
Graphs for one categorical variable 18
The bar chart places the nominal categories A, B, and C at uneven horizontal positions. Lujain says the large empty space before C means that C is “farther from” B than B is from A. Evaluate the statement and describe how the graph should be revised.
Figure for problem 548668

Hints

- Decide whether the horizontal axis represents measurements or category labels. - Ask whether gaps between category positions encode any recorded variable. - Separate a layout choice from the bar heights that represent frequencies.

Solution

1. The x-axis positions represent category placement, not measured numerical distances. 2. The extra space before C therefore has no statistical meaning and can create a false impression of separation. 3. The bars should be evenly spaced while preserving their heights of \(20\), \(35\), and \(25\).

Answer

Lujain's statement is incorrect. Nominal categories have no meaningful numerical distance, so the bars should be evenly spaced. Their frequencies remain A: \(20\), B: \(35\), and C: \(25\).
Graphs for one categorical variable 19
A bar chart for a categorical survey should include four regions and a total of \(150\) responses. The bar for West was accidentally left out. Find the height of the missing West bar and its relative frequency.
Figure for problem 548671

Hints

- Add the heights of all bars that are already shown. - Compare that sum with the stated total number of responses. - Convert the missing count to a share of the full sample.

Solution

1. The displayed frequencies total \(45+35+40=120\). 2. The missing West frequency is \(150-120=30\). 3. Its relative frequency is \(\frac{30}{150}=0.20\), or \(20\%\).

Answer

The West bar should have height \(30\), representing \(20\%\) of the responses.
Graphs for one categorical variable 20
The bars show category frequencies from a survey of \(120\) people, but the y-axis is labeled “Percent.” Identify the labeling error and give the correct percentage represented by each bar.
Figure for problem 548673

Hints

- Compare the sum of the bar heights with the stated sample size. - Decide whether each height is a count or a share. - Convert every count using the same denominator.

Solution

1. The bar heights are counts \(54\), \(36\), and \(30\), not percentages, because they total \(120\). 2. The correct percentages are \(\frac{54}{120}=45\%\), \(\frac{36}{120}=30\%\), and \(\frac{30}{120}=25\%\). 3. The y-axis should either be relabeled “Frequency” or the bar heights should be changed to \(45\), \(30\), and \(25\) on a percent scale.

Answer

The y-axis incorrectly labels counts as percentages. The correct percentages are A: \(45\%\), B: \(30\%\), and C: \(25\%\).
Graphs for one categorical variable 21
A pie chart labels its four sector angles as \(110^\circ\), \(95^\circ\), \(80^\circ\), and \(85^\circ\). Explain why the chart is impossible as labeled. If the first three angles are correct, what should the fourth angle be?

Hints

- Add all displayed sector angles. - Compare the result with one full rotation. - Hold the stated correct angles fixed and find the remainder.

Solution

1. The labeled angles total \(110^\circ+95^\circ+80^\circ+85^\circ=370^\circ\), but a full circle contains only \(360^\circ\). 2. If the first three angles are correct, they total \(285^\circ\). 3. The fourth angle must be \(360^\circ-285^\circ=75^\circ\).

Answer

The labels are impossible because they sum to \(370^\circ\). The fourth angle should be \(75^\circ\).
Graphs for one categorical variable 22
A ballot offers four response options, including “Abstain.” No voter selected “Abstain.” The election office wants a graph that makes every ballot option immediately visible. Compare a bar chart with a pie chart for this purpose, and explain how “Abstain” should be represented.

Hints

- Distinguish whether a graph is mathematically valid from whether it serves the stated communication goal well. - Consider what a zero frequency looks like in each display. - Preserve the category label even though its represented amount is zero.

Solution

1. The “Abstain” category has frequency \(0\), but it is still part of the ballot design. 2. A bar chart can display the “Abstain” label at a position with bar height \(0\), so the option remains immediately visible. 3. A pie chart can list “Abstain” in a complete legend, but a zero-frequency category has no visible sector. Therefore, a bar chart communicates the office’s documentation goal more directly.

Answer

A bar chart is preferable because it can show “Abstain” as a labeled category with bar height \(0\). A pie chart is not inherently invalid if its legend includes the category, but the category would have no visible sector.
Graphs for one categorical variable 23
A pie-chart sector has central angle \(72^\circ\), but its legend labels the category as \(25\%\). Identify the inconsistency. Give the correct percentage for the displayed angle and the correct angle for the stated percentage.
Figure for problem 548679

Hints

- Compare the sector angle with a complete \(360^\circ\) circle. - Convert the displayed angle to a fraction of the whole. - Independently convert the stated percentage to an angle.

Solution

1. The displayed sector represents \(\frac{72}{360}=0.20=20\%\). 2. A \(25\%\) sector should have angle \(0.25(360^\circ)=90^\circ\). 3. Therefore, either the legend should read \(20\%\) or the sector should be redrawn as \(90^\circ\).

Answer

A \(72^\circ\) sector represents \(20\%\), not \(25\%\). A true \(25\%\) sector would have central angle \(90^\circ\).
Graphs for one categorical variable 24
A report uses a tilted three-dimensional pie chart. One sector is labeled \(18\%\) and lies at the front of the disk; another is labeled \(24\%\) and lies near the back. In the rendered chart, the \(18\%\) sector looks larger, and a reader concludes that its category is more common. Evaluate the reader's conclusion. Explain what feature of the three-dimensional design can create the misleading appearance and name a better display choice.

Hints

- Compare the stated percentages before trusting apparent size. - Ask whether depth or viewing position can change apparent area without changing the data. - Choose a replacement display whose visual encoding is not affected by perspective.

Solution

1. The conclusion is false because \(18\%<24\%\); the labels indicate that the back category has the larger relative frequency. 2. Tilting a three-dimensional disk introduces perspective: sectors nearer the viewer can appear larger than sectors farther away even when their data shares are smaller. 3. A flat two-dimensional pie chart removes this depth distortion. A bar chart is also appropriate and makes category comparisons easier.

Answer

The conclusion is false. Perspective in the tilted three-dimensional disk can enlarge the apparent area of a front sector independently of its relative frequency. Use a flat two-dimensional pie chart or a bar chart instead.
Graphs for one categorical variable 25
The bars in the graph touch, so Kyriakos calls the display a histogram. The x-axis categories are Red, Blue, and Green. Is Kyriakos correct? Classify the graph and explain which feature determines the classification.
Figure for problem 548681

Hints

- Look at what the horizontal axis represents rather than only at the spacing between bars. - Decide whether the possible values are categories or numerical intervals. - Classification depends on the variable, not on a decorative choice.

Solution

1. Red, Blue, and Green are categories rather than numerical intervals. 2. A graph of frequencies for categories is a bar chart, regardless of whether the bars happen to touch. 3. Histograms require a quantitative variable divided into ordered numerical intervals.

Answer

Kyriakos is not correct. The graph is a bar chart because its x-axis represents categories, not quantitative intervals.
Graphs for one categorical variable 26
The bar chart shows an ordinal satisfaction variable, but its categories are arranged in an arbitrary order. Give a more informative left-to-right order and explain why that order is preferable to sorting only by bar height.
Figure for problem 548682

Hints

- Decide whether the category names have an inherent ranking. - Think about what a reader should be able to see across adjacent bars. - Frequency order and meaning order serve different purposes.

Solution

1. The response categories have a natural order from least to most satisfied. 2. A suitable order is Poor, Fair, Good, Excellent. 3. This order preserves the meaning of the ordinal scale and makes shifts toward lower or higher satisfaction visible. Sorting by frequency would hide that progression.

Answer

Use Poor, Fair, Good, Excellent. The natural ordinal order communicates the scale's progression better than a frequency ranking.
Graphs for one categorical variable 27
One categorical data set has frequencies \(31\), \(29\), \(22\), and \(18\). A report must make the small difference between the two largest categories easy to compare. Would a bar chart or a pie chart better serve that purpose? Justify the choice using the way each graph encodes frequency.

Hints

- Focus on the report's specific comparison goal. - Consider which visual encoding supports precise judgments between close values. - A graph can be valid yet still be less effective for a particular purpose.

Solution

1. The two largest categories differ by only \(31-29=2\) observations. 2. A bar chart encodes frequency by position along a common numerical scale, so the two heights can be compared directly. 3. A pie chart encodes frequency by sector angle and area, which makes a small difference harder to judge accurately.

Answer

Use a bar chart. Its common baseline and numerical scale make the difference of \(2\) easier to compare than pie-sector angles or areas.
Graphs for one categorical variable 28
The pie chart does not state the survey's sample size. Can the exact number of “Remote” responses be found? Give two different possible sample sizes and counts that match the chart.
Figure for problem 548684

Hints

- A pie chart shows shares of a whole, not necessarily the number of responses. - What other quantity is needed to change a share into a count? - Think of two totals for which the shown percentage gives a whole-number count.

Solution

1. The chart gives a proportion of \(40\%\) for “Remote,” not a total number of responses. 2. A sample of \(5\) could have \(2\) “Remote” responses. 3. A sample of \(100\) could have \(40\) “Remote” responses. 4. Different counts can produce the same sector, so the exact count is unknown.

Answer

No. For example, \(2\) of \(5\) responses or \(40\) of \(100\) responses would give the same sector.
Graphs for one categorical variable 29
The pie chart is intended to display four regions: North, South, East, and West. Inspect the chart and its legend. What labeling defect is present, and what can and cannot be concluded before it is corrected?
Figure for problem 548685

Hints

- Compare the four intended region names with the four legend entries. - Separate information carried by sector size from information carried by the legend. - Do not guess which sector should have the missing region name.

Solution

1. The legend lists East twice and does not list West, so at least one sector has the wrong region label. 2. The sector sizes still show that the observations were divided into four shares. 3. Because the mislabeled sector cannot be identified from the chart alone, the four shares cannot be assigned reliably to all four intended regions until the legend is corrected.

Answer

East appears twice in the legend and West is missing. The chart still shows four sector shares, but the shares cannot all be assigned reliably to North, South, East, and West until the labeling error is corrected.
Graphs for one categorical variable 30
A bar chart of response frequencies includes one bar extending to \(-5\). Explain why the graph cannot represent raw category frequencies as labeled. Give one plausible data-handling issue that should be investigated before the chart is corrected.
Figure for problem 548686

Hints

- Recall the possible values of a count. - Ask whether the vertical axis might actually represent a different quantity. - Identify the inconsistency before proposing a replacement value.

Solution

1. A frequency counts observations, so it cannot be negative. 2. The bar at \(-5\) shows that either the axis label is wrong or the plotted value is not a raw frequency. 3. One plausible issue is that a change from a baseline, such as a net gain or loss, was plotted but labeled as frequency; another is an erroneous subtraction during data processing.

Answer

The chart is invalid as a frequency graph because counts cannot be negative. The analyst should check for a mislabeled change score or a data-processing error.
Graphs for one categorical variable 31
The bar chart summarizes responses from \(120\) people. Does the modal category represent a majority? Do the two most common categories together represent a majority? Support both conclusions numerically.
Figure for problem 548687

Hints

- Identify the tallest bar and compare its count with half of the stated total. - Then combine the two tallest bars. - A mode is the most common category; a majority must exceed half of all observations.

Solution

1. The tallest bar is Category A with frequency \(42\), so its relative frequency is \(\frac{42}{120}=0.35\). It is the mode but not a majority. 2. The two tallest bars are A and B, with combined frequency \(42+33=75\). 3. Their combined relative frequency is \(\frac{75}{120}=0.625\), or \(62.5\%\), so together they represent a majority.

Answer

The modal category is not a majority because it represents \(35\%\). The two most common categories together are a majority because they represent \(62.5\%\).
Graphs for one categorical variable 32
In the pie chart, categories C and D will be recoded as one category. What angle and relative frequency should the merged sector have? What happens to sectors A and B?
Figure for problem 548688

Hints

- Read the two sector angles that are being combined. - Relate the merged angle to the \(360^\circ\) in a full circle. - Combining C and D does not change observations in A or B.

Solution

1. The chart shows C with angle \(72^\circ\) and D with angle \(54^\circ\). 2. The merged angle is \(72^\circ+54^\circ=126^\circ\). 3. Its relative frequency is \(\frac{126}{360}=0.35\), or \(35\%\). 4. The angles and relative frequencies for A and B remain unchanged because only C and D are combined.

Answer

The merged sector has angle \(126^\circ\) and relative frequency \(35\%\). Sectors A and B remain unchanged.
Graphs for one categorical variable 33
Panels a) and b) are equal-size pie charts for two samples. Each chart has the same two categories, Premium and Standard. A reader claims that the Premium relative frequency is the same in both samples because the two complete circles have the same physical size. Is the reader correct? Identify which sample has the larger Premium relative frequency and explain what visual feature of a pie chart represents relative frequency.
Figure for problem 548692

Hints

- Compare the Premium sector with its own full circle in each panel. - Do not use the overall diameter of a pie to judge a category's share. - Focus on the part-to-whole angle represented by the Premium sector.

Solution

1. In panel a), the Premium sector occupies a larger fraction of the circle than the Premium sector in panel b). 2. Therefore, sample a) has the larger Premium relative frequency. 3. The physical size of the whole circle does not encode relative frequency. Within a pie chart, the sector angle, equivalently its fraction of the full circle, represents the category's relative frequency.

Answer

No. Sample a) has the larger Premium relative frequency. Relative frequency is represented by the sector's angle or fraction of the full circle, not by the physical size of the complete pie.
Graphs for one categorical variable 34
A pictograph will display three category frequencies: \(32\), \(28\), and \(16\) responses. The designer wants one full icon to represent \(8\) responses and may use half icons. Specify the icons needed for each category and write a complete legend that makes the meaning of a half icon unambiguous. Explain why stating only “one full icon represents \(8\) responses” is not a complete legend when partial icons are used.

Hints

- Convert each frequency into units of the full-icon value. - Treat a fractional icon as part of the representation that needs its own explicit meaning. - Check that every symbol used in the finished pictograph can be interpreted from the legend alone.

Solution

1. The first category needs \(32\div8=4\) full icons. 2. The second category needs \(28\div8=3.5\) icons, so it can be shown with \(3\) full icons and \(1\) half icon. 3. The third category needs \(16\div8=2\) full icons. 4. A complete legend can state: one full icon represents \(8\) responses, and one half icon represents \(4\) responses. 5. Without the half-icon statement, the reader has to assume how a partial symbol is scaled instead of being told the graph's encoding rule.

Answer

First category: \(4\) full icons. Second category: \(3\) full icons and \(1\) half icon. Third category: \(2\) full icons. Legend: one full icon represents \(8\) responses; one half icon represents \(4\) responses. The half-icon meaning must be stated because a graph should not require readers to guess how partial symbols are encoded.
Graphs for one categorical variable 35
A school club survey produced this table. <table><thead><tr><th>Club</th><th>Students</th></tr></thead><tbody><tr><td>Debate</td><td>\(9\)</td></tr><tr><td>Drama</td><td>\(12\)</td></tr><tr><td>Music</td><td>\(7\)</td></tr><tr><td>Science</td><td>\(12\)</td></tr></tbody></table> Specify a correct categorical bar chart for these data. Use y-axis ticks spaced every \(5\) students and the smallest top tick that includes every bar. In your answer, state the y-axis baseline and tick values, whether neighboring bars should touch, the four bar heights in table order, and the tallest bar or bars.

Hints

- Decide which bar-chart conventions are required because the horizontal variable consists of categories rather than numerical intervals. - Use the stated tick spacing together with the largest frequency to determine the top of the vertical scale. - Map each category's count to a bar height, then compare the completed heights.

Solution

1. A frequency bar chart uses a zero baseline so bar heights represent counts proportionally. 2. With tick spacing \(5\), the smallest top tick above the maximum frequency \(12\) is \(15\), so the y-axis ticks are \(0,5,10,15\). 3. Because the horizontal variable is categorical, neighboring bars should be separated rather than touching. 4. The bar heights are \(9,12,7,12\) for Debate, Drama, Music, and Science. 5. Drama and Science tie for the tallest bar at \(12\).

Answer

Baseline: \(0\). Y-axis ticks: \(0,5,10,15\). Neighboring bars should not touch. Bar heights in table order: \(9,12,7,12\). Drama and Science are tied for tallest at \(12\).
Graphs for one categorical variable 36
A transportation survey of \(80\) students found: Walk \(24\), Bike \(16\), Bus \(28\), Car \(12\). Find the sector angle for each category in a pie chart and identify the largest sector.

Hints

- Every category receives the same share of a circle as its share of the responses. - Change each count into a fraction of \(80\) before finding degrees. - Check that the angles fill exactly one circle.

Solution

1. A full pie is \(360^\circ\), so multiply each count's share of \(80\) by \(360^\circ\). 2. Walk: \(\frac{24}{80}\cdot360^\circ=108^\circ\). 3. Bike: \(\frac{16}{80}\cdot360^\circ=72^\circ\). 4. Bus: \(\frac{28}{80}\cdot360^\circ=126^\circ\). 5. Car: \(\frac{12}{80}\cdot360^\circ=54^\circ\). 6. The angles total \(360^\circ\); Bus is the largest sector.

Answer

Walk: \(108^\circ\); Bike: \(72^\circ\); Bus: \(126^\circ\); Car: \(54^\circ\). Bus is the largest sector.
Graphs for one categorical variable 37
A public library asks visitors to choose one favorite event: poetry, crafts, lectures, or films. The librarian wants to compare the number of votes for each event. Should the librarian use a bar chart or a histogram? Explain.

Hints

- What kind of value would be recorded for each visitor? - Could you sensibly subtract one response label from another? - Think about what the horizontal axis would need to display.

Solution

1. Favorite event is categorical because each answer names a kind of event. 2. A bar chart compares frequencies across named categories. 3. A histogram groups numerical measurements into consecutive intervals, so it does not fit this variable.

Answer

Use a bar chart. Event choices are categories, not numerical intervals for a histogram.
Graphs for one categorical variable 38
A student lunch survey compares wraps and salads. How many more students chose a salad than a wrap? Why might the chart make that difference look much larger than it is?
Figure for problem 563394

Hints

- Read both counts using the axis numbers, not just apparent bar lengths. - Subtract the two frequencies. - Check whether the vertical scale begins at zero.

Solution

1. The chart gives \(82\) for wraps and \(86\) for salads. 2. The difference is \(86-82=4\) students. 3. The vertical scale starts at \(78\), not zero, so the visible bar lengths exaggerate the difference.

Answer

Salads have \(4\) more students. The y-axis begins at \(78\), making the gap look much larger.
Graphs for one categorical variable 39
A school chart shows the number of donations from three drives on a logarithmic vertical axis. Read the counts for Coats and Books. How many times as many books as coats were donated? Do equal distances between the bar tops mean equal numerical increases?
Figure for problem 563434

Hints

- Read the numbers beside the vertical ticks, not just the bar lengths. - Compare two counts using the kind of comparison the question asks for. - Check both the differences and the ratios between neighboring counts.

Solution

1. The Coats bar reaches \(100\) donations and the Books bar reaches \(1000\). 2. The ratio is \(1000\div100=10\), so ten times as many books as coats were donated. 3. The chart uses equal distances for multiplication by \(10\), not equal additions. The increases from Cans to Coats and from Coats to Books are \(90\) and \(900\).

Answer

Coats: \(100\), Books: \(1000\); Books is \(10\) times Coats. The equal vertical steps represent equal ratios, not equal increases.
Graphs for one categorical variable 40
Charts a) and b) show the same categorical variable for two groups. Compare the relative frequency of category \(A\) in the two groups.
Figure for problem 539664

Hints

- Find the total number of observations in each panel. - Convert category \(A\) to a relative frequency within each group. - Compare the two shares rather than the raw counts.

Solution

1. In a), the total is \(48+32+20=100\), so the relative frequency is \(0.48\). 2. In b), the total is \(30+24+6=60\), so the relative frequency is \(0.50\). 3. Category \(A\) has a slightly larger relative frequency in group b).

Answer

Group a): \(0.48\). Group b): \(0.50\). Category \(A\) is relatively more common in b).
Graphs for one categorical variable 41
The graph shows relative frequencies for preferred notification method. A report claims that Text was selected by at least twice the proportion of respondents who selected App. Is the claim correct? State the actual factor.
Figure for problem 539667

Hints

- Read the two relevant relative frequencies from the bars. - A statement about “times as large” requires a ratio rather than a difference. - Compare the ratio with the precise threshold in the claim.

Solution

1. The Text relative frequency is \(0.45\), and the App relative frequency is \(0.20\). 2. The factor is \(\frac{0.45}{0.20}=2.25\). 3. Since \(2.25\ge 2\), the claim is correct.

Answer

Yes. Text was selected at \(2.25\) times the proportion for App.
Graphs for one categorical variable 42
The bar chart shows award categories. The Silver bar represents \(20\%\) of all observations. Use the chart to find the total number of observations.
Figure for problem 539668

Hints

- Read the Silver frequency from the graph and note its stated percentage. - Represent the frequency as the stated share of an unknown total. - Solve for the total and check that the remaining chart frequencies are compatible with it.

Solution

1. The Silver frequency is \(36\). 2. If \(36\) is \(20\%\) of the total \(n\), then \(36=0.20n\). 3. Thus, \(n=180\).

Answer

There are \(180\) observations.
Graphs for one categorical variable 43
Pie chart a) summarizes \(80\) responses, and pie chart b) summarizes \(150\) responses. Which group has the greater number of Approve responses?
Figure for problem 539671

Hints

- A larger percentage does not necessarily mean a larger count when totals differ. - Convert each approval percentage to a count using that chart’s own sample size. - Compare the resulting counts, not the slice angles alone.

Solution

1. Chart a) has \(0.60\cdot 80=48\) Approve responses. 2. Chart b) has \(0.52\cdot 150=78\) Approve responses. 3. Group b) has the greater count.

Answer

Group b), with \(78\) Approve responses compared with \(48\) in group a).
Graphs for one categorical variable 44
If \(6\) additional Plastic items are added to the data represented by the chart, while all other counts stay the same, what is the new relative frequency for Plastic?
Figure for problem 539673

Hints

- Find the original total by combining all chart frequencies. - Increase both the Plastic frequency and the overall total by the added observations. - Form a new relative frequency rather than adjusting the old percentage directly.

Solution

1. The original total is \(40+25+15+40=120\). 2. The new Plastic count is \(40+6=46\), and the new total is \(126\). 3. The new relative frequency is \(\frac{46}{126}=\frac{23}{63}\approx 0.365\).

Answer

The new relative frequency is \(\frac{23}{63}\approx 0.365\), or about \(36.5\%\).
Graphs for one categorical variable 45
Charts a) and b) show the same distribution in different forms. Explain how the bars correspond and determine the sample size.
Figure for problem 539674

Hints

- Use one corresponding pair of bars to recover the common total. - For any matching category, frequency divided by relative frequency gives the same total. - Verify the recovered total with the remaining bar pairs.

Solution

1. Each relative frequency in b) equals the corresponding frequency in a) divided by the same total. 2. For category \(X\), \(50\div 0.50=100\). 3. The sample size is \(100\), and the other pairs also satisfy \(\frac{30}{100}=0.30\) and \(\frac{20}{100}=0.20\).

Answer

Each bar in b) is the corresponding bar in a) divided by \(100\): \(50\leftrightarrow0.50\), \(30\leftrightarrow0.30\), and \(20\leftrightarrow0.20\). The sample size is \(100\).
Graphs for one categorical variable 46
Compare the distributions in a) and b). Which category has the same relative frequency, and how does the Yes–No gap differ?
Figure for problem 539678

Hints

- Compare corresponding category heights on the common scale to find any exact match. - Compute the Yes–No gap separately in each panel. - Compare those two gaps and state which panel has the larger separation.

Solution

1. Unsure is \(0.15\) in both charts. 2. In a), the Yes–No gap is \(0.55-0.30=0.25\). 3. In b), the gap is \(0.50-0.35=0.15\). 4. The gap is larger in a) by \(0.10\).

Answer

Unsure is the same at \(0.15\). The Yes–No gap is \(0.25\) in a) and \(0.15\) in b), so it is \(0.10\) larger in a).
Graphs for one categorical variable 47
The bar chart compares approval rates for Program A and Program B. Its vertical axis begins at \(40\%\) rather than \(0\%\). Explain why the graph can give a misleading visual impression, and state one change that would make the comparison more appropriate.
Figure for problem 539799

Hints

- Read the numerical values before judging the visual size of their difference. - Compare the value-axis baseline with zero. - Ask whether the bar lengths are proportional to the full magnitudes being compared.

Solution

1. The chart shows approval rates of \(48\%\) and \(52\%\), a difference of only \(4\) percentage points. 2. Because the vertical axis shows only the range from \(40\%\) to \(56\%\), the small difference occupies a large fraction of the plotted height and appears exaggerated. 3. A bar chart should begin its value axis at \(0\%\). Alternatively, a point plot could use a clearly labeled nonzero scale without encoding the values as bar lengths.

Answer

The truncated value axis exaggerates the \(4\)-percentage-point difference. Begin the bar-chart axis at \(0\%\), or use a non-bar display with a clearly labeled scale.
Graphs for one categorical variable 48
Panel a) comes from a survey of \(80\) people. Compare the support rates in the two panels. Which group has more support, by how many percentage points, and why are the raw bar heights not directly comparable?
Figure for problem 548670

Hints

- Read the unit on each vertical axis. - Convert a count to a percentage using the first panel's sample size. - Compare only after putting the panels on the same scale.

Solution

1. In panel a), the support count is \(48\) of \(80\), giving \(\frac{48}{80}=60\%\). 2. Panel b) directly shows \(55\%\) support. 3. Panel a) is greater by \(5\) percentage points. 4. The panels use different vertical units: frequency in a) and percent in b).

Answer

Panel a): \(60\%\). Panel b): \(55\%\). Panel a) is higher by \(5\) percentage points. The original bar heights use different units.
Graphs for one categorical variable 49
The nominal categories Apple, Banana, Cherry, and Grape are listed in alphabetical order. Tashi computes cumulative percentages in that order and interprets the cumulative value through Cherry as an intrinsic property of the Cherry category. Explain why cumulative percentages are not meaningful for this nominal variable and state what graph should be used instead.

Hints

- Ask whether the categories have a meaningful least-to-greatest order. - Imagine rearranging the category labels and track what happens to the cumulative values. - Choose a graph whose values do not depend on an arbitrary ordering.

Solution

1. A cumulative value through Cherry includes Apple, Banana, and Cherry only because of the chosen alphabetical order. 2. Nominal categories have no natural order, so rearranging the labels would change every cumulative value without changing the data. 3. An ordinary frequency or relative-frequency bar chart should display each category separately.

Answer

Cumulative percentages are inappropriate because their values depend on an arbitrary alphabetical order. Use an ordinary bar chart of the individual category frequencies or relative frequencies.
Graphs for one categorical variable 50
Two categories have the same frequency, but the chart uses bars of different widths. Explain why the display is misleading. Compare the rectangular areas of the bars using their displayed widths.
Figure for problem 548675

Hints

- In a bar chart, frequency is encoded by height rather than area. - Compare the two heights before reacting to the total shaded region. - Equal-width bars prevent width from introducing a second visual signal.

Solution

1. Both bar heights are \(30\), so the two categories have equal frequency. 2. The displayed widths are \(0.5\) and \(1.2\). Because the heights are equal, the area ratio is \(\frac{1.2}{0.5}=2.4\). 3. The wider bar has \(2.4\) times the visual area even though it represents the same frequency. Bars in one categorical chart should use a common width.

Answer

The categories are equally frequent, but the wider bar has \(2.4\) times the area. Unequal widths falsely make one category look more common.
Graphs for one categorical variable 51
A survey has \(60\) responses in three categories, with frequencies \(10\), \(10\), and \(40\). A graph labels the corresponding pie sectors \(17\%\), \(17\%\), and \(67\%\), which sum to \(101\%\). Is the graph necessarily wrong? Give the exact sector angles that should be used.
Figure for problem 548677

Hints

- Recompute the category shares from the original frequencies. - Distinguish exact proportions from rounded labels. - Base sector angles on the exact fractions rather than on rounded percentages.

Solution

1. The exact relative frequencies are \(\frac{10}{60}=\frac{1}{6}\), \(\frac{1}{6}\), and \(\frac{40}{60}=\frac{2}{3}\). 2. The displayed percentages are each rounded to the nearest whole percent, so their sum of \(101\%\) can result from rounding. 3. The exact angles are \(\frac{1}{6}\cdot360^\circ=60^\circ\), \(60^\circ\), and \(\frac{2}{3}\cdot360^\circ=240^\circ\).

Answer

The graph is not necessarily wrong; the \(101\%\) total is caused by rounding. The exact sector angles are \(60^\circ\), \(60^\circ\), and \(240^\circ\).
Graphs for one categorical variable 52
A bar chart labels Category A as \(40\%\), calculated from \(72\) selections among \(180\) people who answered a survey item. The survey was sent to \(200\) people, including \(20\) who skipped the item. What bar height should be used if the graph is relabeled “Percent of all invitees”? Explain the denominator change.

Hints

- Identify exactly which people are included in each graph label. - Keep the category count fixed while changing the reference total. - Express the revised ratio as a percentage.

Solution

1. The original bar height is \(\frac{72}{180}=0.40\), using only valid responses. 2. Using all invitees gives \(\frac{72}{200}=0.36\). 3. The bar must change because the numerator stays \(72\) while the denominator increases from \(180\) to \(200\).

Answer

The revised bar height should be \(36\%\). The original \(40\%\) used valid responses as the denominator, while the revised graph uses all \(200\) invitees.
Graphs for one categorical variable 53
The bar chart shows defect types recorded among \(50\) inspected products. A product may have more than one defect. Explain why adding the three bar heights does not give the number of defective products.
Figure for problem 548690

Hints

- Read what one unit in each bar represents. - Ask whether the same product can contribute to more than one bar. - Distinguish category occurrences from distinct observational units.

Solution

1. The bars show \(30\) scratches, \(22\) dents, and \(18\) loose-part occurrences, for a total of \(70\) recorded defect occurrences. 2. Because one product can appear in more than one defect category, the categories are not mutually exclusive. 3. Therefore, \(70\) counts defect occurrences rather than distinct defective products. The number of defective products cannot exceed \(50\) and cannot be recovered without overlap information.

Answer

Adding the bars gives \(70\) defect occurrences, not \(70\) distinct products. Because products can contribute to more than one bar, the number of defective products cannot be determined from this chart alone.
Graphs for one categorical variable 54
A \(100\%\) stacked bar represents four mutually exclusive categories. Reading from left to right, the cumulative segment endpoints are \(0.18\), \(0.47\), \(0.76\), and \(1.00\). Find the relative frequency of each category and explain why the endpoint values are not the individual category shares except for the first segment.

Hints

- Treat each labeled endpoint as a cumulative total from the left edge. - Find a segment’s width by subtracting its two boundary positions. - Check that the four segment widths sum to \(1\).

Solution

1. The first segment has relative frequency \(0.18\). 2. Subtract consecutive endpoints: \(0.47-0.18=0.29\), \(0.76-0.47=0.29\), and \(1.00-0.76=0.24\). 3. The endpoint of a later segment includes all segments before it, so it is cumulative rather than an individual share.

Answer

The category relative frequencies are \(0.18\), \(0.29\), \(0.29\), and \(0.24\).
Graphs for one categorical variable 55
A survey chart puts disagreement to the left of zero and agreement to the right. Ardit claims that the bars on the left show impossible negative counts. Explain what the negative direction means, and find the total number of disagreement responses.
Figure for problem 548694

Hints

- Decide what it means to place a category left of the origin. - Read the length of each bar, ignoring the sign used for direction. - Add the two response counts only after interpreting the bar lengths.

Solution

1. The negative horizontal direction shows disagreement, not a negative number of people. 2. The two disagreement bar lengths represent \(18\) and \(27\) responses. 3. Their total is \(18+27=45\) responses.

Answer

The signs show plotting direction, not negative counts. There are \(45\) disagreement responses.
Graphs for one categorical variable 56
A bar chart uses a logarithmic vertical axis and shows category counts \(10\), \(100\), and \(1000\). The tops of the bars are equally spaced vertically. A reader claims the counts increase by the same amount from one category to the next. Evaluate the claim and explain what equal spacing means on this axis.

Hints

- Compare both the differences and the ratios between consecutive counts. - Identify what a logarithmic scale preserves as equal spacing. - Read the axis values rather than treating bar height as a linear measurement.

Solution

1. The numerical increases are \(100-10=90\) and \(1000-100=900\), so the counts do not increase by equal amounts. 2. On a logarithmic axis, equal vertical spacing represents equal multiplicative factors. 3. Each count is \(10\) times the preceding count, so the chart shows equal ratios rather than equal differences.

Answer

The claim is false. Equal spacing on the logarithmic axis means each count is multiplied by \(10\), not increased by the same amount.

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