L-DS-2: Solutions — Data Visualization

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Completing a Frequency Distribution Table

This stepped generator produces a fresh five-class table in one of four contexts: commute time, quiz-completion time, age, or finish time. The numbers change, but the three checked moves are always:

1. Relative frequency: for the named class.
2. Cumulative frequency: add from the first class through the named class.
3. Tail percentage: add the requested upper classes, or subtract the preceding cumulative relative frequency from 1.

Check that the five frequencies sum to before using any proportion.

Problem 2 — Selecting the Correct Graph Type

• Transportation mode: qualitative nominal → bar chart.
• Body temperature: quantitative continuous → histogram.
• Star rating: qualitative ordinal → bar chart in natural order.
• Monthly rainfall over consecutive months: change over time → time-series plot.
• Number of siblings: a small set of discrete counts → separated bars.

The graph follows the variable type and purpose, not the number of displayed groups.

Problem 3 — Reading and Interpreting a Frequency Table

The generator rotates among five question families:

• Class frequency: read from the requested row.
• Relative frequency: compute and report the requested proportion or percentage.
• Cumulative count: add all rows through the stated upper boundary.
• Combined-class percentage: add the two named frequencies, divide by , and multiply by 100.
• Modal class: select the class with the largest .

For a percentage, do not stop at the combined count; the final step is always .

Problem 4 — Identify the Misleading Feature

The chart uses a truncated y-axis: Flagship is $215M and Express is $196M, but the baseline is $180M.

The actual revenue difference is:

The plotted bar-length ratio is:

So a 9.69% revenue advantage looks more than twice as large. Exact value labels would not fix that length distortion; the comparison needs a zero baseline.

Problem 5 — Reading a Histogram

This generator draws four equal-width touching bars and checks three readings:

1. Class width: subtract consecutive class boundaries.
2. Class frequency: read the height of the named bar.
3. Cumulative percentage below a boundary: add every bar to the left of the boundary, divide by the total, and multiply by 100.

The solo option asks for one of those readings without the forced step sequence. A class frequency is one bar; a cumulative frequency is a running total across several bars.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — Build a Complete Frequency Table

The generator draws 15 integers and uses five inclusive integer classes.

1. Find the minimum and maximum.
2. Compute
3. Start the first class at the minimum and advance each lower bound by .
4. Tally each observation to obtain .
5. Compute , as the running count, and .

Verify , , final , and final .

Problem 2 — Interpret a Histogram

The regenerable item asks how many observations fall below 30 minutes. Add the frequencies of the , , and classes. Do not include the class.

For the fixed Figure 11 display, the approximate frequencies are 5, 12, 22, 15, and 6. It peaks at 6,000–7,999 steps. Bars fall on both sides; the higher-step side retains more workers than the lower-step side.

Problem 3 — Select the Best Graph

The generator draws one of four scenario families:

• one qualitative variable → bar chart;
• one quantitative continuous variable → histogram;
• one value tracked over time → time-series plot;
• two quantitative variables studied as a relationship → scatter plot.

State the variable type and purpose before naming the graph.

Problem 4 — Two Variables, One Graph

Use a scatter plot because hours of sleep and cognitive score are both quantitative and the question asks about their relationship. Put sleep on the x-axis as the explanatory variable and score on the y-axis as the response. An upward point cloud would indicate that more sleep tends to accompany higher scores.

Problem 5 — Critique a Misleading Graph

The generator rotates among a truncated baseline, reordered axis, and area-distorting pictogram. For each, name the encoding choice and explain how it changes the perceived magnitude or trend.

For Figure 12:

The decline is real but modest. Use a zero baseline when bar length encodes magnitude. A line chart may use a narrower vertical range when the scale is explicit and the goal is to inspect change; label the values and report “1.6% decline” rather than “plummeted.”

Mixed Review 1 — Variable Classification and Graph Selection

The FindTheError generator varies among three named errors:

• treating category codes or ranks as quantitative measurements;
• choosing a touching-bar histogram for categories;
• discarding the natural order of ordinal categories.

A correct analysis classifies the variable as qualitative, uses separated bars, and preserves the order when the categories are ordinal.

Mixed Review 2 — Sampling Method and Bias

The homepage survey is voluntary response sampling. If frequent food-bank users are more likely to respond than other students, the estimate is pushed upward in expectation. A particular sample can still differ by chance, and undercoverage or reluctance to disclose food insecurity can push in the other direction, so the net direction is unknown without evidence about the competing mechanisms.

A better plan is to draw a simple random sample from the registrar list and invite selected students privately. Random selection addresses self-selection at the invitation stage, but nonresponse can still bias the result. Confidential web, phone, or paper response options plus follow-up with nonrespondents reduce disclosure and access barriers.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Question 1 — Feynman Test

A complete explanation connects the visual gap to data structure: histogram bars touch because their classes are adjacent intervals on a quantitative number line. Category names have no meaningful values between them, so separated bar-chart bars communicate that the groups are distinct.

Question 2 — Design a Clinic Dashboard

• Waiting time vs. satisfaction: scatter plot, waiting time on x and satisfaction on y. A downward cloud would support the claim that longer waits accompany lower satisfaction.
• Waiting time by week: summarize waiting time consistently for each week, then use a time-series plot with week on x and the summary on y. A sustained downward trend indicates improvement.

Audit axis ranges, missing weeks, date-window selection, and outliers before trusting either visual summary.

Question 3 — Error Analysis

The class widths are unequal, so frequency density rather than raw frequency must determine height:

The class contains more plants: 11 rather than 9. But the bar should be taller because . With unequal widths, bar area represents frequency while height represents frequency density; the modal class by count and the tallest bar need not be the same.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

🔬 Path A — The Analyst

The proposed 2–4, 5–7, 8–10, 11–14 classes are unequal. With inclusive integer data:

Use 3–5, 6–8, 9–11, and 12–14.

Class
3–5	6	0.24	6	0.24
6–8	9	0.36	15	0.60
9–11	7	0.28	22	0.88
12–14	3	0.12	25	1.00
Total	25	1.00	—	—

The histogram peaks at 6–8 books. Fewer than 9 books includes the first two classes:

The board can describe moderate engagement and investigate support for the lower group, but these data cannot show the program caused the pattern because there is no baseline or comparison group.

🏗️ Path B — The Architect

Revenue: a nonzero baseline exaggerates bar length, and the wider Q4 icon exaggerates area.

Use equal-width bars from zero with value labels. Q4 is 42.5% above Q1, but Q2 and Q3 are not shown, so these data cannot identify the year’s strongest quarter.

Satisfaction: raw-frequency height is invalid with unequal class widths.

Scores from 80 through 100 account for:

That is not a majority, so “most customers” is unsupported. Use frequency density with the existing classes, or return to raw scores before rebuilding equal-width classes.

Market share: the tilted perspective adds apparent volume, while the exploded Electronics slice gives its 18% share disproportionate emphasis over nearby values. Replace the pie with a flat horizontal bar chart ordered by share, starting at zero and labelling all seven bars directly.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Challenge 1 — The Ogive

The corrected frequency table has cumulative proportions 0.15, 0.45, 0.75, 0.95, and 1.00. The 50th percentile therefore lies between and :

So the grouped ogive estimates the median at about 4.83 defects. The exact median of the original, ungrouped observations is 5 defects.

Challenge 2 — Does Bin Width Matter?

The fine-bin frequencies are 5, 7, 14, 8, 4, and 2. Exact adjacent-pair aggregation gives:

The total remains 40. Wider bins preserve total frequency but hide within-pair structure. The regenerable rematch uses the same rule with fresh frequencies: add adjacent pairs and do not shift the original boundaries.

Using 18 one-year classes would spread 40 observations across 18 bins, an average of about 2.2 observations per class. The final three-year class contains only two observations, so at least one of its three one-year classes must be empty. The display therefore risks looking sparse and jagged, but the grouped counts do not determine the exact one-year heights or overall shape; those require the raw ages.

Challenge 3 — The Double Y-Axis Debate

Two independently scaled axes can be tuned to create or erase apparent alignment. A dual-axis display is defensible only when both units and scales are explicit and the layout does not imply a direct proportional comparison.

The statistical error is treating association as proof of causation. Hot weather is one plausible common cause: it could increase both ice-cream sales and swimming exposure. The graph alone does not establish that explanation, but the possibility of a confounder makes the causal claim unjustified.