L-DS-2: Data Visualization

Module 1 · Descriptive Statistics

Section 1: Introduction

▾

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.

In May 2020, the Georgia Department of Public Health posted a bar chart showing COVID-19 cases in the most affected counties over two weeks. The dates were not in chronological order, and county positions changed from one date to the next. That layout created the visual impression of a steady decline even though the sequence did not support one. Georgia Public Broadcasting documented the error, and the department later announced clearer charts, labels, and supporting documentation.

Same data. Two very different stories.

This is what visualization literacy is really about: not just making a graph, but understanding what a graph communicates — and what it can hide. Every design choice (what goes on each axis, where the axis starts, how wide the bars are, whether to use a pie chart or a histogram) changes the story the viewer perceives. Make the wrong choice by accident and you mislead your audience. Make it deliberately and you deceive them.

By the end of this lesson you’ll have the tools to both create honest, informative graphs and read suspicious ones critically.

In L-DS-1, you learned to classify variables and summarize their frequencies. Here you will use those same variable types and counts to decide which graph fits, construct it accurately, and judge whether its visual story is honest.

After this lesson, you will be able to:

Build a frequency distribution table showing absolute, relative, and cumulative frequencies
Select the correct graph type for a given variable — histogram for quantitative data, bar chart for qualitative data
Construct and interpret histograms, bar charts, stem-and-leaf plots, scatter plots, and time-series graphs
Identify the techniques used to make graphs misleading — truncated axes, inconsistent scales, distorted areas — and explain why they deceive

These skills matter well beyond this course. Every field — medicine, economics, journalism, engineering, social science — presents data graphically. Being able to read and evaluate those graphs is a form of literacy as important as reading text.

Section 2: Prerequisites

▾

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.

Building accurate visual representations requires a clear understanding of the data types and frequency summaries you identified in L-DS-1.

From L-DS-1 — qualitative vs. quantitative: You can distinguish category labels from numerical measurements. That distinction determines whether today’s display needs separated category bars or a continuous number line.
From L-DS-1 — discrete vs. continuous: You can distinguish countable values from measurements that can take any value in an interval. That distinction helps you interpret what touching bars mean.
From L-DS-1 — frequency and relative frequency: You can read a raw count and convert it to a share of the whole using . Those two summaries become possible vertical scales today.
From L-DS-1 — ordered data: You can recognize when values have a meaningful order. Today you will preserve that order when a graph encodes time, ranks, or numerical intervals.

Retrieval Checkpoint

A dataset contains the heights (in cm) of 50 students. You want to visualize the distribution of these heights using a chart.

Which category does this height data belong to?

I can tell a category label from a numerical measurement. I can explain why a recorded rounded value may still come from a continuous variable. I can compute a relative frequency as count divided by total.

Bridge to today’s decisions: L-DS-1 classification tells you what kind of variable you have; it does not automatically choose or validate the finished graph. In L-DS-2 you will decide how to encode that variable, choose class intervals when needed, and check whether axes, widths, and ordering tell the truth.

Section 3: Core Concepts

▾

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.

Your C1–C6 workflow:

C1 — Organize: turn observations into frequencies and running totals.
C2–C4 — Encode: match quantitative distributions, categories, relationships, and time to the right visual form.
C5 — Decide: use variable type and purpose to select the graph.
C6 — Audit: inspect axes, widths, areas, ordering, and context for distortion.

C1 — Frequency Distributions

Before drawing any graph, we organize raw data into a frequency distribution — a table that tallies how often each value (or range of values) appears in the dataset. It’s the scaffolding behind almost every graph you’ll encounter.

Frequency Distribution

A frequency distribution is a table that summarizes data by recording:

Class (or value): the category or interval
Absolute frequency : the raw count of observations in that class
Relative frequency : the proportion of the total in that class, computed as

Cumulative frequency : the running total of absolute frequencies up to and including that class
Cumulative relative frequency : the running total of relative frequencies — equivalently, the proportion of observations at or below that class

Here’s what a frequency distribution looks like for a dataset of 20 exam scores:

Class (Score)	Absolute Freq.	Relative Freq.	Cumulative Freq.	Cumul. Rel. Freq.
	2	0.10	2	0.10
	5	0.25	7	0.35
	8	0.40	15	0.75
	4	0.20	19	0.95
	1	0.05	20	1.00
Total	20	1.00	—	—

Read it like this: 8 students scored in the range (absolute frequency). That’s 40% of the class (relative frequency). And 75% of students scored below 80 (cumulative relative frequency at the class).

Cumulative relative frequency ≠ relative frequency. The relative frequency for the class is 0.40 (40% scored in that range). The cumulative relative frequency is 0.75 (75% scored below 80 — including everyone in earlier classes too). These are completely different quantities. The cumulative column always ends at 1.00 (100%).

Figure 1: What “cumulative” means. Left: the five classes as separate bars — ordinary frequency. Right: the same classes stacked into one column (colours match). Drag the slider to add classes one at a time and watch the running total climb: the left axis reads it as a count () and the right axis as a proportion (). The stack always tops out at , — the cumulative scale ends at 100%.

C2 — Histograms (for Quantitative Data)

A histogram turns a frequency distribution into a picture. Each class becomes a bar. With equal-width classes, bar height may represent frequency or relative frequency. When class widths differ, height must represent frequency density or relative-frequency density so that bar area remains proportional to frequency. One feature distinguishes histograms from other bar-based graphs: the bars touch.

Histogram

A histogram is a graph of a frequency distribution for quantitative (numerical) data. Each bar represents one class interval:

Horizontal axis (x): the numerical values — class boundaries are marked at the edges of each bar
Vertical axis (y): frequency or relative frequency when class widths are equal; frequency density or relative-frequency density when widths differ
Bars touch — because the x-axis is a continuous numerical scale; there are no gaps between classes
Each bar’s area is proportional to its frequency; when class widths are equal, height alone is also proportional to frequency

Here is the histogram for the exam score data above. Notice how the bars share their edges — the right edge of the bar is the same as the left edge of the bar.

Figure 2: Histogram of 20 exam scores (class width = 10). Bars touch — the x-axis is continuous.

Histogram shapes tell a story. Where does the data cluster? Are there a few unusually high or low values that extend one side? Practice describing what you see: “most values fall between X and Y, with a few as high as Z.” You will give these patterns formal names and measures in L-DS-5 (Position and Distribution Shape).

Histogram intervals are not discrete categories. The bar represents every score from 70 up to (but not including) 80. There’s no “70 category” and “79 category” — the x-axis is a continuous number line divided into intervals. Reading the bars as isolated discrete bins is a misinterpretation; the bars represent ranges on a continuum.

Figure 3: Type your own numbers and drag the Number of classes slider. Watch the frequency table (, , , ) and the histogram rebuild together — every bar height is a row in the table. Notice how the same data can look smooth with few classes and jagged with many: that is the bin-width effect you’ll revisit in the Challenge Problems.

C3 — Bar Charts (for Qualitative Data)

When the data is categorical rather than numerical, we use a bar chart. Bar charts look similar to histograms at a glance, but there’s one crucial difference that isn’t merely cosmetic — it reflects the nature of the data itself.

Bar Chart

A bar chart displays the frequency or relative frequency of each category in a qualitative variable. Key features:

Horizontal axis (x): category labels
Vertical axis (y): frequency or relative frequency
Bars do NOT touch — there is a gap between bars because categories are distinct (there is no value “between” Bakery and Dairy)
Bar order is arbitrary for nominal data; use the natural order for ordinal data
A bar chart sorted in descending order of frequency — most common category first — is called a Pareto chart, widely used in quality control to quickly identify the largest sources of defects

Figure 4: Bar chart of favourite sports (qualitative nominal data). Bars have gaps — the categories are distinct.

The Key Difference — One Rule to Remember

Histogram bars touch. Bar chart bars don’t.

This isn’t an aesthetic choice. In a histogram, the x-axis is a continuous number line — scores of 69.9 and 70.0 are adjacent, so the bars share an edge. In a bar chart, the x-axis shows distinct categories — “Soccer” and “Basketball” are not adjacent on any meaningful scale, so there’s a gap to reflect that separation.

Use a histogram for quantitative data. Use a bar chart for qualitative data. Using a histogram for categorical data is always wrong.

Counting categories does not make the variable quantitative. A table may report 12 soccer fans and 9 basketball fans, but the variable is still “favourite sport” — a set of names. The counts determine bar heights; they do not turn the category axis into a number line.

C4 — Other Graph Types

Histograms and bar charts cover most of what you’ll build in this course. But four other graph types appear regularly in statistics — each exists because a particular question about data cannot be answered with a bar. Here’s what each one is for.

Pie Chart

A pie chart divides a circle into sectors, where each sector’s angle is proportional to its relative frequency (). Best used when:

Data is qualitative (usually nominal)
There are only a few categories; five is a useful readability guideline, not a mathematical cutoff
Part-to-whole relationships are the main message

Limitation: Humans are poor at estimating angles. When categories are similar in size, a bar chart is usually clearer. Many statisticians prefer bar charts to pie charts almost universally.

Pie chart

Bar chart — same data

Figure 5: Same data, two graph types. The pie chart encodes frequency as sector angle — difficult to compare when sectors are similar in size (try judging “Basketball vs. Swimming” precisely). The bar chart encodes frequency as bar height — immediately comparable. For most purposes, prefer the bar chart.

Stem-and-Leaf Plot

A stem-and-leaf plot displays quantitative data by splitting each value into a stem (leading digit(s)) and a leaf (trailing digit). It looks like a sideways histogram but preserves the individual data values:

Stem | Leaf
  5  | 3 7
  6  | 1 4 4 8 9
  7  | 0 2 3 5 5 7 8
  8  | 1 6 9
  9  | 4

Best for small datasets (n ≤ 50) where you want to see the distribution and keep the raw data visible. Each row is a class; leaves within a row are individual observations.

Scatter Plot

A scatter plot displays pairs of quantitative measurements as points on a two-dimensional graph. Each point represents one observation; its x-coordinate is one variable and its y-coordinate is another.

Use a scatter plot when you want to explore the relationship between two quantitative variables — does one tend to increase as the other does? Is there a linear pattern? Are there outliers? You’ll use scatter plots extensively in L-REG-1 (Correlation Analysis).

Figure 6: Scatter plot of study hours versus exam score (n = 15). Each dot is one student. The upward trend — more hours, higher scores — is visible as a whole-cloud pattern, not a rule for any individual point. This is what “relationship between two quantitative variables” looks like. You will quantify this trend precisely in L-REG-1.

Time-Series Plot

A time-series plot (line graph) shows how a quantitative variable changes over time. Time goes on the x-axis; the variable of interest on the y-axis. Points are connected by lines to emphasize the change from one time period to the next.

Use a time-series plot when the x-variable specifically represents time (days, months, years) and the trend over time is the story you want to tell.

Figure 7: Time-series plot of monthly average temperature over one year. The connected line emphasizes the trend — rise through summer, fall through winter. Points are connected because the x-axis is time: consecutive months are adjacent, so tracking the change from one to the next is meaningful. Compare this to a bar chart of the same data: a bar chart would show the same heights but lose the sense of continuous progression.

A time-series plot is not just any connected scatter plot. Connect points only when the horizontal axis is an ordered time sequence and the changes between adjacent times matter. Connecting unrelated paired measurements suggests a path that the data do not contain.

C5 — Choosing the Right Graph

Every graph-choice decision starts with the same question: What type of variable is this? Use the decision chain below.

Graph Selection Decision Chain

One variable?
- Qualitative: bar chart (the default); a pie chart can work for a small number of categories when part-to-whole is the focus
- Quantitative: histogram for continuous or many-valued discrete data; separated frequency bars for a few discrete count values; stem-and-leaf for a small dataset when the raw values matter
Two variables?
- Both quantitative: scatter plot; if one is time, use a time-series plot
- One qualitative, one quantitative: compare the quantitative distributions across categories with side-by-side boxplots or faceted/grouped histograms (covered in later courses). Use bars only for a clearly defined summary, such as a mean or proportion, and show its uncertainty when appropriate.

Using a histogram for qualitative data is always wrong. A histogram’s x-axis is a continuous number line — it implies that values between bars are possible. For qualitative data (e.g., “Bakery,” “Dairy”), no such “between” exists. Use a bar chart. Similarly, using a bar chart for quantitative data with many distinct values (like a weight distribution) will produce a cluttered mess where a histogram would be clean and clear.

C6 — Graph Misrepresentation

Not all misleading graphs are accidents. Knowing the common techniques helps you spot them — and avoid creating them yourself.

Truncated Y-Axis

A truncated y-axis starts above zero, making small differences look large. The bars or line still reflect the true data values, but the visual impression is distorted.

When it’s a problem: bar charts and histograms should (almost) always start at 0. A bar that represents 97% vs. 94% disappears entirely when the y-axis starts at 0 — fine. But when the axis starts at 93%, that 3-point difference looks enormous.

When it’s acceptable: time-series plots often legitimately start above zero (e.g., tracking temperature variations around 15°C — starting at 0 would waste most of the chart). The key is whether the starting point is disclosed and whether the visual impression matches the magnitude of the difference.

A nonzero axis is not automatically dishonest. The question is whether the encoding and labels let the viewer judge the magnitude accurately. Bar lengths normally need a zero baseline because length itself represents value; a clearly labelled time-series axis may zoom in when the purpose is to inspect small changes.

Here is the same fictional data — approval ratings over four quarters — displayed two ways. The only difference between the two graphs is where the y-axis starts.

✓ Honest: y-axis starts at 0

The change looks small — as it should.

✗ Misleading: y-axis starts at 93

Looks like a collapse — same 3-point drop.

Figure 8: Same data, two y-axes. The left graph starts at 0%; the right graph starts at 93%. The 3-percentage-point decline from Q1 to Q4 looks negligible on the left and catastrophic on the right. This is the truncated y-axis trick.

Figure 9: Drag the slider yourself. The data never changes: approval slips from 97% to 94%, a 3-point decline. As you raise where the y-axis starts, watch the red “visual drop” readout balloon from a few percent to “looks 75% shorter” — while the green “actual change” stays fixed at 3 points. The gap between those two numbers is the deception.

Other Common Misrepresentation Techniques

Inconsistent class widths in histograms: If classes are different widths, the bar height alone is misleading — a wider class captures more observations. The correct display uses frequency density (frequency ÷ class width) on the y-axis, ensuring area rather than height encodes frequency.
3D effects and distorted areas: 3D pie charts tilt and expand the slice nearest the viewer, making it look larger than its true proportion. Pictograms (where a doubled icon also doubles in width, quadrupling area) distort relative sizes.
Selective date ranges in time-series: Choosing a start date that captures only the upward part of a trend creates the impression of consistent growth when the full history shows volatility.
Omitting the sample size n: “67% of customers prefer our brand!” — based on a survey of 3 customers — may be numerically correct, but it is extremely imprecise and cannot support a stable generalization without context.

Quick checklist for evaluating any graph you encounter:

Does the y-axis start at 0? If not, why not — and does the distortion matter?
Are all bars/slices drawn to a consistent scale?
Is the graph type appropriate for the variable type?
What is n? Is the sample large enough to support the claim?
Is the time range shown cherry-picked?

Concept Check

Before the worked examples, confirm the two distinctions students most often blur.

A researcher has recorded the eye colour (brown, blue, green, hazel) of 300 students and wants to display how the four colours are distributed.

Which graph is appropriate for the eye-colour data, and why?

In a frequency table, the class minutes has relative frequency and cumulative relative frequency .

What does the cumulative relative frequency 0.66 tell you?

Section 4: Worked Examples

▾

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.

Let’s walk through four examples — each exercises a different core concept, and the scaffolding gradually fades so you’re doing more of the thinking by Example 4.

Example 1 — Building a Frequency Distribution Table (C1)

Scenario: A quality-control inspector records the number of defective items found in each of 20 production batches:

3, 7, 2, 5, 8, 4, 6, 3, 5, 9, 1, 4, 6, 7, 5, 2, 8, 3, 6, 4

Organize this into a frequency distribution with 5 classes of width 2 (classes: 1–2, 3–4, 5–6, 7–8, 9–10). Compute the absolute frequency, relative frequency, cumulative frequency, and cumulative relative frequency.

Step 1: Tally each class.

Go through the data values one by one and mark which class each belongs to:

1–2: values 3, 7, 2, 5, 8, 4, 6, 3, 5, 9, 1, 4, 6, 7, 5, 2, 8, 3, 6, 4 → count = 3
3–4: 3, 7, 2, 5, 8, 4, 6, 3, 5, 9, 1, 4, 6, 7, 5, 2, 8, 3, 6, 4 → count = 6
5–6: 3, 7, 2, 5, 8, 4, 6, 3, 5, 9, 1, 4, 6, 7, 5, 2, 8, 3, 6, 4 → count = 6
7–8: 3, 7, 2, 5, 8, 4, 6, 3, 5, 9, 1, 4, 6, 7, 5, 2, 8, 3, 6, 4 → count = 4
9–10: 3, 7, 2, 5, 8, 4, 6, 3, 5, 9, 1, 4, 6, 7, 5, 2, 8, 3, 6, 4 → count = 1

Step 2: Compute relative frequency. Divide each count by .

Step 3: Compute cumulative frequencies. Running totals from top to bottom.

Result:

Class
1–2	3	0.15	3	0.15
3–4	6	0.30	9	0.45
5–6	6	0.30	15	0.75
7–8	4	0.20	19	0.95
9–10	1	0.05	20	1.00
Total	20	1.00	—	—

Interpretation: 75% of batches had 6 or fewer defects ( at the 5–6 class). Only 1 batch had 9–10 defects (5% of all batches).

Sanity checks: the column must sum to . The column must sum to 1.00. The final row of must equal . The final row of must equal 1.00. Run these checks every time.

Example 2 — Constructing a Histogram (C2)

Scenario: Using the frequency table from Example 1, sketch the histogram. Before looking at the worked steps, predict which class or classes will form the highest point. Then write the bar heights as you move outward from that high region — first to the left, then to the right.

Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.

Show Solution

Step 1: Draw and label the axes.

Horizontal axis: because the recorded defect counts are whole numbers, use half-unit class boundaries at 0.5, 2.5, 4.5, 6.5, 8.5, and 10.5. These boundaries turn the inclusive classes 1–2, 3–4, and so on into adjacent intervals with no gaps or overlap.
Vertical axis: frequency; mark from 0 to 6 (the maximum frequency)

Step 2: Draw bars for each class, touching at the class boundaries.

Figure 10: Histogram of defective items per batch (n = 20). The two highest bars form a plateau across the 3–4 and 5–6 classes.

Shape: The two highest bars are the 3–4 and 5–6 classes, both with , so the centre forms a short plateau rather than one unique peak. Moving outward from the plateau, the left side goes from to in the 1–2 class. The right side goes from to and then in the 7–8 and 9–10 classes. In L-DS-5 you will give shapes like this formal names and measures; for now, practice describing the visible bar-height sequence rather than assigning a formal shape name.

Example 3 — Choosing the Right Graph (C3 + C5)

Choose the most appropriate graph for each scenario. Record one graph choice and one variable-type reason for each before revealing the answers. When you compare afterward, check both: a correct graph with the wrong reason is not yet a reliable decision rule.

Scenario A: A researcher records the preferred music genre of 200 university students (Pop, Rock, Hip-hop, Classical, Other).

Scenario B: A nurse records the resting heart rate (beats per minute) of 50 patients.

Scenario C: An economist tracks the unemployment rate each month for 5 years.

Show answer for Scenario A

Answer: Bar chart. Music genre is a qualitative nominal variable — categories with no natural order. A bar chart with one bar per genre and frequency on the y-axis is correct. A pie chart would also be acceptable here (5 categories fits the pie chart guideline), but a bar chart makes relative sizes easier to compare. Not a histogram — genre is not a number; there is no meaningful “between Pop and Rock.”

Show answer for Scenario B

Answer: Histogram (or stem-and-leaf). Heart rate is quantitative continuous. A histogram groups the values into class intervals (e.g., , bpm) and shows the distribution shape. For a small dataset (n = 50), a stem-and-leaf plot would also work and preserves individual values. Not a bar chart — the x-axis is a continuous number line.

Show answer for Scenario C

Answer: Time-series plot (line graph). The x-variable is time (months), and showing the trend over time is the point. Connecting the points with a line emphasizes the change from month to month. A histogram would destroy the temporal ordering of the data — it would show the distribution of rates but not how rates evolved.

Example 4 — Spot the Misleading Graph (C6)

This example tests whether you can recognize the misrepresentation techniques from C6 in a new context — not just describe them, but identify them when they appear. For each graph, name the misleading technique and write one sentence explaining how it changes the viewer’s impression. Then compare both your diagnosis and your explanation with the model answer.

Graph A — Monthly sales

Graph B — “Stock price surges”

Two published-looking displays. Diagnose what each visual encoding makes the viewer perceive before opening either critique.

Identify the misrepresentation in Graph A

Misrepresentation: Distorted area in a pictogram. Store B has twice the sales of Store A, so the icon is drawn twice as tall — but it was also made twice as wide! Area = height × width. A 2× taller and 2× wider icon has 4× the area of Store A’s icon. Store B therefore looks 4× as large even though its sales are 2× as large; the area encoding doubles the intended 2:1 ratio.

Fix: Use bars of equal width or use a simple bar chart without pictogram icons.

Identify the misrepresentation in Graph B

Misrepresentation: Truncated y-axis creating visual exaggeration. Starting the y-axis at $82 rather than $0 makes a 7.2% rise look like a near-vertical climb. On an axis from $0 to $90, the same data would appear as a very shallow upward slope.

Note: For a time-series plot of a stock, starting at $0 is sometimes impractical (the line would be nearly flat in an invisible region). The honest fix is to clearly label the axis start, report the 7.2% change directly, and avoid claiming the visual slope itself represents the magnitude of the change. Whether “surges” is justified requires a stated benchmark; the graph alone cannot establish that description.

Section 5: Guided Practice

▾

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.

Time to try it yourself — with support. Each problem below gives you immediate feedback. If you get something wrong, read the rationale to understand why before moving on.

Problem 1 — Completing a Frequency Distribution Table (C1)

Work through one frequency table at a time — relative frequency, then cumulative frequency, then a tail percentage. Each step is checked before the next unlocks, so you can see exactly which move (if any) trips you up. Click “Try a similar problem” for a fresh context.

Problem 2 — Selecting the Correct Graph Type (C3 + C5)

Apply the decision chain: identify the variable type, then select the graph. Click “Try a similar problem” to practice with a different scenario.

A city planner surveys 300 residents about their primary mode of transportation to work (Car, Bus, Bike, Walk, Work from home).

What is the most appropriate graph for displaying these results?

A nurse records the body temperature (°C) of each of 80 patients at admission.

What is the most appropriate graph for displaying the distribution of temperatures?

An online review platform collects star ratings (1 star, 2 stars, 3 stars, 4 stars, 5 stars) from 500 diners for a restaurant.

What is the most appropriate graph for displaying the distribution of ratings?

A meteorologist records the total monthly rainfall (mm) in Montréal over 36 consecutive months.

What is the most appropriate graph for showing how rainfall changed over time?

A sociologist records the number of siblings each student in a class has (0, 1, 2, 3, 4, or 5+).

What is the most appropriate graph for displaying the sibling-count distribution?

Problem 3 — Reading and Interpreting a Frequency Table (C1)

A fresh frequency table appears each time. Read the value it asks for — sometimes a count, sometimes a proportion or percentage, sometimes the modal class — and type your answer (or select it, for the modal-class question). A wrong entry that matches a common slip returns the reason it is wrong. Click “Try a similar problem” for a new table.

Problem 4 — Identify the Misleading Feature (C6)

A supermarket chain releases a bar chart comparing the annual revenue of its three store formats. Inspect the chart itself before answering.

Annual revenue by store format. Answer the question using the published view, then use the comparison button to see the same values on a zero-baseline scale.

What is the primary misleading feature of this chart?

Show full explanation

The actual difference between Flagship ($215M) and Express ($196M) is $19M — only about 9.69% of the Express revenue. On a y-axis running from $0 to $220M, the Flagship bar would be 97.7% of the chart height, the Express bar 89.1% — visually almost identical.

By starting at $180M, the chart compresses the data range to $40M. Now Express ($196M) appears at of the axis height, while Flagship ($215M) appears at . The plotted bar lengths have ratio , so Flagship looks more than twice as large even though its revenue is only 9.69% higher.

Problem 5 — Reading a Histogram (C2)

By now the steps should feel familiar, so this one lets you choose your support. Answer the whole problem at once if you can; break it into steps only if you want them. A new histogram is drawn each time.

Before moving to Independent Practice, check your confidence on these key skills:

I can compute relative and cumulative frequencies from a table
I can identify the correct graph type for a given variable
I can recognize a truncated y-axis as a form of misrepresentation

Section 6: Independent Practice

▾

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.

No hints here — these problems are yours to work through. Use scratch paper as needed. Show the solution when you’re ready to check your work.

Interleaving tip: These problems mix concepts intentionally. Don’t expect every problem to test the same skill as the one before — that’s by design. Research shows interleaved practice builds stronger long-term memory than blocked practice. If you get stuck, the concept tag in each problem header — e.g., (C2) — tells you exactly which subsection of Section 3 to revisit.

Problem 1 — Build a Complete Frequency Table (C1)

A new dataset is generated each time you click “Generate new problem.” The absolute frequencies are supplied. Complete the three missing columns — relative frequency, cumulative frequency, and cumulative relative frequency — using 5 equal-width classes.

Problem 2 — Interpreting a Histogram (C2)

The histogram below shows the distribution of daily steps walked by 60 office workers over one month.

Figure 11: Histogram of daily step counts for 60 office workers.

Study the histogram for ten seconds, then make two predictions:

Which class has the highest frequency?
Is most of the data concentrated toward lower step counts, higher step counts, or roughly in the middle?

Reading values off a histogram is a cold skill here — no worked support. The problem below draws a fresh commute-time histogram and asks for the cumulative count below 30 minutes. Type your answer; a common misreading returns the reason it is wrong.

Then, in your own words: describe the shape of the Figure 11 distribution above — where does it peak, and how do the bars change as step count increases or decreases from the peak?

Show model answer — shape of Figure 11

The distribution peaks at the 6,000–7,999 class (the tallest bar, f ≈ 22). From the peak, bars decrease on both sides: the 8,000–9,999 class (f ≈ 15) and 10,000–11,999 class (f ≈ 6) step down to the right, while the 4,000–5,999 class (f ≈ 12) and 2,000–3,999 class (f ≈ 5) step down to the left. The right side (higher step counts) retains more workers than the left side (lower step counts), so the distribution sits somewhat higher in the step-count range — not quite balanced around the centre.

Problem 3 — Select the Best Graph (C5)

Each scenario below describes a dataset — click through several. For each one, commit to the graph you think fits best; a wrong choice explains why the variable type rules it out. Click “Try a similar problem” for a new scenario.

Problem 4 — Two Variables, One Graph (C4 + C5)

A public health researcher collects two measurements for each of 85 participants in a study:

Hours of sleep per night (continuous, 4.5 to 9.5 hours)
Score on a cognitive performance test (continuous, 0–100)

She asks: “Does more sleep correlate with better cognitive performance?”

Choose the display, assign its axes, and identify the pattern that would support the researcher’s question. Solve the whole problem at once or open the step-by-step route when you want it.

Show model explanation

(a) Graph: Scatter plot. Both variables are quantitative continuous. When both variables are quantitative and the goal is to explore the relationship between them, a scatter plot is the correct choice. Each participant is one point on the plot; the x-coordinate is hours of sleep and the y-coordinate is the test score.

(b) Axes: Hours of sleep (the explanatory variable — the factor we think might influence performance) goes on the x-axis. Cognitive performance score (the response variable — the outcome we’re measuring) goes on the y-axis. Convention: the variable you think explains variation in the other goes on x.

(c) Pattern suggesting positive association: If more sleep correlates with better performance, the points should form an upward pattern from left to right — participants with fewer hours of sleep (x small) tend to have lower scores (y small), and those with more sleep (x large) tend to have higher scores (y large). This is called a positive association and usually appears as points trending upward from the lower-left to the upper-right of the scatter plot.

Problem 5 — Critique a Misleading Graph (C6)

A political party releases the following graph to support its campaign message “Crime has plummeted under our leadership.”

Figure 12: “Crime Rate Over Six Years” — published by a political party. Data: Year 1: 958, Year 2: 955, Year 3: 952, Year 4: 950, Year 5: 947, Year 6: 943 incidents per 100,000. Use the button only after you’ve answered the questions below.

(a) Diagnose Figure 12. What feature makes the published graph exaggerate the decline, and how does that feature change the visual impression?

Rematch with a new graph. Each generated problem below shows a different real-world chart. Identify its distortion before reading the feedback.

Now go deeper on Figure 12:

(b) Compute the true percentage decrease from Year 1 (958) to Year 6 (943). Type the percentage without the percent sign.

(c) Describe how the graph should be redrawn to represent the data honestly.

Show Solution — Figure 12

Technique: Truncated y-axis. The y-axis starts at 940 instead of 0 (or at least a much lower value). This compresses the visual scale so that the decline from 958 to 943 — which spans only 15 units — takes up nearly the entire height of the chart. The line appears to drop almost vertically, suggesting a massive collapse in crime.

(b) True percentage decrease:

(c) Honest redraw: A zero-baseline version is useful when the goal is to compare the change with the full magnitude of the rate. A zoomed line chart may also start above zero without a broken-axis symbol, provided the linear scale and its starting value are prominent and the caption reports the 15-point, 1.57% decrease directly. In either version, include the actual values and replace “plummeted” with neutral wording.

Mixed Review — Retrieval from Earlier Lessons

These problems draw on concepts from L-DS-1. Attempting them without re-reading that lesson is the point — retrieval practice strengthens long-term memory more than re-reading.

Review Problem 1 — Variable Classification and Graph Selection

A classmate has picked a graph for a variable and written out their reasoning. One step is where the choice goes wrong. Find it — this is the exact L-DS-1 misconception (using a histogram for a categorical variable) applied to graph selection. A new case appears each time.

Review Problem 2 — Sampling Method and Bias

A university wants to estimate the proportion of its 12,000 students who use the campus food bank. The student affairs office proposes the following approach: place a link to an online survey on the campus homepage for one week and count responses.

Which sampling method is this?

Suppose students who use the food bank respond at a higher rate than students who do not.

What direction does that response mechanism push the estimated proportion?

Then, in writing: name a second source of bias (and its direction), and suggest a better sampling method, explaining why it reduces bias.

Show Solution

Sampling method: Voluntary response sampling (also called self-selection). Only students who notice the link and choose to respond are included — the sample is not randomly selected from the population.

Source of bias 1 — Voluntary response bias: If students who use the food bank respond at a higher rate than non-users, they are overrepresented and that mechanism pushes the estimate upward in expectation. A realized sample can still land above, below, or exactly on the true value by chance.

Source of bias 2 — Undercoverage and nonresponse: Students without reliable internet access or who rarely visit the homepage can be missed, and selected or exposed students may decline to disclose food insecurity. The first mechanisms have an uncertain direction; reluctance among users pushes downward. Because these mechanisms compete, the survey’s net bias direction is unknown without information about response rates in each group.

Better method: Simple random sampling (SRS). Randomly select, say, 600 student ID numbers from the registrar’s complete list and invite each selected student privately. Random selection removes self-selection from the invitation stage and a complete registrar list reduces undercoverage, but neither guarantees that selected students will respond. Offer confidential web, phone, or paper response options and follow up with nonrespondents to reduce access and nonresponse bias.

Section 7: Mastery Check

▾

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.

These questions ask you to recall and apply what you’ve learned independently.

Question 1 — Feynman Test

Imagine a classmate missed the lesson on histograms and bar charts. They’ve been told that “a histogram is just a bar chart without gaps” and don’t understand why that matters.

Explain — in your own words, as if to that classmate — why you cannot use a histogram for categorical data, and what the touching bars actually represent. Don’t just state the rule; explain the reasoning behind it.

Compare with a model answer

A histogram’s x-axis is a continuous number line. The bars touch because the classes are adjacent intervals on that line — there is no gap between and because the first ends exactly where the second begins. Every value on the number line belongs to exactly one bar, and the bars cover the line with no holes.

For categorical data (like colour, gender, or city), there is no number line. “Red,” “Blue,” and “Green” are not adjacent on any scale — there is nothing “between” red and blue. Drawing the bars touching would imply that some value exists between “Red” and “Blue,” which is nonsense. The gap in a bar chart signals: these are distinct categories with no in-between.

So the touching vs. gap distinction isn’t cosmetic — it tells the viewer whether the x-axis is a continuous scale (histogram) or a set of unrelated categories (bar chart).

Question 2 — Design a Clinic Dashboard

A community clinic records three values for each of 120 visits: the appointment date, the waiting time in minutes, and the patient’s satisfaction score from 0 to 100.

The director wants the dashboard to answer two questions:

Do longer waits tend to accompany lower satisfaction?
Are waiting times improving or worsening from week to week?

Without using the graph-selection table, design both displays. Decide the graph types, axes, answering patterns, and audit checks. Work solo first or ask for the step-by-step route.

Show model design

Display 1 — waiting time and satisfaction: Use a scatter plot. Each visit is one point, waiting time goes on the x-axis as the explanatory variable, and satisfaction goes on the y-axis as the response. A downward cloud would show that longer waits tend to accompany lower satisfaction. Audit both axis ranges and look for an outlier or subgroup before summarizing the overall pattern.

Display 2 — waiting time over weeks: First summarize waiting time consistently for each week, then use a time-series plot with week on the x-axis and the chosen weekly summary on the y-axis. A sustained downward trend would indicate improvement; an upward trend would indicate worsening. Audit the date range and make sure missing weeks or a cherry-picked start date do not manufacture the trend.

Question 3 — Error Analysis

A student creates a histogram to display the following frequency table for the height (cm) of 30 plants. Treat the continuous classes as half-open intervals:

Height class (cm)
	4
	9
	11
	6

The student draws four bars — all the same width — with heights proportional to the frequencies 4, 9, 11, and 6. They correctly note that contains the most plants, then infer that it must also be the tallest bar in a valid histogram.

The student’s submitted graph. Inspect both the class labels and the way width and height are encoded.

Identify the error, compute the frequency densities for the and classes, and state both which class contains more plants and which bar should be taller on a correct frequency-density histogram. Work solo or use the step-by-step route; the calculation steps use typed answers.

Show full explanation

The class has a width of 20 cm, while the , , and classes each have a width of 10 cm. The class is twice as wide as the others.

In a histogram, a bar’s area — not its height — represents frequency. When class widths are equal, height and area are proportional, so height works fine. But when class widths differ, you must plot frequency density (= frequency ÷ class width) on the y-axis, not raw frequency:

For the class: frequency density = 11 ÷ 20 = 0.55 per cm. For the class: frequency density = 9 ÷ 10 = 0.90 per cm.

The class still contains more plants: 11 versus 9. But on a frequency-density histogram, the bar would be taller because 0.90 plants per cm exceeds 0.55 plants per cm. The student’s graph therefore confuses greatest count with greatest concentration per unit of the horizontal scale.

Corrected display: class widths match their numerical spans, while bar height is frequency density so each bar’s area represents frequency.

Self-Assessment

How confident are you with the material in this lesson?50% confident

Still confusedReady for the Boss Fight

Section 8: Boss Fight

▾

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.

You’ve reached the Boss Fight — a substantial challenge that asks you to bring everything together. Two paths, equal in difficulty, different in approach. Choose the one that fits how you like to think.

🔬 The Analyst

Turn raw reading-program data into an honest evidence brief and a defensible recommendation.

🏗️ The Architect

Audit three annual-report claims, quantify what the visuals hide, and specify honest redesigns.

🔬 Path A: The Analyst

A school board recorded the number of books read by each of 25 students in a summer program:

8, 3, 12, 5, 9, 7, 11, 4, 6, 10, 8, 14, 3, 7, 9, 6, 13, 5, 8, 11, 7, 4, 10, 6, 9

Task 1 — Decide the structure. Someone proposes classes 2–4, 5–7, 8–10, and 11–14. Decide whether that scheme supports an honest histogram. If not, replace it with four equal-width classes that include every observation and explain your rule.

Task 2 — Build the evidence. Construct , , , and for your classes. Then describe the histogram’s peak and how the bars change on each side. Calculate the percentage of students who read fewer than 9 books.

Task 3 — Advise the board. Write a two- or three-sentence evidence brief: what did a typical level of engagement look like, what group might need attention, and what limitation prevents you from claiming the program caused the reading pattern?

Show complete Path A solution

Task 1: The proposed last class spans four integers while the others span three, so raw bar heights would not be comparable. Using the inclusive-integer width rule, , gives 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	—	—

Task 2: The histogram peaks at 6–8 books (). Frequencies fall to 6 on the lower side and to 7 then 3 on the higher side. Fewer than 9 books includes the first two classes:

Task 3: Most students read in the moderate range, with the modal class at 6–8 books; 60% read fewer than 9, while 3 students reached 12–14. The board could investigate supports for the lower-engagement group, but these observations alone cannot show that the program caused any change because there is no before-program comparison or control group.

Reflection: Which decision carried the most uncertainty: defining classes, interpreting the shape, or limiting the claim? What would you check first on a fresh dataset?

🏗️ Path B: The Architect

You are reviewing three displays from a retailer’s annual report. Inspect the charts before reading the tasks.

Revenue — “A remarkable year”

Satisfaction — “Most are highly satisfied”

Market share — seven-category 3D pie

Three annual-report displays. Treat them as evidence: inspect scale, dimensions, class treatment, perspective, and labelling before deciding what each claim supports.

The published figures are:

Revenue: Q1 = $41.2M and Q4 = $58.7M.
Satisfaction: (), (), (), (), and ().
Market share: Electronics = 18%, Furniture = 17%, Clothing = 16%, Groceries = 15%, Beauty = 13%, Sports = 11%, and Books = 10%.

Task 1 — Audit the claims. For each graph, name the primary visual flaw and explain how it changes what the viewer perceives.

Task 2 — Quantify the truth. Calculate the Q1-to-Q4 revenue increase, compare the frequency densities of satisfaction classes and , and determine what share scored from 80 through 100. Decide whether “most customers” is supported.

Task 3 — Specify the redesign. Give a concrete honest redesign for each graph, including the encoding, scale or class treatment, and labels. End with a two-sentence note to the marketing team that corrects the claims without accusing anyone of intent.

Show complete Path B solution

Revenue: The nonzero baseline exaggerates length, and the wider Q4 icon adds area distortion. The actual increase is

Use equal-width bars from a zero baseline and label each value. Q4 is 42.5% above Q1; because Q2 and Q3 are not supplied, these data do not identify the year’s strongest quarter. The redesigned display no longer implies a several-fold jump.

Satisfaction: With unequal widths, area must represent frequency, so height must be frequency density.

The scores from 80 through 100 contain of 90 responses:

That is the largest combined upper band, but it is not a majority, so “most customers” is unsupported. Use frequency density with the existing unequal classes, or return to the raw scores and rebuild equal-width classes such as . The aggregated table cannot be redistributed into those new classes without the raw scores.

Market share: The tilted 3D perspective adds apparent volume, and pulling the Electronics slice forward gives its 18% share disproportionate visual emphasis over Furniture at 17% and Clothing at 16%. The seven legend-linked sectors also require less accurate angle-and-area comparisons. Replace the pie with a horizontal bar chart ordered by share, use a common zero baseline, and label all seven bars directly.

Suggested note: “The underlying values are worth highlighting, but the current encodings exaggerate some differences and the satisfaction caption overstates the count. The revisions above preserve the results while making the comparisons proportional and auditable.”

Reflection: Which correction changed only presentation, and which changed the verbal conclusion itself? How would you explain that distinction to a nontechnical colleague?

Section 9: Challenge Problems

▾

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.

Optional stretch material. These problems go beyond the lesson objectives. They’re here if you’re curious, ambitious, or just enjoy a harder challenge. None of the material below is required for L-DS-3 — but C1 (the ogive) will reappear in L-DS-5.

Challenge 1 — The Ogive (C1 + extension)

An ogive (pronounced “OH-jive”) is a graph of the cumulative frequency or cumulative relative frequency. Instead of bars, it connects consecutive points with straight line segments — the x-value of each point is the upper class boundary and the y-value is the cumulative frequency up to that boundary.

Use the frequency table from Example 1 (Section 4) to build a cumulative relative frequency ogive:

Class	Upper boundary
1–2	2.5	0.15
3–4	4.5	0.45
5–6	6.5	0.75
7–8	8.5	0.95
9–10	10.5	1.00

The ogive starts at (0.5, 0.00) — the lower boundary of the first class with cumulative frequency 0 — and ends at (10.5, 1.00).

Question: Using the ogive, estimate the value below which approximately 50% of the observations fall (the median). Draw a horizontal line at , find where it intersects the ogive, and read off the x-value. Type the interpolated estimate.

Show Solution

Between the 3–4 class () and the 5–6 class (), the cumulative relative frequency passes through 0.50. Linear interpolation between the two points (4.5, 0.45) and (6.5, 0.75):

The grouped ogive therefore estimates the median at about 4.83 defects. Because grouping hides the positions inside each class, this is an interpolation estimate; the exact median of the original 20 observations is 5 defects. You’ll formalize percentile estimation like this in L-DS-5.

Preview: The ogive is the graphical tool for reading percentiles directly. The 25th percentile (Q1) corresponds to , the 75th (Q3) to , and the 50th (Q2 = median) to . You’ll use this in L-DS-5 when studying position in a distribution.

Challenge 2 — Does Bin Width Matter? (C2)

Here are two histograms of the same dataset: ages at first employment for 40 recent graduates, ranging from 18 to 35 years. The only difference is the number of classes used.

Histogram A: 3 classes (width = 6)

Shape appears: one broad peak at

Histogram B: 6 classes (width = 3)

Shape appears: sharper peak at , then a thinning higher-age tail

Figure 13: The same 40 observations at two bin widths. Histogram A is the exact adjacent-pair aggregation of Histogram B:

, and

Want to feel this for yourself? Scroll up to the Histogram Builder in Section 3 (C2), paste in a dataset, and drag the Number of classes slider from 3 up to 8. Depending on the data, more classes may reveal localized structure, create a more jagged display, or leave the broad pattern largely unchanged. Compare what changes and what remains stable.

Answer the following:

a. Both histograms display the same 40 data values. Why do they appear to tell different stories about the shape of the distribution? b. Which histogram would you trust more, and why? c. Why would using 18 one-year classes risk producing a sparse, noisy-looking histogram? What can — and can’t — you determine without the raw ages?

Show Solution

(a) The bin width controls how much detail is visible. With 3 wide classes, the class lumps together two patterns that Histogram B separates: a high cluster in and a secondary bar in . Wide bins smooth out the distribution and can make an uneven concentration look balanced. Narrow bins reveal more localized structure, but that structure may reflect either a real concentration or random variation.

(b) Neither histogram is automatically more trustworthy. Histogram A is useful for the broad shape; Histogram B is useful when the question requires more local detail. There is a tradeoff: too many bins can leave many bars with only a few observations — and some with none — making random variation (noise) look like a pattern. The choice of bin width requires judgment about the sample size and the question being asked. A common rule of thumb (Sturges’ rule) is , suggesting 6–7 bins as a reasonable starting point for , not proof that one display is the true one.

(c) With 18 classes (each 1 year wide), gives an average of about 2.2 observations per class. The final three-year class in Histogram B contains only two observations, so at least one of its three one-year classes must be empty. More generally, splitting 40 observations across 18 classes risks a sparse, jagged display in which random variation looks important. However, the grouped counts do not determine the exact one-year bar heights or the overall 18-bin shape; that requires the raw ages. Too few bins can hide detail, while too many can make noise look like structure.

Regenerable rematch

Try a fresh pair of fine-bin frequencies. Decide what information survives when adjacent bins are combined and what detail disappears.

Challenge 3 — The Double Y-Axis Debate (C6)

A common graph type in business and journalism is the dual y-axis plot (also called a secondary axis chart). It overlays two different variables on the same graph, with one y-axis on the left and a different y-axis on the right.

Example: A financial news article shows monthly ice cream sales (in thousands of units, left y-axis) and monthly drowning rates per 100,000 (right y-axis) on the same graph. The two lines move almost perfectly together. The article implies this suggests a causal link.

The two data series remain fixed. Move only the right-axis maximum and observe what happens to their apparent alignment.

a. Use the scale explorer to explain why this graph is potentially misleading, independent of the correlation–causation issue. b. Under what conditions is a dual y-axis graph a legitimate and useful tool? c. What is the underlying statistical error in claiming ice cream sales cause drowning rates?

Show Solution

(a) Why the graph can mislead: The scales on the two y-axes are chosen independently by the designer. Changing their ranges can make slopes and vertical movements look unusually similar or very different, even though the underlying data have not changed. Rescaling does not create or erase the actual statistical correlation — positive linear rescaling preserves it — but it can strongly change the viewer’s impression of how closely the magnitudes and slopes match.

(b) When dual y-axes are legitimate: They can be useful when two variables are measured in genuinely different units and both are relevant to the same story (e.g., overlaying temperature in °C and precipitation in mm on a climate plot, where both axes are clearly labelled and the reader understands they cannot be compared directly). The key conditions: clearly labelled axes, no implication of a direct comparison between the two y-scales, and no manipulation of scale to create a false impression of alignment.

(c) Correlation ≠ causation: The graph shows association, not evidence that ice cream sales cause drownings. Summer heat is one plausible confounding variable (sometimes called a lurking variable): it could increase both ice cream consumption and the number of people swimming, creating more opportunities to drown. The graph does not establish that explanation, but it demonstrates why a causal claim requires evidence beyond two lines moving together. You’ll study this formally in L-REG-1 (Correlation Analysis).

Section 10: Solutions Reference

▾

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.

Complete, step-by-step solutions for all problems in Sections 5–9 are available on the solutions page. Solutions include worked arithmetic, common mistakes to watch for, and interpretation guidance.

View Full Solutions →

If you’re stuck: Re-read the relevant Core Concept in Section 3, then find the Worked Example that maps to that concept (e.g., Example 1 maps to Concept 1). The solutions page shows the reasoning behind every step, not just the final answer.

Quick-Reference Formulas

Class Width (for frequency distributions): For inclusive integer classes: where is the number of classes. For continuous measurements, choose convenient boundaries that cover the observed range and state which endpoint convention you use.

Relative Frequency:

Graph Type	Best Used For	Key Feature
Histogram	Quantitative distributions, especially continuous or many-valued discrete data	Bars touch across adjacent intervals
Bar Chart	Qualitative, categorical data	Bars do not touch
Pareto Chart	Categorical, finding largest factor	Bars sorted descending by frequency
Pie Chart	Parts of a whole (relative freq)	Angles proportional to frequency

Common Misleading Features	Why it’s a problem
Y-axis not starting at 0	Exaggerates small differences (mostly an issue for bar charts)
3D effects / perspective	Distorts areas and makes values hard to read
Pictograms without proportional area	Uniform scaling changes both height and width, so icon area grows with the square of the scale factor