AP Statistics Topic 2.1: Tabular and Graphical Representations for Two Categorical Variables

01 · Start here

The concept in 60 seconds

With one categorical variable, we describe how observations are split among categories. With two categorical variables measured on the same individuals, we can ask whether the distribution of one variable changes across categories of the other.

For example, record each student’s grade level and main travel mode. A two-way table keeps both pieces together. A graph then makes it easier to compare travel patterns in Grade 9 and Grade 10.

The key question: within each grade, what share of students use each travel mode? Comparing raw counts alone can be misleading when the grades contain different numbers of students.

The same count can represent a different share

Grade 9 ·

80

students $24$ walkers out of $80$

$30\%$ of this grade walk.

Grade 10 ·

120

students $24$ walkers out of $120$

$20\%$ of this grade walk.

Each track represents $100\%$ of its grade. The walker counts match, but the group proportions do not.

If the category proportions differ across groups, the data show an association. That describes a relationship in the observed data. It does not, by itself, establish that one variable causes the other or that a population relationship has been proved.

02 · Keep the observations paired

A school records grade and travel mode

In an original fictional example, a school records $1$ main travel mode for each of $200$ participating students. There are $80$ Grade 9 students and $120$ Grade 10 students. Each student belongs to $\text{exactly }1\text{ listed grade and }1\text{ listed mode}$ : bus, walk or car ride.

Two-way frequency table: $\text{grade level}\times\text{main travel mode}$ . Entries are numbers of students.
Grade level	Bus	Walk	Car ride	Row total
Grade 9	$40$	$24$	$16$	$80$
Grade 10	$36$	$24$	$60$	$120$
Column total	$76$	$48$	$76$	$200$

Two-way table

A table showing combinations of categories for two categorical variables. It is also called a contingency table.

Rows identify grade; columns identify travel mode.

Cell

An entry at the intersection of a row and column.

$40$ means $40$ participating Grade 9 students travel mainly by bus.

Margins

The row and column totals, which summarize one variable at a time.

$80$ is the Grade 9 total; $76$ is the bus total across both grades.

Grand total

The number of observations in the table, $\text{counted }1\text{ time each}$ .

$80 + 120 = 200$ ; also $76 + 48 + 76 = 200$ .

Read a cell by combining its two labels. Read a margin by identifying which label has been kept and which has been combined. The row totals and column totals each add to $200$ ; adding both sets together would count every student $2$ times.

Swapping rows and columns gives another valid orientation of the same data. It does not change which students are counted together. Keep the category labels clear, because the denominator for a comparison follows the question, not a rule that rows are always correct.

Think first: do the separate grade totals, $80$ and $120$ , and travel totals, $76$ , $48$ and $76$ , tell you exactly how many Grade 9 students walk?

Check your prediction

No. Those margins do not specify the combinations. Different cell arrangements can have the same row and column totals. The two-way table preserves the paired information needed to see that $24$ Grade 9 students walk.

Grade level is categorical here: 9 and 10 name the groups being compared. A table that records two quantitative measurements, such as height and mass, calls for other displays. A table of grade counts alone contains only one categorical variable.

03 · Decide what the question compares

Counts and proportions answer different questions

A frequency is a count. A relative frequency is a count divided by a specified total, written as a proportion or percentage. A two-way table can display either, so check its caption and labels before interpreting an entry.

Number of studentsWhich grade has more car riders?

Grade 10 has $60$ ; Grade 9 has $16$ . Counts answer this question directly.

Share within a gradeWhich grade has a larger car-rider proportion?

Grade 10: $\frac{60}{120} = 50\%$ . Grade 9: $\frac{16}{80} = 20\%$ . Use the relevant grade total for each comparison.

\text{Within-group percentage}=\frac{\text{category count}}{\text{group total}}\times100\%

For Grade 9 walkers: $\frac{24}{80}\times100\%=30\%$ .

A group distribution lists all of its category shares. Grade 9’s travel distribution is $50\%$ bus, $30\%$ walk and $20\%$ car ride. Grade 10’s is $30\%$ bus, $20\%$ walk and $50\%$ car ride. Each list adds to $100\%$ because the modes are mutually exclusive and cover every student in its grade.

Dividing every cell by $200$ instead produces shares of the entire data set. For example, $\frac{24}{200} = 12\%$ says that $12\%$ of all participating students are Grade 9 walkers. It does not say that $12\%$ of Grade 9 students walk.

Ask “out of whom?” An unlabeled percentage is incomplete. A table of percentages of all $200$ students and a table of percentages within each grade describe different reference groups.

This lesson uses simple within-group percentages to read graphs and compare patterns. Topic 2.2 develops the joint, marginal and conditional relative-frequency calculations in more detail.

04 · Check the scale before the pattern

Side-by-side and segmented bar charts

Side-by-side bar chart: compare a category across groups

A side-by-side bar chart, also called a grouped bar chart, places the groups’ bars next to each other within each category. Bars can run vertically or horizontally. Their heights or lengths show counts or relative frequencies, as stated on the scale.

Side-by-side bar chart: numbers of students

Grade 9 ·

n = 80

Grade 10 ·

n = 120

Number of students · common scale from

0

60

0

20

40

60

Bus

40

36

Walk

24

24

Car ride

16

60

The bars show counts, not within-grade proportions. $\text{All }6\text{ values}$ come from the frequency table above.

The count chart shows that both grades have $24$ walkers. It also shows $60$ car riders in Grade 10 compared with $16$ in Grade 9. The grades have unequal totals, so compare within-grade percentages when the question is about travel patterns rather than the numbers using each mode.

Side-by-side bar chart: percentage within each grade

Grade 9 ·

n = 80

Grade 10 ·

n = 120

Percentage of students within the labeled grade

0\%

25\%

50\%

75\%

100\%

Bus

50\%

30\%

Walk

30\%

20\%

Car ride

20\%

50\%

Each grade uses its own total as the denominator. Compare the same travel category across grades.

The percentage chart uses each grade as its own reference group. For bus travel, compare $50\%$ in Grade 9 with $30\%$ in Grade 10. For walking, compare $30\%$ with $20\%$ . For car travel, compare $20\%$ with $50\%$ . A matching category and a common scale make the contrasts readable.

On this side-by-side percentage chart, the $3\text{ bars for a given grade}$ collectively add to $100\%$ . The $2\text{ bars within }1\text{ travel category}$ do not need to add to $100\%$ : they refer to different grades.

Segmented bar chart: compare each group’s whole distribution

A segmented bar chart divides a bar into category pieces. Here we use a $100\%$ segmented bar chart: each grade has an equal-length bar representing its full distribution, and the segment lengths show its category percentages.

100\%

segmented bar chart: travel distribution within each grade

BusWalkCar ride

Grade 9 ·

n = 80

50\%

30\%

20\%

Grade 10 ·

n = 120

30\%

20\%

50\%

0\%

25\%

50\%

75\%

100\%

Both full bars represent $100\%$ of their own grade. Their equal lengths do not imply equal numbers of students.

Grade 9 has a larger bus segment; Grade 10 has a larger car-ride segment. Equal full bar lengths help compare proportions without the larger grade dominating the display. The labels $n = 80$ and $n = 120$ still tell you how many students those $100\%$ bars represent.

Read a segment’s length, not its ending position. In the Grade 9 bar, walking runs from $50\%$ to $80\%$ , so its share is $80\%-50\%=30\%$ . It is not $80\%$ .

A stacked bar chart can instead use counts, making each full bar’s length the group’s total. Check the units before assuming that every segmented display is standardized to $100\%$ . Keep the same category order and legend when comparing groups.

05 · Width, height and area have different jobs

Read a mosaic plot

A mosaic plot divides a rectangle into group columns and category pieces. In the version below, we first split by grade. Column width represents that grade’s share of all $200$ students. Within a column, segment height represents the travel-mode share within that grade.

Mosaic plot: grade widths and travel-mode heights

BusWalkCar ride

Vertical scale: percentage within each grade. Width: share of all students. Bus is at the base, Walk in the middle, Car ride at the top.

Grade 9

n = 80

· width

40\%

Grade 10

n = 120

· width

60\%

100\%

80\%

60\%

40\%

20\%

0\%

Bus $50\%$

40

students

Walk $30\%$

24

students

Car $20\%$

16

students

Bus $30\%$

36

students

Walk $20\%$

24

students

Car $50\%$

60

students

The complete rectangle represents all $200$ students. Segment labels give within-grade percentages; areas give whole-table shares.

Widths: Grade 9 is $\frac{80}{200} = 40\%$ of the total; Grade 10 is $\frac{120}{200} = 60\%$ . The widths are in the ratio $2:3$ .
Heights within a column: the bus segment occupies $50\%$ of the Grade 9 column and $30\%$ of the Grade 10 column.
Area: each cell’s area represents its share of the complete data set. Grade 9 bus area is $40\% \times 50\% = 20\%$ of the plot, corresponding to $40$ of the $200$ students.

The Grade 10 car-ride cell covers $60\% \times 50\% = 30\%$ of the plot, corresponding to $60$ students. A broad column tells you that the group is large, not that every category has a high proportion within it.

Equal areas need not mean equal within-group percentages. Both grades have $24$ walkers, so the walker cells each occupy $12\%$ of the plot. Their heights differ: $30\%$ in the narrower Grade 9 column and $20\%$ in the wider Grade 10 column.

Compare the heights of the same category across columns to assess travel patterns. With a consistent category order, differently placed internal boundaries reveal different distributions. You can reverse the initial split and build a travel-mode-first mosaic; that changes which groups the column widths describe, so read the labels again.

This mosaic has proportional widths and no decorative gaps between columns. In a mosaic with equal-width grade columns, the group-size information would be lost. An ordinary $100\%$ segmented chart can use equal-sized bars because it has a different purpose.

06 · Compare distributions, then state the limit

Recognize an association between categories

Two categorical variables are associated in a data set when the distribution of one variable differs across categories of the other. In the school data, knowing a student’s grade changes the observed travel distribution: the car-ride share is $20\%$ in Grade 9 and $50\%$ in Grade 10.

Compare matching category proportions. Do not decide from the fact that one group is bigger, that the table has unequal counts, or that one category is common overall. A bus total of $76$ tells us how common bus travel is overall, but not how its share varies by grade.

Different distributions versus matching distributions

Association in the school dataTravel-mode shares differ

BusWalkCar ride

Grade 9 ·

n = 80

50\%

30\%

20\%

Grade 10 ·

n = 120

30\%

20\%

50\%

Car ride is $20\%$ in Grade 9 and $50\%$ in Grade 10. The within-grade patterns differ.

A separate no-association exampleYes/no shares match

YesNo

Group A ·

n = 50

60\%

40\%

Group B ·

n = 100

60\%

40\%

Group A: $30$ yes, $20$ no. Group B: $60$ yes, $40$ no. Both are $60\%$ yes and $40\%$ no.

These are $100\%$ bars. The second example is a different fictional data set, not another view of the school table.

The counterexample has different group sizes but matching distributions of $60\%$ yes and $40\%$ no. Its count bars would differ, yet its $100\%$ bars have the same internal boundary. In those constructed data there is no observed association between group membership and the yes/no response.

If at least one matching category share differs, the full distributions differ. Conversely, finding that one share matches does not establish that every share matches when there are $3\text{ or more categories}$ . Check the entire pattern.

Keep claims at the right level: a graph can show an association in the observed students. Establishing a population association requires appropriate collection and statistical analysis. Establishing a causal effect needs a suitable experimental design; the travel table provides no random assignment.

For nominal categories, avoid describing the relationship with a correlation coefficient or a “positive slope.” Say which group has a higher proportion in which category. An association is a distribution comparison, not a claim that all members of a group behave alike.

07 · Follow the reasoning

Five worked examples

These are original fictional teaching examples. The graph values come from the displayed tables; they are not results from a real school survey.

Example 1: Complete and interpret a two-way table

Task: the Grade 10 total is $120$ . Its bus count is $36$ and walk count is $24$ . Find its car-ride count, then interpret the school-wide walker total.

Use the row’s total: the $3\text{ travel modes}$ cover all students in that grade, with $1\text{ main mode per student}$ .
Find the missing count: $120 – 36 – 24 = 60$ car riders.
Combine the grades for walkers: $24 + 24 = 48$ walkers.
Interpret the margin: $48$ participating students, across both grades, travel mainly by walking. It is not the count in each grade.

Check: Grade 10’s row sums to $120$ , and the column totals $76 + 48 + 76$ sum to $200$ . A table’s margins are totals, not additional observations.

Example 2: Repair a count-based comparison

Task: a student says, “Both grades have $24$ walkers, so walking is equally common in both grades.” Assess the statement.

Clarify “common”: the walker numbers match. The proportions need not match because the grade totals differ.
Compare Grade 9’s share: $\frac{24}{80} = 30\%$ .
Compare Grade 10’s share: $\frac{24}{120} = 20\%$ .
Write a supported comparison: the same number walk in each grade, but the walking proportion is higher in Grade 9 by $10$ percentage points.

A side-by-side percentage chart or $100\%$ segmented chart reveals this contrast. “ $10$ percentage points” is the subtraction $30\%-20\%$ ; it is different from a $10\%$ relative increase.

Example 3: Read a $100\%$ bar’s boundaries

Task: the Grade 9 segmented bar has a bus/walk boundary at $50\%$ and a walk/car boundary at $80\%$ . Find the walking and car-ride shares and their counts.

Locate the walker segment: it extends from $50\%$ to $80\%$ .
Subtract endpoints: $80\%-50\%=30\%$ walking.
Read the remaining final segment: $100\% – 80\% = 20\%$ car ride.
Use the stated grade size: $0.30 \times 80 = 24$ walkers; $0.20 \times 80 = 16$ car riders.

Without the grade total, the percentage bar alone would not tell you the counts. Segment labels and the group-size note supply different information.

Example 4: Interpret a mosaic cell

Task: compare the $2\text{ walker rectangles}$ in the school mosaic. Why do they have equal areas even though their heights differ?

Grade 9 width: $\frac{80}{200} = 0.40$ . Its walker height is $\frac{24}{80} = 0.30$ .
Grade 9 walker area share: $0.40 \times 0.30 = 0.12$ .
Grade 10 width: $\frac{120}{200} = 0.60$ . Its walker height is $\frac{24}{120} = 0.20$ .
Grade 10 walker area share: $0.60 \times 0.20 = 0.12$ .
Connect area to counts: each rectangle represents $12\%$ of $200$ , or $24$ students. The narrower column needs a taller segment for the same area.

Takeaway: use height for the within-grade walker proportion, width for the grade share, and area for the combination’s share of all students.

Example 5: Unequal counts without an observed association

Task: Group A has $50$ respondents: $30$ yes and $20$ no. Group B has $100$ respondents: $60$ yes and $40$ no. A student claims an association because Group B has $2\text{ times as many yes responses}$ .

Check group sizes: B also has $2\text{ times as many respondents}$ overall.
Compare response distributions: A is $\frac{30}{50} = 60\%$ yes and $40\%$ no; B is $\frac{60}{100} = 60\%$ yes and $40\%$ no.
Identify the graphical pattern: both $100\%$ bars have the same split at $60\%$ . In a group-first mosaic, B’s column is wider, but corresponding segment heights match.
Assess the claim: the count difference does not show an association here. The response distribution is the same in these $2\text{ observed groups}$ .

This constructed example does not prove that two variables are independent in every possible population. It illustrates why matching within-group distributions, rather than matching raw counts, are the relevant comparison.

08 · Give evidence, not just a verdict

Explain a comparison in context

A good explanation names the variables, identifies the groups being compared, and gives evidence from a matching category on a suitable scale. Finish with the claim that the evidence supports.

Build an association explanation in three steps

1 · Name the comparisonTravel mode across grade levels

Compare the mode distribution within Grade 9 with that within Grade 10.

2 · Give matching evidenceCar ride: $20\%$ versus $50\%$

The shares are $\frac{16}{80}$ for Grade 9 and $\frac{60}{120}$ for Grade 10.

3 · State the supported claimAn association in these data

The mode distribution differs by grade. The table does not establish a causal effect.

Writing frame: “Among [observed units], [category] accounts for [percentage] of [group A] and [percentage] of [group B]. Because the distributions differ across [grouping variable], the data show an association between [variable 1] and [variable 2].”

Applied to the example: “Among the participating students, $20\%$ of Grade 9 students and $50\%$ of Grade 10 students travel mainly by car ride. Travel mode therefore has a different distribution in the two grades, showing an association between grade level and main travel mode in these data.”

For a claim about numbers, use numbers: “There are $60$ Grade 10 car riders and $16$ Grade 9 car riders.” For a claim about a group’s share, use its denominator: “ $\frac12\text{ of Grade 10 students}$ are car riders.” Neither sentence says that $\frac12\text{ of all car riders}$ are in Grade 10.

Say what the graph cannot tell you

The table does not explain why the travel distributions differ. Distance from school, family arrangements or other features could be related to both grade and travel mode. Grade was not randomly assigned, and the data do not justify a cause-and-effect claim.

The example describes participating students. Before generalizing to all students, inspect how participants were selected and how accurately travel mode was recorded. Before claiming convincing evidence of a population association, use appropriate inference rather than a visual impression alone.

A descriptive difference can be real in the displayed data without establishing a statistically significant population relationship. This topic focuses on reading representations and explaining their evidence.

09 · Find the wrong turn

Common mistakes and how to fix them

Check the labels, reference group and graphical dimensions.
Mistake	Why it fails	Better approach
Comparing raw counts to decide which group has the higher proportion.	A larger group can have more observations in a category even with a smaller category share.	Divide by each relevant group total, or read a properly labeled percentage display.
“ $\frac{24}{200}$ is the Grade 9 walking percentage.”	$200$ includes both grades.	Use $\frac{24}{80}$ for the share within Grade 9; $\frac{24}{200}$ describes Grade 9 walkers among everyone.
Adding row totals and column totals to find the sample size.	Each set already counts every observation once.	Add one set of margins, or use the grand total.
Reading a segment’s upper endpoint as its percentage.	An interior segment begins $\text{above }0$ .	Subtract its starting endpoint from its ending endpoint.
Assuming equal $100\%$ bar sizes mean equal group sizes.	Standardizing removes group-size differences from the full bar lengths.	Read the $n$ labels or the frequency table.
Making unequal-sized groups equally wide in a proportional mosaic.	The widths no longer represent their shares of the total.	Use proportional widths; here $40\%$ and $60\%$ .
Using mosaic area as the percentage within a group.	Area combines group share and within-group category share.	Use segment height within that column for the category proportion.
“One category matches, so there is no association.”	Other category shares may differ.	Check the full distributions, especially with $3\text{ or more categories}$ .
“Different bars prove causation.”	The display describes a relationship without supplying a causal design.	State an association in context and inspect the study design before making a causal claim.
Comparing different legends, category orders or scales as if they match.	A color or position may represent something different in each group.	Match the category labels and use a consistent scale, order and legend.

When checking a graph, look for a clear title, group labels, a count or percentage scale, and an unambiguous legend. Count bar lengths need a $0\text{ baseline}$ to represent counts proportionally. A $100\%$ bar must cover exactly the complete group, allowing small rounding discrepancies in printed labels.

10 · Work first, then check

Eight practice questions with hints and solutions

All situations are fictional. Write your comparison before opening the solution. When using a percentage, state the reference group.

1. Identify the two variables

A school records each participating student’s grade level, Grade 9 or Grade 10, and main travel mode, bus, walk or car ride. Name the observational unit and the two variables. Does using 9 and 10 as labels make grade level a quantitative measurement here?

Hint

What does one row of the original student data describe? Do the numbers measure an amount, or identify groups?

Solution and explanation

The unit is $1\text{ participating student}$ . The variables are grade level and main travel mode, both categorical in this comparison. The grade labels identify ordered categories rather than a quantitative amount measured on each student. Their numerical appearance does not change the table’s purpose.

2. Complete the table and choose a denominator

A café records preferred drink for $100$ respondents. Of $60$ morning respondents, $24$ prefer coffee and the rest prefer tea. Of $40$ afternoon respondents, $12$ prefer tea and the rest prefer coffee. Find the missing counts and the overall drink totals. What percentage of morning respondents prefer tea?

Hint

Use each session total before combining sessions. For the final percentage, who is the reference group?

Solution and explanation

Morning tea: $60 – 24 = 36$ . Afternoon coffee: $40 – 12 = 28$ . Overall tea: $36 + 12 = 48$ ; overall coffee: $24 + 28 = 52$ . Among morning respondents, tea accounts for $\frac{36}{60} = 60\%$ . Dividing $36$ by $100$ would instead give the share of all respondents who are morning tea-preferrers.

3. A taller count bar can have a smaller proportion

A count chart shows $48$ yes responses in Group A and $36$ in Group B. Group A contains $120$ respondents; Group B contains $60$ . A student says A has the larger yes proportion because $48$ exceeds $36$ . Assess the claim and give a suitable graph choice.

Hint

Compare yes responses out of each group’s own total.

Solution and explanation

A’s yes proportion is $\frac{48}{120} = 40\%$ ; B’s is $\frac{36}{60} = 60\%$ . A has more yes responses but the smaller yes proportion. A side-by-side within-group percentage chart or $100\%$ segmented chart makes the proportional comparison clearer. The count chart is appropriate for the different question of which group has more yes responses.

4. Read an interior segment

A $100\%$ bar displays bus, then walk, then car ride. Its first boundary is $35\%$ and its second boundary is $75\%$ . State all $3\text{ mode shares}$ . Is the walking share $75\%$ ?

Hint

The walking segment starts at $35\%$ , $\text{not }0$ .

Solution and explanation

Bus: $35\%$ . Walk: $75\% – 35\% = 40\%$ . Car ride: $100\% – 75\% = 25\%$ . The walking share is not $75\%$ ; that endpoint combines bus and walk. The shares sum to $100\%$ .

5. Separate mosaic width, height and area

A mosaic represents $200$ responses. Group A occupies $25\%$ of its width and Group B $75\%$ . The yes segment occupies $80\%$ of A’s height and $40\%$ of B’s height. Find the yes proportion within each group, the yes count in each, and each yes rectangle’s share of the whole plot.

Hint

Width gives a group’s share of $200$ . Height gives the yes share of that group. Multiply them for area.

Solution and explanation

A contains $0.25 \times 200 = 50$ respondents and B contains $150$ . A is $80\%$ yes, giving $40$ yes responses; B is $40\%$ yes, giving $60$ . The yes areas are $0.25 \times 0.80 = 20\%$ of the plot for A and $0.75 \times 0.40 = 30\%$ for B. A has the higher yes proportion but the smaller yes count and area.

6. Evaluate a cause-and-effect claim

Using the school travel table, a student says, “Moving to Grade 10 causes students to travel by car because the car-ride percentage rises from $20\%$ to $50\%$ .” Give a supported interpretation and explain the overstatement.

Hint

Was grade randomly assigned? What does the table actually show?

Solution and explanation

The observed car-ride share is $20\%$ in Grade 9 and $50\%$ in Grade 10, so grade level and travel mode are associated in these data. Grade was not randomly assigned, and other differences could contribute to travel choices. The table does not establish that changing grade causes a change in travel mode, or that a particular student would change mode.

7. Check a claimed $100\%$ chart

A bar labeled “Percentage within Group A” has $2\text{ nonoverlapping pieces}$ labeled $70\%$ yes and $40\%$ no. Every respondent is in $\text{exactly }1\text{ of those categories}$ , and the chart gives no indication that the percentages are rounded. What is wrong? What should you check before repairing it?

Hint

Can a complete group occupy $110\%$ of itself?

Solution and explanation

The labels total $110\%$ , which is incompatible with a complete $2\text{-category within-group distribution}$ . Check the original counts, the Group A total and whether both percentages use that same reference group. Then recompute and redraw the pieces. Do not silently change $40\%$ to $30\%$ without verifying the underlying values; either entry or its labeling could be wrong.

8. One matching category is not the whole distribution

In Group A, $40\%$ choose red, $35\%$ blue and $25\%$ green. In Group B, $40\%$ choose red, $20\%$ blue and $40\%$ green. A student says there is no association because the red proportions match. Assess the claim using both distributions.

Hint

Check blue and green as well as red.

Solution and explanation

The full distributions differ. Blue is $35\%$ in A versus $20\%$ in B, and green is $25\%$ versus $40\%$ . These differences show an association between group and color choice in the displayed data, even though the red shares match. A $100\%$ segmented chart would have the same red boundary but a different blue/green boundary.

11 · Keep the essentials

Quick revision checklist

Two-way tableKeep two categorical measurements on the same units together.

Cell + labelsCombine the row and column categories to interpret the count.

Count or share?Choose the scale that matches the claim; state “out of whom?”

Side-by-side barsCompare matching category bars on the same labeled scale.

$100\%$ segmented barsEach full bar is a group. Segment lengths give its category shares.

Mosaic columnsWidth gives group share; height gives within-group share; area gives whole-table share.

AssociationDifferent group distributions show a relationship in the observed data.

Supported claimName groups, matching category evidence, context and scope.

Quick questions students often ask

Can a two-way table have more than two rows or columns?

Yes. “Two-way” refers to two variables, not two categories. A grade-by-mode table can have $2$ grade rows and $3$ mode columns, plus totals. Each variable can have several categories.

Must the row percentages or the column percentages be used?

Use the reference groups specified by the question. To compare travel within grades when grades are rows, divide by row totals. If the same table is transposed so grades are columns, divide by column totals. Orientation changes the calculation’s position, not its meaning.

Can two groups have different counts but the same distribution?

Yes. Group A’s $30$ yes and $20$ no, and Group B’s $60$ yes and $40$ no, both produce $60\%$ yes and $40\%$ no. The count difference reflects different group sizes, while the category proportions match.

Is a mosaic plot the same as a

100\%

segmented chart?

They can show the same within-group pattern. The mosaic also encodes group size through proportional widths. A $100\%$ segmented chart usually gives equal-sized full bars, so obtain the group counts from labels or the table rather than bar size.

12 · Put the ideas together

Final understanding check

A fictional learning center records the course schedule and main study location of $150$ survey respondents. Each learner selects $1\text{ location}$ . The data are:

Numbers of respondents by course schedule and main study location.
Schedule	Library	Home	Other	Total
Day	$36$	$12$	$12$	$60$
Evening	$27$	$45$	$18$	$90$
Total	$63$	$57$	$30$	$150$

Name the observational unit and both categorical variables.
Write the study-location distribution within each schedule.
Choose a graph for comparing those distributions and specify what its scale represents.
Give the Day and Evening widths in a schedule-first mosaic. Find the whole-plot area share for the Day-library cell.
Explain whether the observed variables are associated, using matching category evidence.
Assess “Evening classes cause learners to study at home.”

Reveal the full solution

1. Variables: $1\text{ survey respondent}$ is the unit. The variables are course schedule, Day or Evening, and main study location, Library, Home or Other.

2. Distributions: Day: $\frac{36}{60} = 60\%$ library, $\frac{12}{60} = 20\%$ home, and $20\%$ other. Evening: $\frac{27}{90} = 30\%$ library, $\frac{45}{90} = 50\%$ home, and $\frac{18}{90} = 20\%$ other. Each schedule’s percentages add to $100\%$ .

3. Graph: a $100\%$ segmented bar chart, with $1\text{ equal-length bar per schedule}$ , uses segment lengths to show within-schedule location percentages. Keep category order and colors consistent, and label the group sizes $60$ and $90$ . A side-by-side within-schedule percentage chart is also appropriate.

4. Mosaic: Day width is $\frac{60}{150} = 40\%$ ; Evening width is $\frac{90}{150} = 60\%$ . The Day-library cell has height $60\%$ within Day, so its area share is $0.40 \times 0.60 = 24\%$ of the plot. This equals $\frac{36}{150}$ . It is different from the $60\%$ library share within Day.

5. Association: among respondents, the home share is $20\%$ for Day and $50\%$ for Evening. Library shares also differ, $60\%$ versus $30\%$ . Thus study-location distributions differ by schedule, showing an association in these data. The matching Other share does not undo the differences elsewhere.

6. Scope: the survey does not randomly assign class schedules. Other differences between learners could be related to their study choices. Describe an association among the respondents; the table alone does not establish a causal effect or justify a claim about all learners.

Ready to move on? You should be able to read a table entry in context, choose a graph that answers the question, identify what each graphical dimension represents, and support an association claim with an appropriate comparison.

13 · Keep building

Continue learning

Up next: Topic 2.2

Topic 2.2: Summary Statistics for Two Categorical Variables →

Calculate and interpret joint, marginal and conditional relative frequencies, and use those summaries to compare categorical distributions.

Previous: Topic 1.13 — Experimental Design · Review bar charts · Review the mosaic · Back to the lesson overview

0000003998 : Abdad Alam Shamim Alam

This is the bestest institution I have ever come to and I love it very much.

Aliki S

Ali

Sly is him

Jitendra Kumar Kumawat

Raahil Hasan

Tina Mudarres

The classes are amazing and my child learnt so much

alisha gadoya

I have experienced a lot of good things, it has taught me so many things and from B/Cs I have gone to an A! This is wonderful and Mavish’s class is awesome.

smasher 123

It’s sensational my kid went. First he only would get C now it’s all A’s really good and recommend

Mosa Al- Samaraie

AMAZING PRICES GREAT TEACHERS STRAIGHT FORWARD LEARNING STEADY PACE IN TUTORING

Cael Dagnelie

I had a great experience learning math with Ms. Mavish. She explains complex topics in a very clear and simple way, which made it easier for me to understand and enjoy the subject. Her patience and dedication really stood out, and she always made sure that everyone in the class was keeping up. I especially appreciated how approachable she was — I never felt afraid to ask questions, and she was always willing to help. Thanks to her teaching, my confidence in math has grown a lot. I’m really thankful for the effort she puts into every lesson!

The concept in 60 seconds

A school records grade and travel mode

Two-way table

Cell

Margins

Grand total

Counts and proportions answer different questions

Side-by-side and segmented bar charts

Side-by-side bar chart: compare a category across groups

Segmented bar chart: compare each group’s whole distribution

Read a mosaic plot

Recognize an association between categories

Five worked examples

Example 1: Complete and interpret a two-way table

Example 2: Repair a count-based comparison

Example 3: Read a 100%100\% bar’s boundaries

Example 4: Interpret a mosaic cell

Example 5: Unequal counts without an observed association

Explain a comparison in context

Say what the graph cannot tell you

Common mistakes and how to fix them

Eight practice questions with hints and solutions

1. Identify the two variables

2. Complete the table and choose a denominator

3. A taller count bar can have a smaller proportion

4. Read an interior segment

5. Separate mosaic width, height and area

6. Evaluate a cause-and-effect claim

7. Check a claimed 100%100\% chart

8. One matching category is not the whole distribution

Quick revision checklist

Quick questions students often ask

Final understanding check

Continue learning

Example 3: Read a $100\%$ bar’s boundaries

7. Check a claimed $100\%$ chart