Describe association between two categorical variables

What should be done first when comparing two categorical groups?

What should be done to find the endpoint after a segment?

Predict what changes when the same travel counts are displayed as percentages. Compare view 1 (stacked counts) with view 2 (100% stacked bars). Check both total heights and the Bus segments

Why do the whole bars become equal in view 2?

Percentage segmented bar charts

A percentage segmented bar divides one complete bar into category segments. Calculate the percentage width of each segment, add the widths in order to find endpoints, and check that the bar finishes at 100%

1. 1Find each percentage Divide each category frequency by the sample size and multiply by 100. \(\displaystyle p_i=\frac{f_i}{n}\times100\%\)
2. 2Keep the category order Place the segments in the stated order because the order is part of the display \(\displaystyle A\;\rightarrow\;B\;\rightarrow\;C\)
3. 3Accumulate the endpoints Add each segment percentage to the endpoint already reached \(\displaystyle C_i=C_{i-1}+p_i,\quad C_0=0\%\)
4. 4Check the complete bar The last cumulative endpoint must be 100% so every record is represented \(\displaystyle C_k=100\%\)

Category frequency

\(\displaystyle 8\)

Sample size

\(\displaystyle 40\)

A category has frequency 8 in a sample of 40. Find its percentage frequency

First segment

\(\displaystyle 30\%\)

Second segment

\(\displaystyle 20\%\)

The first segment is 30% and the next segment is 20%. Find the endpoint after the second segment

Using a segment length as an endpoint. One of these lines is right. Which one?

Produce two percentage segmented bars

In an invented travel survey, the morning session has 12 Bus and 8 Walk responses; the afternoon session has 20 Bus and 20 Walk responses. Each person gives one response. Draw one complete 100% segmented bar per session, with Bus followed by Walk from zero

1. 1Choose and use a function Use a 100% stacked bar or column function and explain why it suits comparing shares in different-sized groups. Record the counts, group totals, data orientation, category order and percentage-scale settings. Include labels and a legend
2. 2Verify and interpret Calculate every segment percentage and cumulative endpoint by hand to check the tool. Compare Bus using both counts and percentage points. State why equal total bar lengths do not establish equal numbers of respondents
3. 3Complete before continuing Use its model and checklist for self-review. Your drawing, report and use of an external tool are not automatically marked

Model response: Produce two percentage segmented bars

Compare your saved response with this model. Other correct layouts, technology functions and explanations are possible

1. 1Percentages and endpoints Morning total: 20. Bus is 12/20×100%=60%; Walk is 40%. Endpoints from zero are 60%, 100%. Afternoon total: 40. Bus and Walk are each 50%; endpoints are 50%, 100%. Use the same category order in both bars
2. 2Complete display and technology record Enter sessions as the two groups and Bus/Walk as the two series, then choose a 100% stacked chart. Check the tool has made one bar for Morning and one for Afternoon; switch rows and columns if necessary. Use a 0–100% scale, label Morning (n=20) and Afternoon (n=40), and include a Bus/Walk legend. The total lengths are equal because each represents its own complete group
3. 3Independent verification Check 60+40=100 and 50+50=100, and compare each boundary with its cumulative total. The morning Bus share is 60−50=10 percentage points higher. The afternoon has more Bus respondents, 20 compared with 12. A stacked count chart would show totals 20 and 40 instead of two complete percentage bars
4. 4Self-review checklist Check both totals, all four percentages, segment order and endpoints, equal whole-bar lengths, labels, sample sizes, legend, a reproducible tool record and a comparison that distinguishes counts from shares

This model uses a shared percentage scale. What do its equal total bar heights establish?

Describing categorical association

Use conditional percentages to compare the same response category across the explanatory groups. State what the displayed sample shows and do not claim that the grouping variable causes the response

1. 1Use the group total For a percentage within one explanatory group, divide by that group's total \(\displaystyle \frac{18}{40}\times100=45\%\)
2. 2Hold the response category fixed Compare the same response category in each explanatory group \(\displaystyle \text{Group A: }45\%,\quad\text{Group B: }40\%\)
3. 3Compare percentages, not counts Conditional percentages allow a fair comparison when group sizes differ \(\displaystyle 18\text{ of }40\to45\%,\quad30\text{ of }75\to40\%\)
4. 4Describe the whole conditional distribution Quantify sample differences using the same response category and each group’s own denominator. Inspect all response categories before describing the overall association. A small difference in one category does not show that the other categories agree. Sample differences do not establish a population effect or causation, and no universal percentage-point threshold supplies a significance test

Given values

\(\displaystyle n_A=40,\quad n_{A,\mathrm{target}}=18\)

A target response appears 18 times in Group A, which has 40 records. Find its conditional percentage

Given values

\(\displaystyle n_B=75,\quad n_{B,\mathrm{target}}=30\)

A target response appears 30 times in Group B, which has 75 records. Find its conditional percentage

The same response occurs 18 times among 40 Group A records and 30 times among 75 Group B records. Which line makes the appropriate conditional comparison?

Morning and Evening both have 60% Repair. Does that establish equal response distributions?

Given values

\(\displaystyle p_M=60\%,\quad p_A=40\%\)

Repair is selected by 60% of Morning and 40% of Afternoon participants. Find the percentage-point difference

Repair shares are 60%,40%,60% for Morning, Afternoon, Evening. Which description fits all three groups?

Write an association report with evidence and limits

Name the two variables and the conditioning direction. Cite like conditional percentages, quantify differences in percentage points, and compare the whole response profile. Do not use raw counts to compare shares in groups of different sizes, or compare unlike response categories

1. 1Describe the whole conditional distribution Quantify sample differences using the same response category and each group’s own denominator. Inspect all response categories before describing the overall association. A small difference in one category does not show that the other categories agree. Sample differences do not establish a population effect or causation, and no universal percentage-point threshold supplies a significance test
2. 2Read bars as lengths A segment’s width is its percentage; an internal boundary is the cumulative endpoint. Keep the response order, legend, common 0–100% scale and group sample sizes visible. Equal total bar lengths represent 100% of each group, not equal counts
3. 3Limit your inference Describe these observations in context. Similar profiles give little descriptive separation in this sample; they do not prove population independence. Rounded displayed percentages may conceal small differences. Sampling variation, selection effects and confounding matter when extending a claim beyond the observed records

Produce and report three conditional distributions

In a fictional convenience sample, each participant selects one reuse action. In the order Repair, Borrow, Recycle, Morning has 24, 12 and 4 responses; Afternoon has 24, 24 and 12; Evening has 12, 2 and 6. Construct the complete count table and conditional percentage table. Draw three 100% segmented bars, ordered Repair,Borrow,Recycle from zero, with one common scale, legend, group labels and sample sizes

1. 1Write the report Compare Repair across all groups and compare Morning with Evening using the whole profile. Explain why equal Repair counts in Morning and Afternoon do not mean equal shares, and why equal Repair shares in Morning and Evening do not mean equal distributions. Give a sample-based conclusion with quantitative evidence and limitations. Separately consider group A with 6, 3 and 1 responses and group B with 12, 6 and 2 responses, in the same category order: explain what their equal profiles do and do not show
2. 2Choose and verify technology Choose a 100% stacked bar/column function and a conditional-percentage calculation. Record the orientation and category order. Verify all group totals, percentages and cumulative endpoints by arithmetic and check the plotted bars against those values. Record how the convenience sample limits population claims
3. 3Complete before continuing Save your complete table, labelled displays, report and technology record before opening the model response. Complete writing, drawing and external-tool execution are not automatically marked; use self or teacher review

Model response: compare all categories and groups

Group totals are 40,60,20; response totals are 60,38,22 and the grand total is 120. Conditional profiles (Repair,Borrow,Recycle) are Morning=(60%,30%,10%), Afternoon=(40%,40%,20%), Evening=(60%,10%,30%). Each exact profile totals 100%. Cumulative bar endpoints are Morning=(0,60,90,100), Afternoon=(0,40,80,100), Evening=(0,60,70,100)

1. 1A complete contextual report In these sampled participants, reuse-action distributions differ by session. Repair is 60% in Morning and Evening versus 40% in Afternoon, a 20 percentage-point difference. Morning and Evening have equal Repair shares but differ by 20 percentage points in Borrow (30% versus 10%) and Recycle (10% versus 30%). Morning and Afternoon each have 24 Repair responses, but their totals 40 and 60 produce different shares. The convenience sample does not establish population patterns or show that session attendance causes the response
2. 2A proportional comparison A=(6,3,1) and B=(12,6,2) have totals 10 and 20 and identical profiles 60%,30%,10%. Their bars coincide despite different counts. There is no observed profile difference in these data, which does not prove population independence or eliminate sampling uncertainty
3. 3One reproducible workflow Enter the three counts for each labelled session and calculate its total. Divide each count by that total; use a 100% stacked display with Repair,Borrow,Recycle in the same order. Keep sample sizes beside the labels. Verify 24/40=60%,24/60=40%,12/20=60% and cumulative sums. The second segment ends at 90%,80%,70% but its widths are 30%,40%,10%. Check the actual output, preserve a screenshot or export, and independently reconcile both margin sums to 120.

A table shows only a 1 percentage-point difference for one of four response categories. What can you conclude about the whole association?