Report on the distribution of a categorical variable

Walk responses

\(\displaystyle 12\)

Bus responses

\(\displaystyle 21\)

Car responses

\(\displaystyle 9\)

Bike responses

\(\displaystyle 8\)

Students asked

\(\displaystyle 50\)

A survey asked 50 students how they get to school. The bar chart shows the counts for each way. What is the mode of this distribution?

Move the bus count slider from low to high, past the walk marker. Watch which bar is tallest and what happens to the walk percentage as the sample grows

As the bus count rose past 14, what happened to the mode?

Describing a categorical distribution

A description of a categorical distribution states how many responses were recorded, names the category that occurs most, and backs that claim with its count or percentage

1. 1State the sample size Add every category count to give the total number of responses \(\displaystyle \text{n}=12+21+9+8=50\)
2. 2Name the modal category The mode is the category with the greatest count. Name the category, not its count \(\displaystyle 21>12,9,8\quad\Rightarrow\quad\text{mode}=\text{Bus}\)
3. 3Quote a supporting figure Give the count or the percentage of the whole sample that sits in the modal category \(\displaystyle \frac{21}{50}\times100=42\%\)
4. 4Respect category order Keep ordered categories in their natural order and report briefly rather than reciting every category \(\displaystyle \text{Very unsatisfied}<\text{Unsatisfied}<\text{Satisfied}<\text{Very satisfied}\)

Poor ratings

\(\displaystyle 6\)

Fair ratings

\(\displaystyle 14\)

Good ratings

\(\displaystyle 36\)

Excellent ratings

\(\displaystyle 24\)

Customers asked

\(\displaystyle 80\)

Rating order

\(\displaystyle \text{Poor}<\text{Fair}<\text{Good}<\text{Excellent}\)

A cafe asked 80 customers to rate their visit as Poor, Fair, Good or Excellent. What percentage of the customers gave the modal rating?

Find which rating has the greatest count, then divide that count by all 80 customers and multiply by 100

Poor ratings

\(\displaystyle 6\)

Fair ratings

\(\displaystyle 14\)

Good ratings

\(\displaystyle 36\)

Excellent ratings

\(\displaystyle 24\)

Customers asked

\(\displaystyle 80\)

Rating order

\(\displaystyle \text{Poor}<\text{Fair}<\text{Good}<\text{Excellent}\)

Using the same cafe ratings, what percentage of the customers rated the visit Good or Excellent?

Add the two counts, then divide by all 80 customers

Walk responses

\(\displaystyle 12\)

Bus responses

\(\displaystyle 21\)

Car responses

\(\displaystyle 9\)

Bike responses

\(\displaystyle 8\)

Which line correctly names the mode?

Write a complete distribution report

At an invented optional workshop, each attendee chose one first activity. Morning counts are Repair 9, Make 12, Learn 9; afternoon counts are Repair 20, Make 15, Learn 15. Compare the activity distributions for these two sessions in a short report

1. 1Produce the evidence Create a labelled frequency and percentage table for each session. Choose a useful chart for comparing the distributions, justify using counts or percentages, and produce it using a suitable technology function. Record the formula or settings, the input and the output
2. 2Write the report State both sample sizes and modal categories with supporting percentages. Compare Make using counts and percentage points, identify any equal shares, and distinguish a modal category from a majority. Finish with a limit on what the data establish
3. 3Verify independently Check both totals by hand and verify a percentage using the exact count divided by its own session total. Check each number and category in your report against your table. Explain why one large count alone cannot compare shares across unequal groups

Model response: Write a complete distribution report

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

1. 1Complete numerical evidence Morning total 30: Repair 9 (30%), Make 12 (40%), Learn 9 (30%). Afternoon total 50: Repair 20 (40%), Make 15 (30%), Learn 15 (30%). Each session’s counts sum to its own total; each set of percentages sums to 100%
2. 2One technology record and rationale Enter the three category counts for each session, sum each session column, and calculate each category percentage from its count and session total without intermediate rounding. Select the resulting percentages and insert a grouped bar chart with the three activity labels, a Morning/Afternoon legend and a zero-based percentage axis. Percentages compare shares despite the totals 30 and 50. A correctly labelled 100% segmented display is another suitable choice
3. 3Model report Among the 30 morning attendees, Make was the modal first activity at 40%. Among the 50 afternoon attendees, Repair was the mode at 40%. Neither mode was a majority. Make had more afternoon choices, 15 compared with 12, but its share was 30% rather than 40%, a decrease of 10 percentage points. Learn accounted for 30% in both sessions. These results describe the attendees’ first choices at this workshop. Optional attendance and a single event do not establish preferences in a wider population or show that session time caused the differences
4. 4Independent verification The morning counts add to 30 and the afternoon counts add to 50. Make accounts for 40% of the morning session and 30% of the afternoon session, a difference of 10 percentage points. Relative to its morning share, this is a 25% decrease. Check the report’s labels as well as its numbers
5. 5Self-review checklist Check complete tables and a labelled comparison chart, both sample sizes, modal categories and supporting percentages, the count/share distinction, equal Learn shares, mode versus majority, a justified and recorded tool choice, an independent numerical check and a defensible limitation

This model table uses each session’s own denominator. Which statement correctly compares Make?