Unit 1 · Topic 1.4 Beta

Graphical Representations for One Categorical Variable

Bar charts and pie charts display one categorical variable.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

A school report shows two bars for the percent of students passing a state exam: one school at 52%, the other at 58%. The second bar looks four times as tall. Is the second school four times better? No: the vertical axis starts at 50%. Graphs of categorical data are simple, but they have rules, and those rules are what keep a picture honest.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. In a sample of 200, the relative frequency of a category is 0.15. What is its count?

  1. 30
  2. 15
  3. 0.15
  4. 1,333
Show the answer

Count = relative frequency × total = 0.15 × 200 = 30.

  • Correct: 30:
  • 15:
  • 0.15:
  • 1,333:

2. When do the relative frequencies of a categorical variable add to 1?

  1. When each individual is in exactly one category
  2. Always, whatever the survey asked
  3. Only when there are two categories
  4. When the sample is large
Show the answer

Then the categories together make up the whole sample.

  • Correct: When each individual is in exactly one category:
  • Always, whatever the survey asked:
  • Only when there are two categories:
  • When the sample is large:

Part 4 · See it

See it first

Left: a bar chart of how 250 students get to school, with relative frequency on the vertical axis from 0 to 0.4: Bus 0.38, Car 0.32, Walk 0.18, Bike 0.08, Other 0.04. Right: a pie chart of the same data, with slices Bus 38%, Car 32%, Walk 18%, Bike 8%, Other 4%.
The same relative frequencies as a bar chart and as a pie chart. Bars are easier to compare; the pie shows parts of one whole. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Each category of a categorical variable gets one barthe bar’s height is the category’s count or relative frequency, measured from 0
  2. Categories have no number orderbars can go in any order and are drawn with gaps between them
  3. When each individual is in exactly one categorythe parts make one whole, so a pie chart can show them as slices
  4. A cut-off axis or bars from groups of different sizesthe picture misleads, so start at 0 and graph relative frequencies to compare groups

Part 6 · Key ideas

Key ideas

  • A bar chart shows one bar per category, with height equal to the count or relative frequency. The vertical axis starts at 0.
  • A pie chart shows each category as a slice of one circle, sized by its relative frequency. It needs categories that do not overlap and cover everyone.
  • Bar charts make it easier to compare categories; pie charts show how parts make up a whole.
  • To compare groups of different sizes, graph relative frequencies, and describe what the graph shows with numbers, in context.

Part 7 · Misconception

A common mistake

The wrong idea: School B’s bar is four times as tall, so its pass rate is four times school A’s.

What actually happens: The axis starts at 50%. The actual rates are 58% and 52%, a difference of 6 percentage points.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Favorite kind of film

A random sample of 200 teens in a county named their favorite kind of film. "Docs" stands for documentaries (invented data).
00.10.20.3Relative frequencyComedyActionDramaHorrorDocsOther

1. Which kind of film did 15% of the teens choose?

  1. Horror
  2. Drama
  3. Other
  4. Action
Show the answer

Read the height of each bar on the relative frequency axis: drama is at 0.15.

  • Horror: The horror bar reaches 0.12, so 12% chose horror.
  • Correct: Drama: Right: the drama bar reaches 0.15, which is 15%.
  • Other: The other bar reaches 0.10, so 10% chose another kind.
  • Action: The action bar reaches 0.25, so 25% chose action.

2. How many of the 200 teens chose action films?

  1. 25
  2. 60
  3. 800
  4. 50
Show the answer

Count = relative frequency × total = 0.25×200=500.25 \times 200 = 50.

  • 25: The bar shows a relative frequency, 0.25. Multiply by the total, 200.
  • 60: That is comedy: 0.30 × 200 = 60.
  • 800: A count from a relative frequency is a multiplication: 0.25 × 200.
  • Correct: 50: Right: 0.25×200=500.25 \times 200 = 50 teens.

3. Which claim does the bar chart support?

  1. Comedy was chosen about twice as often as drama
  2. Most teens chose comedy
  3. Action films are better liked than drama films by each teen
  4. More teens chose horror than chose documentaries and other together
Show the answer

Compare bar heights: comedy at 0.30 is twice drama at 0.15. "Most" would need more than 0.50.

  • Correct: Comedy was chosen about twice as often as drama: Right: 0.30 is twice 0.15.
  • Most teens chose comedy: Comedy is the most common choice, but 0.30 is less than half, so not most.
  • Action films are better liked than drama films by each teen: The chart shows how many teens chose each kind as a favorite, not each teen’s ranking of all kinds.
  • More teens chose horror than chose documentaries and other together: Horror is 0.12, and documentaries plus other is 0.08 + 0.10 = 0.18.

4. Could these data also be shown in a pie chart?

  1. No: a pie chart can show four categories or fewer
  2. No: pie charts are used for counts, and these are relative frequencies
  3. Yes: each teen chose one kind, so the six parts add to 1
  4. Yes, but just after the bars are sorted from tallest to shortest
Show the answer

A pie chart works when every individual is in exactly one category, so the parts add to the whole. Here they add to 1.

  • No: a pie chart can show four categories or fewer: A pie chart can have many slices; what it needs is parts that make up one whole.
  • No: pie charts are used for counts, and these are relative frequencies: A pie chart shows each category as a part of the whole, which is exactly a relative frequency.
  • Correct: Yes: each teen chose one kind, so the six parts add to 1: Right: 0.30 + 0.25 + 0.15 + 0.12 + 0.08 + 0.10 = 1, so the slices make one whole.
  • Yes, but just after the bars are sorted from tallest to shortest: Sorting is a design choice; the parts already make up one whole.

Diagram

A bar chart from a school report

A report shows the percent of students who passed a state exam at two schools: 52% at school A and 58% at school B. Its bar chart is below.

50%52%54%56%58%60%School ASchool BPercent passing

5. Why does this bar chart give a misleading picture?

  1. It uses different colors for the two schools’ bars
  2. It shows percents instead of numbers of students passing
  3. Its vertical axis starts at 50%, not at 0%
  4. It puts A on the left
Show the answer

A bar’s height (or length) must be its value, measured from 0. Starting the axis at 50% exaggerates the difference.

  • It uses different colors for the two schools’ bars: Colors help tell the bars apart; they do not change the heights.
  • It shows percents instead of numbers of students passing: Percents are the right choice for schools of different sizes.
  • Correct: Its vertical axis starts at 50%, not at 0%: Right: with the bottom cut off, bar heights no longer match the values they stand for.
  • It puts A on the left: The order of categories in a bar chart is a free choice.

6. In this chart, how many times as tall does school B’s bar look compared with school A’s?

  1. 4 times
  2. About 1.1 times
  3. 6 times
  4. The same height
Show the answer

The visible heights are 58−50=858 - 50 = 8 and 52−50=252 - 50 = 2, so B looks 82=4\frac{8}{2} = 4 times as tall, though 58% is only about 1.1 times 52%.

  • Correct: 4 times: Right: above 50%, school A’s bar is 2 points tall and school B’s is 8, a ratio of 4.
  • About 1.1 times: That is how the actual percents compare. The drawn bars start at 50, so they look 8/2 = 4 times as tall.
  • 6 times: 6 is the difference in percents, not the ratio of the drawn heights.
  • The same height: The bars are drawn to different heights: 2 and 8 points above the axis.

Part 9 · Summary

Summary

Bar charts and pie charts display one categorical variable. A bar’s height is its count or relative frequency measured from 0; a pie’s slices are parts of one whole, so every individual must be in exactly one category. Compare groups of different sizes with relative frequencies, and check axes before trusting a picture.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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