AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-3)
Unit 1 · Topic 1.3
1.3 Tabular Representation and Summary Statistics for One Categorical Variable
For one categorical variable, the basic summary is a table that counts how many units fall in each category. Turning counts into proportions lets you compare groups of different sizes and back up claims with numbers.
Key terms
- frequency table
- relative frequency
- proportion
- percentage
Frequency tables
A frequency table lists each category and how many observational units fall in it. The counts are called frequencies, and they add up to the total number of units, n.
For example, 80 students named their favorite school lunch: pizza 28, tacos 20, salad 12, sandwich 20. That's a frequency table, and 28 + 20 + 12 + 20 = 80.
Relative frequency tables
A relative frequency is a category's share of the total: its count divided by n. Relative frequencies are always between 0 and 1 and add up to 1 (allowing for rounding).
Proportions, percentages and relative frequencies are three ways of saying the same thing. 28 out of 80 is a proportion of 0.35, a relative frequency of 0.35 and 35%.
| Lunch | Frequency | Relative frequency |
|---|---|---|
| Pizza | 28 | 28/80 = 0.35 |
| Tacos | 20 | 20/80 = 0.25 |
| Salad | 12 | 12/80 = 0.15 |
| Sandwich | 20 | 20/80 = 0.25 |
| Total | 80 | 1.00 |
Why proportions matter
Counts are fine for describing one group. As soon as you compare groups of different sizes, you need proportions. If 30 students at a small school and 45 at a large school walk to school, you can't say walking is more common at the large school until you divide by each school's total.
Ratios also carry the same information. "Pizza was chosen 7 times for every 3 salad choices" (28 : 12 reduces to 7 : 3) says something about the relative sizes of two categories.
Using counts and proportions to support a claim
A claim about a categorical variable needs a number behind it. "Pizza is the most popular lunch" is supported by "35% of the 80 students chose pizza, more than any other option."
Be careful with words like "most" and "majority." Pizza is the most common choice, but 35% isn't a majority. A majority means more than 50%.
Reading a table carefully
Check what the total is before you trust any percentage. "60% chose pizza" means something very different if 5 students answered than if 500 did. A good table shows the counts and the total along with the proportions.
Rounded percentages may add to 99% or 101%. That's fine, as long as the unrounded values add to 1. If they're far off, a category is missing or a count was copied wrong.
Watch for an "other" category. If it's large, the table may be hiding categories that matter for your claim.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Build a relative frequency table
A shelter records the type of each of 150 animals adopted last month: 66 dogs, 57 cats, 18 rabbits and 9 other. Make a relative frequency table and state what proportion of adoptions were not dogs or cats.
Show the solutionHide the solution
- Step 1: Divide each count by the total, 150.
- Step 2: Dogs: 66/150 = 0.44. Cats: 57/150 = 0.38. Rabbits: 18/150 = 0.12. Other: 9/150 = 0.06.
- Step 3: Check: 0.44 + 0.38 + 0.12 + 0.06 = 1.00.
- Step 4: Not dogs or cats means rabbits or other: 0.12 + 0.06 = 0.18.
Answer: Dogs 0.44, cats 0.38, rabbits 0.12, other 0.06. About 18% of last month's adoptions were animals other than dogs and cats.
- Example 2Calculator allowed
Trap: comparing counts from groups of different sizes
At School A, 90 of 200 surveyed students ride the bus. At School B, 15 of 50 surveyed students ride the bus. A student says bus riding is six times as common at School A. Is that right?
Show the solutionHide the solution
- Step 1: 90 is six times 15, but the schools surveyed different numbers of students, so the counts can't be compared directly.
- Step 2: Convert to proportions: School A 90/200 = 0.45; School B 15/50 = 0.30.
- Step 3: Bus riding is more common at School A, but by a factor of 0.45/0.30 = 1.5, not 6.
Answer: No. 45% of School A's sample rides the bus, compared with 30% at School B. That's 1.5 times as common, not six times.
Common mistakes
- Comparing raw counts between groups of different sizes. Convert to proportions first.
- Calling the largest category a "majority" when it's under 50%.
- Forgetting to check that relative frequencies add to 1 (or to 100%). If they don't, there's an arithmetic error or a missing category.
On the exam
- Questions often give a table of counts and ask which statement is supported. Convert to proportions before choosing.
- When you justify a claim, quote the proportion and the group it comes from, in context.
Connected topics
Videos
Check yourself
3 questions on 1.3 Tabular Representation and Summary Statistics for One Categorical Variable. Pick an answer to see if you got it, and why.
In a survey of 640 adults about how they get to work, 15% walk, 55% drive alone, 12% carpool, and the rest take public transit. How many of the surveyed adults take public transit?
| Lunch choice | Number of students |
|---|---|
| Pizza | 70 |
| Tacos | 48 |
| Salad | 34 |
| Sandwich | 28 |
| Pasta | 20 |
Invented data: a survey of 200 students about their favorite cafeteria lunch
What proportion of the surveyed students chose pizza?
What is the difference between the proportion of surveyed students who chose tacos and the proportion who chose salad?
0 of 3 answered