Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A categorical variable places each individual in exactly one category.
- A frequency table gives the count of individuals in each category.
- A relative frequency table gives the proportion in each category.
- Proportions, relative frequencies, percentages and ratios carry the same information in different forms.
- Summarise a categorical variable with counts, proportions and the most common category, never a mean or median.
- Counts and proportions can justify a claim in context, if the claim’s wording matches the numbers.
Key relationships
| Idea | How to find or check it |
|---|---|
| Frequency (count) | number of individuals in the category; all counts add to n |
| Relative frequency (proportion) | count ÷ n; between 0 and 1; all add to 1 |
| Percentage | proportion × 100; all add to 100% (allow small rounding differences) |
| Count from a proportion | proportion × n (you must know n) |
| Group of categories | add their proportions (categories do not overlap) |
| “Not” a category | 1 − proportion in that category |
| Part-to-whole ratio | “a out of n” = a ÷ n |
| Part-to-rest ratio | a : (n − a); turn back with a ÷ (a + (n − a)) |
| “A majority” | proportion > 0.5 |
| “Most common” | largest count or proportion, which can be below 0.5 |
Assumptions and conventions
- Each individual is counted in exactly one category. If people could choose several answers, the table is not the distribution of one categorical variable.
- List categories from largest to smallest count, unless they have a natural order.
- Round proportions to 3 decimal places (or percentages to 1 decimal place) unless exact. Say when a total differs from 1 because of rounding.
- A table from a survey describes that sample. Its proportions are statistics, not certain facts about everyone.
Mistakes to avoid
- Dividing by 100 instead of n to get a relative frequency.
- Forgetting to check that counts add to n and proportions add to 1.
- Calling the most common category a majority when its share is 50% or less.
- Comparing counts from groups of different sizes. Compare proportions instead.
- Mixing up ratio types: 9 : 16 (part to rest) is not 9 out of 16.
- Inventing a count from a percentage when n is not given.
- No context: name the category, the individuals and the group surveyed.
Quick self-check
- In a fictional survey, 18 of 45 shoppers paid by card. What is the relative frequency of “card”? (18 ÷ 45 = 0.40)
- A relative frequency table shows 0.15 for “Late” and n = 320 trains. How many trains were late? (0.15 × 320 = 48 trains)
- The ratio of “Yes” to “No” answers is 2 : 3, and there are no other answers. What proportion said “Yes”? (2 ÷ (2 + 3) = 0.40)
Next: practice questions.