Topic 1.3 Representing a Categorical Variable with Tables: build and interpret frequency and relative frequency tables for a single categorical variable, and read proportions and percentages from them.

What this topic is asking

The College Board (Topic 1.3) wants you to organize a single categorical variable into a frequency table (counts) and a relative frequency table (proportions or percentages), to move fluently between counts and proportions, and to interpret what the table says in context.

Frequency tables

For example, surveying $50$ people on their favorite fruit might give: apple $18$, banana $14$, orange $12$, other $6$. The counts sum to $18 + 14 + 12 + 6 = 50$, confirming every individual is accounted for once.

Relative frequency tables

From the fruit data: apple $18/50 = 0.36$, banana $14/50 = 0.28$, orange $12/50 = 0.24$, other $6/50 = 0.12$. These sum to $0.36 + 0.28 + 0.24 + 0.12 = 1.00$.

Why relative frequencies matter for comparison

Counts answer "how many," but they are misleading when groups have different sizes. If School A surveys $400$ students and School B surveys $120$, a raw count of "car users" will almost always be larger at School A simply because it surveyed more people. Converting to relative frequencies removes the effect of total size: comparing "$40\%$ at A versus $35\%$ at B" is fair in a way that "$160$ versus $42$" is not. This is the single most important reason the exam keeps asking for relative frequencies, and it returns in Unit 2 when you compare conditional distributions in two-way tables. Whenever a question asks you to compare two groups of unequal size, your instinct should be to switch to proportions or percentages. A related discipline is to always report the total $n$ alongside the proportions, because a percentage from a tiny sample is far less trustworthy than the same percentage from a large one, and stating the $n$ shows the reader the basis for your figures.

Reading and interpreting in context

A table is only useful if you can translate it back into a sentence about the situation. The exam rewards interpretations that name the proportion or percentage, the category, and the group it refers to. "Banana: $0.28$" by itself earns little; "$28\%$ of the $50$ people surveyed chose banana as their favorite fruit" earns full credit because it ties the number to its meaning. When you build a relative frequency table under exam conditions, lay it out clearly with a column for category, a column for count, and a column for relative frequency, and write the total row so the marker can see the proportions sum to one. Rounding sensibly (two or three decimal places, or whole percentages) keeps the table readable, but keep enough precision that the proportions still add to one or to within a rounding whisker of it, and say so if rounding makes them sum to, say, $1.01$.

Try this

Q1. In a class of $40$, $16$ prefer math. State the relative frequency as a percentage. [1 point]

• Cue. $\frac{16}{40} = 0.40 = 40\%$.

Q2. Explain why relative frequencies are better than counts for comparing two surveys of different sizes. [2 points]

• Cue. Relative frequencies divide by each survey's own total, putting both on a per-total scale, so differences in sample size do not distort the comparison.