Unit 1 · Topic 1.3 Beta

Tabular Representation and Summary Statistics for One Categorical Variable

A frequency table counts each category; a relative frequency divides each count by the total, so relative frequencies are between 0 and 1, add to 1, and compare groups of different sizes.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

A city newspaper asks 160 adults where they get their news. Fifty-six say social media and 40 say television. A second city asks 500 adults and 130 say television. Which city leans more on television? The counts 40 and 130 cannot answer it, because the samples are different sizes. Dividing each count by its total can.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is 40 out of 160 as a decimal?

0.25
0.4
4
0.04

Show the answer

Part over whole: 40/160 = 0.25.

Correct: 0.25:
0.4:
4:
0.04:

2. A survey records each adult’s main news source. What kind of variable is that?

Categorical
Quantitative and discrete
Quantitative and continuous
A statistic

Show the answer

Each adult is placed in a group (a source), so the variable is categorical.

Correct: Categorical:
Quantitative and discrete:
Quantitative and continuous:
A statistic:

Part 4 · See it

See it first

Left: 250 students drawn as dots, one dot for every 5 students, in five rows: Bus 95, Car 80, Walk 45, Bike 20, Other 10. Right: the frequency table with each count over 250: Bus 0.38, Car 0.32, Walk 0.18, Bike 0.08, Other 0.04, total 1.00. — Counting students in each category gives the frequencies; dividing each by the total, 250, gives the relative frequencies, which add to 1. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Each individual falls in one category of a categorical variablecounting each category gives a frequency table
Divide each count by the totalthe relative frequencies are parts of the whole, between 0 and 1, and they add to 1
Relative frequencies all divide by their own totalssamples of different sizes can be compared fairly
The relative frequency of a category in a sample is $\hat{p}$ it is a statistic that estimates p, the proportion in the whole population

Part 6 · Key ideas

Key ideas

A frequency table lists each category of a categorical variable and its count (frequency).
A relative frequency is a count divided by the total. Relative frequencies are between 0 and 1 and add to 1 when categories do not overlap.
Count = relative frequency × total, so a relative frequency table needs the total to get back to counts.
The sample proportion $\hat{p}$ is the relative frequency of a category in a sample, a statistic. The population proportion $p$ is the same quantity for the whole population, a parameter.

Part 7 · Misconception

A common mistake

The wrong idea: The second city has 130 television viewers against 40, so a larger share of its adults watch television.

What actually happens: Compare relative frequencies: $\frac{40}{160} = 0.25$ and $\frac{130}{500} = 0.26$ . The shares are almost the same; the second sample is simply bigger.

Part 8 · Check yourself

Check yourself

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

Table

Where adults get their news

A random sample of 160 adults in a city were asked for their main source of news (invented data).

Main news source of 160 adults
News source	Frequency
Social media	56
Television	40
News websites	36
Radio	16
Print newspaper	12
Total	160

1. What is the relative frequency of adults whose main source is television?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

$\frac{40}{160} = 0.25$ : a quarter of the sampled adults get their news mainly from television.

Answer: 0.25

2. What proportion of the sampled adults get their news mainly from social media or television?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

$\frac{56 + 40}{160}$ $= \frac{96}{160}$ $= 0.60$ .

Answer: 0.6

3. Which statement about the relative frequencies of the five news sources is true?

They add to 160, the number of adults who were asked
They add to 1, because each adult is in exactly one category
They are equal to one another, since each adult chose a single source
They add to 100, because each one is a percent of the sample

Show the answer

When categories do not overlap and cover everyone, the relative frequencies add to 1 (or 100%).

They add to 160, the number of adults who were asked: The frequencies add to 160. Each relative frequency is a count over 160, so together they add to 1.
Correct: They add to 1, because each adult is in exactly one category: Right: every adult chose one main source, so the five parts make up the whole sample.
They are equal to one another, since each adult chose a single source: Each adult chose one source, but the sources were chosen different numbers of times.
They add to 100, because each one is a percent of the sample: As percents they add to 100%. As relative frequencies (decimals) they add to 1.

4. For print newspapers, $\hat{p} = \frac{12}{160} = 0.075$ . What does $\hat{p}$ describe?

The city’s adults, as a population proportion
The 12 adults
The news sources that the city’s adults can choose from
The 160 adults in the sample

Show the answer

A sample proportion $\hat{p}$ is the part of the sample in a category. It is a statistic, used to estimate the population proportion p.

The city’s adults, as a population proportion: That would be p, the population proportion, which is unknown here.
The 12 adults: The 12 are the part; $\hat{p}$ compares them with the whole sample of 160.
The news sources that the city’s adults can choose from: The sources are the categories; $\hat{p}$ is a number about adults in one category.
Correct: The 160 adults in the sample: Right: $\hat{p}$ is a sample proportion, a statistic. It estimates p for the city.

5. A gym asked 60 members which class they like best: 12 chose yoga, 18 chose cycling and 30 chose weights. What is the relative frequency of cycling?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

$\frac{18}{60} = 0.30$ , so 30% of the members asked chose cycling.

Answer: 0.3

6. A survey asked 200 students, "Which of these apps do you use? Check every one you use." The relative frequencies add to 1.85. What explains this?

Someone added the relative frequencies incorrectly
The survey asked more students than it reported in the total
Students could check more than one app, so categories overlap
Relative frequencies are supposed to add to 2 for a survey

Show the answer

Relative frequencies add to 1 only when each individual falls in exactly one category. With "check every one", students fall in several.

Someone added the relative frequencies incorrectly: The sum is correct; it is over 1 because the categories overlap.
The survey asked more students than it reported in the total: The total is 200; overlap explains the sum.
Correct: Students could check more than one app, so categories overlap: Right: one student can be counted in several categories, so the parts add to more than the whole.
Relative frequencies are supposed to add to 2 for a survey: They add to 1 only when each individual is in exactly one category.

Part 9 · Summary

Summary

A frequency table counts each category; a relative frequency divides each count by the total, so relative frequencies are between 0 and 1, add to 1, and compare groups of different sizes. In a sample, the relative frequency of a category is the sample proportion

\hat{p}

, a statistic that estimates the population proportion

p

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic