FREQUENCY TABLES, RELATIVE FREQUENCY, TWO-WAY TABLES AND BAR CHARTS IN AP STATISTICS
The Box-Divide-Say Method: How to Read Categorical Data, Calculate Conditional Percentages and Identify Association (US and UK Statistics Guide)
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THE LIBRARY THAT LOOKED EQUALLY POPULAR
A school surveys 200 students about where they usually do their homework.
The results show:
24 younger students study in the school library.
22 older students study in the school library.
At first glance, the difference seems insignificant. The library appears almost equally popular among both groups.
But there is a problem.
The school surveyed 120 younger students and only 80 older students.
Now compare the percentages:
Younger students:
24 ÷ 120 = 0.20 = 20%
Older students:
22 ÷ 80 = 0.275 = 27.5%
The older students in this sample are 1.375 times as likely to choose the library as the younger students.
The original comparison was misleading because it compared counts from groups of different sizes.
This is one of the most important ideas in introductory statistics:
When comparing groups of different sizes, compare proportions rather than relying on raw counts.
Frequency tables, relative frequency tables, two-way tables, conditional distributions, bar charts, segmented bar charts and mosaic plots all help us understand categorical data.
This guide develops these ideas using a simple method called Box-Divide-Say. It also explains how to recognize association, avoid misleading statistical comparisons and write stronger AP Statistics examination answers.
The examples use original fictional datasets created for this guide.
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THE ESSENTIAL STATISTICS VOCABULARY
Before calculating percentages, understand these terms.
Individual: The person or object being studied. In our example, each student is an individual.
Variable: A characteristic recorded for each individual.
Categorical variable: A variable that places individuals into groups, such as preferred subject, transport method or study location.
Quantitative variable: A numerical variable that measures or counts something, such as age, height or number of books read.
Category: One possible group or value of a categorical variable.
Frequency: The number of individuals belonging to a particular category.
Relative frequency: The frequency divided by the total number of individuals, expressed as a proportion or percentage.
Frequency distribution: A summary showing how observations are distributed across categories or values.
Frequency table: A table listing categories and their corresponding counts.
Relative frequency table: A table showing the proportion or percentage belonging to each category.
Two-way table: A table that classifies the same individuals according to two categorical variables.
Contingency table: Another name for a two-way table.
Joint relative frequency: The proportion of the entire sample belonging to a particular combination of two categories.
Marginal relative frequency: The proportion belonging to a category of one variable, calculated using a row or column total.
Conditional relative frequency: The proportion belonging to a category within a specified group.
Conditional distribution: The complete distribution of one categorical variable within a particular category of another variable.
Association: A relationship between two categorical variables in which the distribution of one variable differs across the categories of the other.
Segmented bar chart: A chart in which each bar represents a group and is divided into segments showing the percentages of different categories.
Mosaic plot: A graphical display of two categorical variables in which rectangle areas represent frequencies or proportions.
One distinction is particularly important:
A frequency is a count. A relative frequency is a proportion. A conditional relative frequency is a proportion calculated within a specified group.
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PART 1: FREQUENCY TABLES AND CATEGORICAL DATA
What is a frequency table in statistics?
A frequency table organizes observations into categories and records how many individuals belong to each category.
Suppose a school surveys 200 students about where they usually complete their homework.
The results are:
Study location | Frequency
Home | 104
School library | 46
Café | 30
Friend's house | 20
TOTAL | 200
This is a frequency distribution for one categorical variable.
The variable is study location, and its four categories are Home, School Library, Café and Friend's House.
The table immediately tells us that studying at home is the most common response.
However, a reliable frequency table must satisfy three conditions.
1. Every observation must be counted.
The frequencies must add up to the total sample size.
104 + 46 + 30 + 20 = 200
2. Categories must be mutually exclusive.
An individual should not be counted in two categories when the question requires one response.
3. Categories must cover all possible responses.
Every student must have a suitable category.
These conditions prevent missing observations, overlapping categories and double counting.
Frequency tables for quantitative data
Frequency tables are not restricted to categorical variables.
They can also summarize quantitative data when the number of distinct values is manageable.
Suppose 25 students report how many books they read last month.
Books read | Frequency | Cumulative frequency
0 | 3 | 3
1 | 7 | 10
2 | 8 | 18
3 | 5 | 23
4 | 2 | 25
Cumulative frequency is the running total of observations.
For example:
The number of students who read two books or fewer is:
3 + 7 + 8 = 18
Therefore, the cumulative frequency at two books is 18.
When quantitative data contain many possible values, observations are often grouped into class intervals. Histograms and grouped frequency distributions are then useful.
Remember:
A categorical frequency table summarizes categories. A quantitative frequency table summarizes numerical values or intervals.
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PART 2: RELATIVE FREQUENCY TABLES AND PERCENTAGES
What is relative frequency?
Relative frequency tells us what fraction of the entire sample belongs to a particular category.
The formula is:
RELATIVE FREQUENCY = FREQUENCY ÷ TOTAL FREQUENCY
To convert a relative frequency into a percentage:
PERCENTAGE = RELATIVE FREQUENCY × 100
Return to the homework survey.
| Study location | Frequency | Relative frequency |
|---|---|---|
| Home | 104 | 0.52 |
| School library | 46 | 0.23 |
| Café | 30 | 0.15 |
| Friend's house | 20 | 0.10 |
| Total | 200 | 1.00 |
Now convert each proportion into a percentage.
Home:
104 ÷ 200 = 0.52 = 52%
School library:
46 ÷ 200 = 0.23 = 23%
Café:
30 ÷ 200 = 0.15 = 15%
Friend's house:
20 ÷ 200 = 0.10 = 10%
The relative frequency distribution is:
Home: 52%
School library: 23%
Café: 15%
Friend's house: 10%
The percentages add to 100%.
This is the first important checking rule:
The relative frequencies for all categories of one complete variable should add to 1, or 100%.
Small differences caused by rounding are acceptable.
Why relative frequency is more useful than frequency when comparing samples
Imagine another school surveys 500 students.
Its results show that 240 students study at home.
School A:
104 ÷ 200 = 52%
School B:
240 ÷ 500 = 48%
School B has more students studying at home in absolute numbers.
However, School A has a higher proportion of students choosing home study.
The difference matters because the sample sizes are unequal.
This is why relative frequency is essential when comparing survey results, population groups, customer preferences and educational statistics.
Relative frequency versus percentage
These expressions describe the same information in different forms.
0.25 = 25%
0.60 = 60%
0.075 = 7.5%
A decimal proportion is useful in calculations, while a percentage is often easier to interpret in written answers.
Do not confuse relative frequency with frequency density. Frequency density is used when constructing histograms, particularly when class intervals have unequal widths.
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PART 3: BAR CHARTS FOR CATEGORICAL DATA
What is a bar chart in statistics?
A bar chart is a graphical representation of categorical data.
Each category is represented by a separate bar.
The height of the bar represents its frequency or relative frequency.
For our homework survey:
Home: 104 students
Library: 46 students
Café: 30 students
Friend's house: 20 students
A frequency bar chart would use these counts on the vertical axis.
A relative frequency bar chart would use 52%, 23%, 15% and 10%.
Both are valid representations of the same sample.
How to construct a bar chart correctly
Follow these rules:
Give the graph a descriptive title.
Label both axes.
Identify whether the vertical axis represents frequency or relative frequency.
Start the vertical axis at zero for an ordinary bar chart.
Keep bar widths equal.
Leave gaps between categories.
Use an appropriate ordering of categories.
For nominal categories, you can use a meaningful order, alphabetical order or descending frequency order.
For ordinal categories, preserve their natural order.
For example, satisfaction ratings should normally appear in this sequence:
Very dissatisfied → Dissatisfied → Neutral → Satisfied → Very satisfied.
Bar chart versus histogram: What is the difference?
This distinction is frequently tested in introductory statistics.
| Bar chart | Histogram |
|---|---|
| Represents categorical data | Represents quantitative data grouped into intervals |
| Bars are separated by gaps | Bars normally touch |
| Each bar represents a category | Each bar represents an interval |
| Bar height represents frequency or relative frequency | Frequency or frequency density may be represented, depending on the vertical scale |
Example of a bar chart:
Favourite subjects: Mathematics, Science, English and History.
Example of a histogram:
Students' examination scores grouped into intervals of 0–20, 20–40, 40–60 and so on.
For a histogram using frequency density:
Frequency density = Frequency ÷ Class width
When class widths differ, the area of each histogram bar represents frequency if the vertical axis is frequency density.
This is particularly important in UK GCSE and A-level statistics questions.
Common ways bar charts mislead readers
A graph can be mathematically constructed yet communicate a misleading impression.
Watch for:
Truncated vertical axes: Starting above zero exaggerates differences in bar heights.
Three-dimensional effects: Perspective can distort apparent bar sizes.
Unequal bar widths: Wider bars can attract attention even when their heights are similar.
Missing sample sizes: Percentages without the number of observations can conceal important information.
Comparing unequal groups using counts: A larger group can produce more responses even when its percentage is lower.
Always inspect the axis, units, categories and sample size before interpreting a statistical graph.
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PART 4: TWO-WAY TABLES AND CONTINGENCY TABLES
What is a two-way table?
A frequency table describes one variable.
A two-way table describes two categorical variables simultaneously.
It helps answer questions such as:
Do younger and older students prefer different study locations?
Is transport preference associated with age group?
Does product preference vary by customer category?
Are voting preferences distributed differently across demographic groups?
Let's reorganize our original survey.
There are 200 students:
120 younger students.
80 older students.
The study-location data are:
| Study location | Younger | Older | Total |
|---|---|---|---|
| Home | 78 | 26 | 104 |
| School library | 24 | 22 | 46 |
| Café | 6 | 24 | 30 |
| Friend's house | 12 | 8 | 20 |
| Total | 120 | 80 | 200 |
This is a two-way frequency table, also known as a contingency table or cross-tabulation.
Each interior cell represents individuals belonging to one category of each variable.
For example:
The number 24 in the Café row and Older column represents older students who study at cafés.
Understanding marginal totals and the grand total
The right-hand column contains the row totals.
These are called marginal totals.
The bottom row contains the column totals.
The number in the bottom-right corner is the grand total.
There are three useful ways to read this table:
Row totals: The number of students in each study-location category.
Column totals: The number of students in each age group.
Interior cells: The number of students belonging to both specified categories.
A useful checking rule:
Every row must add to its row total, every column must add to its column total, and the marginal totals must agree with the grand total.
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PART 5: JOINT, MARGINAL AND CONDITIONAL RELATIVE FREQUENCY
This is one of the most important sections for AP Statistics students.
All three types of relative frequency involve division, but they answer different questions.
1. Joint relative frequency
Joint relative frequency describes the proportion of the entire sample belonging to a particular combination of categories.
Formula:
JOINT RELATIVE FREQUENCY = CELL FREQUENCY ÷ GRAND TOTAL
Example:
How many students are both younger and café users?
There are 6.
Joint relative frequency:
6 ÷ 200 = 0.03 = 3%
Therefore, 3% of all surveyed students are younger students who study at cafés.
The denominator is the entire sample.
2. Marginal relative frequency
Marginal relative frequency describes the proportion belonging to one category of either variable.
Formula:
MARGINAL RELATIVE FREQUENCY = MARGINAL TOTAL ÷ GRAND TOTAL
Example:
What percentage of all students study at home?
104 ÷ 200 = 0.52 = 52%
Another example:
What percentage of students belong to the younger age group?
120 ÷ 200 = 0.60 = 60%
Again, the denominator is the grand total.
3. Conditional relative frequency
Conditional relative frequency describes the proportion within a specified group.
Formula:
CONDITIONAL RELATIVE FREQUENCY = CELL FREQUENCY ÷ TOTAL OF THE GIVEN GROUP
Example:
What percentage of older students study at cafés?
There are 24 older students who study at cafés.
There are 80 older students altogether.
24 ÷ 80 = 0.30 = 30%
Therefore, 30% of older students in the sample study at cafés.
Notice that the denominator is no longer 200.
It is 80 because the question restricts our attention to older students.
This is the distinction that causes many mistakes in two-way table examination questions.
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PART 6: THE BOX-DIVIDE-SAY METHOD
A simple way to avoid choosing the wrong denominator is to use three steps.
BOX
Identify the group specified in the question.
Look for phrases such as:
Of the older students
Among library users
Given that a student studies at home
Of all students who prefer cafés
Among people aged 40 and over
Mentally box the entire row or column representing that group.
DIVIDE
Find the required cell and divide it by the total of the boxed group.
SAY
Write the answer as a complete sentence that identifies the group, category and percentage.
This final step matters because a percentage without context may answer a different question.
Worked example 1: Percentage of café users who are older
Question:
What percentage of students who study at a café are older?
BOX:
The specified group is café users.
The Café row total is 30.
DIVIDE:
24 ÷ 30 = 0.80 = 80%
SAY:
Of all students who study at a café, 80% are older students.
Worked example 2: Percentage of older students who study at cafés
Question:
What percentage of older students study at cafés?
BOX:
The specified group is older students.
The Older column total is 80.
DIVIDE:
24 ÷ 80 = 0.30 = 30%
SAY:
Of the older students, 30% study at cafés.
Why are the answers different?
Both questions use the same cell: 24.
But their denominators are different.
24 ÷ 30 = 80%
24 ÷ 80 = 30%
Neither calculation is wrong.
They answer different statistical questions.
This is the most important habit to develop when solving conditional relative frequency problems.
Never choose the denominator before identifying the group named in the question.
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PART 7: CONDITIONAL DISTRIBUTIONS AND ASSOCIATION
What is a conditional distribution?
A conditional distribution shows the complete distribution of one categorical variable within a particular category of another variable.
For example, we want to compare study-location preferences among younger and older students.
We must calculate percentages using each age group's own total.
Younger students: Total = 120
Home:
78 ÷ 120 = 65%
School library:
24 ÷ 120 = 20%
Café:
6 ÷ 120 = 5%
Friend's house:
12 ÷ 120 = 10%
Total = 100%
Older students: Total = 80
Home:
26 ÷ 80 = 32.5%
School library:
22 ÷ 80 = 27.5%
Café:
24 ÷ 80 = 30%
Friend's house:
8 ÷ 80 = 10%
Total = 100%
Now the two groups can be compared fairly.
| Study location | Younger | Older |
|---|---|---|
| Home | 65% | 32.5% |
| Library | 20% | 27.5% |
| Café | 5% | 30% |
| Friend's house | 10% | 10% |
| Total | 100% | 100% |
Each column represents a conditional distribution.
How do you identify association between two categorical variables?
Two categorical variables are associated when the conditional distribution of one variable differs across the categories of the other variable.
In our example:
65% of younger students study at home, compared with 32.5% of older students.
5% of younger students study at cafés, compared with 30% of older students.
20% of younger students use the library, compared with 27.5% of older students.
These differences suggest an association between age group and study location in the sample.
A strong statistical conclusion would be:
"Study location and age group appear to be associated in this sample. Older students are more likely to study at cafés than younger students, with percentages of 30% and 5%, respectively."
What if the conditional percentages are nearly identical?
Imagine that the conditional distributions were:
| Study location | Younger | Older |
|---|---|---|
| Home | 52% | 52% |
| Library | 23% | 23% |
| Café | 15% | 15% |
| Friend's house | 10% | 10% |
The distributions are identical.
This would indicate no association between the two categorical variables in the observed sample.
In real datasets, distributions may be approximately rather than exactly equal.
Remember:
Association is about comparing conditional distributions, not simply comparing raw frequencies.
Association does not prove causation
Suppose older students are more likely to study in cafés.
Does being older cause students to choose cafés?
Not necessarily.
Other factors might include:
Greater independence.
Different school schedules.
Access to transport.
Different study habits.
Availability of money.
A two-way table identifies a pattern of association. It does not establish the cause of that pattern.
This distinction is essential in statistical interpretation.
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PART 8: COMPLETING A TWO-WAY TABLE WITH MISSING VALUES
AP Statistics and other school-level statistics courses may ask students to complete an incomplete frequency table.
The principle is straightforward:
Every row and column must add to its corresponding total.
Consider a survey of 100 students.
| Club membership | Walks | Rides | Total |
|---|---|---|---|
| Has club | 14 | ? | 40 |
| No club | ? | 22 | ? |
| Total | ? | ? | 100 |
Step 1: Complete the first row.
14 + ? = 40
Missing value = 26.
Step 2: Calculate the second row total.
100 − 40 = 60.
Step 3: Complete the second row.
? + 22 = 60
Missing value = 38.
Step 4: Calculate column totals.
Walks:
14 + 38 = 52.
Rides:
26 + 22 = 48.
Completed table:
| Club membership | Walks | Rides | Total |
|---|---|---|---|
| Has club | 14 | 26 | 40 |
| No club | 38 | 22 | 60 |
| Total | 52 | 48 | 100 |
Final check:
52 + 48 = 100.
All row totals, column totals and the grand total agree.
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PART 9: SEGMENTED BAR CHARTS AND 100% STACKED BAR CHARTS
What is a segmented bar chart?
A segmented bar chart displays two categorical variables simultaneously.
Each bar represents one category of a selected grouping variable.
The bar is divided into segments representing the categories of the second variable.
Every bar has the same total length: 100%.
This makes segmented bar charts particularly useful for comparing conditional distributions.
They are also called 100% stacked bar charts.
How to construct a segmented bar chart
Using our homework survey:
Step 1: Choose the variable that defines the bars.
We want to compare younger and older students, so age group determines the bars.
Step 2: Calculate the conditional percentages for each age group.
Younger:
Home 65%, Library 20%, Café 5%, Friend's house 10%.
Older:
Home 32.5%, Library 27.5%, Café 30%, Friend's house 10%.
Step 3: Draw two equal-length bars.
Each bar represents 100%.
Step 4: Divide each bar into segments according to the percentages.
Step 5: Use consistent colours or shading for the same categories.
Step 6: Add a title, percentage axis and legend.
Interpreting the segmented bar chart
Younger students:
Home 65% | Library 20% | Café 5% | Friend's house 10%
Older students:
Home 32.5% | Library 27.5% | Café 30% | Friend's house 10%
The Home segment is considerably larger for younger students.
The Café segment is considerably larger for older students.
The Friend's house segment is identical.
The overall distribution changes substantially between the two groups.
This is visual evidence of association.
The Line-Up Test for association
Here is a quick way to remember how segmented bar charts help identify association.
Imagine drawing horizontal lines at the boundaries between categories.
If the boundaries occur at approximately the same percentages in each bar, the conditional distributions are similar.
If the boundaries shift considerably, the conditional distributions differ.
In our example, the end of the Home segment occurs at:
Younger: 65%
Older: 32.5%
That is a substantial difference.
The boundaries therefore do not line up.
The variables appear to be associated.
Important: A single matching segment does not mean that the variables are independent. Compare the entire conditional distributions.
Segmented bar chart versus side-by-side bar chart
Both graphs can display two categorical variables, but they emphasize different comparisons.
A side-by-side bar chart places separate bars next to each other for each combination of categories.
A segmented bar chart places the categories inside equal-length bars representing groups.
For unequal group sizes, segmented bars are particularly useful because they emphasize conditional percentages rather than raw counts.
The AP Statistics framework also includes mosaic plots as a graphical method for displaying two categorical variables.
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PART 10: MOSAIC PLOTS AND TWO-WAY CATEGORICAL DATA
What is a mosaic plot in statistics?
A mosaic plot is another graphical representation of the relationship between two categorical variables.
Instead of using equal-length bars, it uses rectangles.
The width of each section represents the relative frequency of a category of the first variable.
The subdivisions within each section represent the conditional distribution of the second variable.
The area of each rectangle is proportional to the frequency or proportion of the corresponding combination of categories.
This is an important distinction.
A segmented bar chart gives every group an equal-width bar.
A mosaic plot can use different widths to represent different group sizes.
When should you use a mosaic plot?
A mosaic plot is useful when you want to display:
The relative sizes of different groups.
The distribution of categories within each group.
Differences in conditional percentages.
The relationship between two categorical variables.
For example, if one age group contains 120 students and another contains 80, a mosaic plot can represent their different sizes as well as their study-location distributions.
Segmented bar chart versus mosaic plot
| Feature | Segmented bar chart | Mosaic plot |
|---|---|---|
| Group widths | Equal | May differ |
| Conditional distributions | Shown by segment lengths | Shown by subdivisions |
| Group size | Not represented by bar width | Represented by width |
| Main purpose | Compare percentages across groups | Display group sizes and category proportions |
Both graphs can help identify association.
When interpreting either graph, remember to consider the actual percentages and sample sizes rather than relying entirely on visual appearance.
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PART 11: HOW TO WRITE STRONG AP STATISTICS ANSWERS
A correct calculation is not always enough.
Statistical questions frequently require students to describe, compare and interpret results in context.
Use this three-part structure:
CONTEXT + NUMBERS + COMPARISON
1. Context
Name the variables and groups.
Avoid vague phrases such as "the first group" or "the second category".
2. Numbers
Include the relevant percentages or frequencies.
Identify the correct denominator when it helps clarify the comparison.
3. Comparison
Explain how the distributions differ.
Use phrases such as:
Higher than.
Lower than.
Approximately twice as high.
Similar proportions.
Noticeably different distributions.
Model AP Statistics answer
"Study location and age group appear to be associated in this sample. Thirty percent of older students study at cafés, compared with only 5% of younger students. Younger students are also more likely to study at home, with 65% compared with 32.5% of older students."
This answer includes context, numerical evidence and a comparison.
A weaker answer would be:
"Older students prefer cafés."
That statement does not provide enough statistical evidence to demonstrate the difference.
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PART 12: THE MOST COMMON EXAMINATION MISTAKES
Mistake 1: Using the grand total for a conditional percentage.
If the question says "of older students", divide by the older-student total, not the grand total.
Mistake 2: Reversing the condition.
"Of café users, what percentage are older?" is different from "Of older students, what percentage use cafés?"
Mistake 3: Comparing raw counts from unequal groups.
Compare relative frequencies when group sizes differ.
Mistake 4: Confusing joint and conditional relative frequencies.
Joint uses the grand total. Conditional uses the total of the specified group.
Mistake 5: Confusing a bar chart with a histogram.
Categorical data use bar charts. Quantitative data grouped into intervals use histograms.
Mistake 6: Forgetting to check totals.
Frequency totals must agree with the sample size.
Mistake 7: Claiming causation from association.
A difference between conditional distributions does not prove that one variable causes the other.
Mistake 8: Ignoring sample size.
A percentage based on 10 observations should not automatically be treated as equally informative as the same percentage based on 10,000 observations.
Mistake 9: Misreading a segmented bar chart.
Each bar represents a conditional distribution and should total 100%.
Mistake 10: Writing an answer without context.
A statistical conclusion should identify the variables, groups and numerical evidence.
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PART 13: AP STATISTICS PRACTICE QUESTIONS WITH ANSWERS
Consider a survey of 150 adults about their preferred book format.
There are 70 adults under 40 and 80 adults aged 40 or over.
The results are:
| Book format | Under 40 | 40 and over | Total |
|---|---|---|---|
| 18 | 46 | 64 | |
| E-book | 30 | 22 | 52 |
| Audiobook | 22 | 12 | 34 |
| Total | 70 | 80 | 150 |
Use this table to answer the following questions.
Question 1
What percentage of all surveyed adults prefer e-books?
Question 2
What percentage of adults under 40 prefer audiobooks?
Question 3
What percentage of audiobook users are aged 40 or over?
Question 4
What percentage of all surveyed adults are aged 40 or over and prefer printed books?
Question 5
Calculate the conditional distribution of book-format preference for each age group.
Question 6
Is there an association between age group and book-format preference? Explain using numerical evidence.
Question 7
A student says:
"12 of the 80 older adults prefer audiobooks, but 12 out of 34 audiobook users are older. Therefore, 12 ÷ 34 = 35.3% of older adults prefer audiobooks."
Identify the error and calculate the correct percentage.
Question 8
A bar chart displays 92 library visitors in March and 96 in April. However, the vertical axis begins at 90.
Explain why the chart could be misleading.
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FULL ANSWERS
Answer 1: Marginal relative frequency
E-book users = 52.
Total adults = 150.
52 ÷ 150 = 0.3467.
Answer: Approximately 34.7%.
Answer 2: Conditional relative frequency
The specified group is adults under 40.
Their total is 70.
Audiobook users = 22.
22 ÷ 70 = 0.3143.
Answer: Approximately 31.4%.
Answer 3: Conditional relative frequency
The specified group is audiobook users.
Their total is 34.
Older audiobook users = 12.
12 ÷ 34 = 0.3529.
Answer: Approximately 35.3%.
Answer 4: Joint relative frequency
Older adults who prefer print = 46.
Total adults = 150.
46 ÷ 150 = 0.3067.
Answer: Approximately 30.7%.
Answer 5: Conditional distributions
Under 40, total = 70.
Print:
18 ÷ 70 = 25.7%
E-book:
30 ÷ 70 = 42.9%
Audiobook:
22 ÷ 70 = 31.4%
Total = 100%.
Aged 40 or over, total = 80.
Print:
46 ÷ 80 = 57.5%
E-book:
22 ÷ 80 = 27.5%
Audiobook:
12 ÷ 80 = 15%
Total = 100%.
Answer 6: Association
Yes, the conditional distributions differ.
For example, 57.5% of adults aged 40 or over prefer print books, compared with approximately 25.7% of adults under 40.
E-books are preferred by approximately 42.9% of younger adults and 27.5% of older adults.
The distributions are noticeably different, suggesting an association between age group and preferred book format in this sample.
Answer 7: Incorrect denominator
The student has reversed the condition.
12 ÷ 34 = 35.3% represents the percentage of audiobook users who are older.
The question asks for the percentage of older adults who prefer audiobooks.
The correct denominator is 80.
12 ÷ 80 = 0.15 = 15%.
Answer: 15%.
Answer 8: Misleading vertical axis
The vertical axis begins at 90 instead of zero.
The actual difference is only four visitors.
However, the truncated axis makes the bars appear dramatically different.
An ordinary bar chart should use a zero baseline so that differences in bar heights are not exaggerated.
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FREQUENTLY ASKED QUESTIONS ABOUT FREQUENCY TABLES AND TWO-WAY TABLES
1. What is the difference between frequency and relative frequency?
Frequency is the number of observations in a category. Relative frequency is the frequency divided by the total number of observations.
2. How do you calculate relative frequency from a frequency table?
Divide the frequency of the category by the total frequency. Multiply by 100 if you want the answer as a percentage.
3. What is a two-way frequency table?
A two-way frequency table organizes observations according to two categorical variables. It shows the frequency for every combination of categories.
4. What is a contingency table in statistics?
A contingency table is another name for a two-way table. It helps summarize and compare two categorical variables.
5. How do you calculate joint relative frequency?
Divide the frequency in the selected cell by the grand total of the table.
6. How do you calculate marginal relative frequency?
Divide the appropriate row or column total by the grand total.
7. How do you calculate conditional relative frequency?
Divide the frequency in the selected cell by the total of the group specified in the question.
8. What is the difference between joint and conditional relative frequency?
Joint relative frequency uses the entire sample as its denominator. Conditional relative frequency uses only the specified subgroup.
9. What is a conditional distribution?
It is the distribution of one categorical variable within a particular category of another categorical variable.
10. How do you determine whether two categorical variables are associated?
Compare their conditional distributions. If the percentages differ across groups, the variables show an association.
11. What is a segmented bar chart?
A segmented bar chart represents groups using equal-length bars divided into sections showing the conditional percentages of another categorical variable.
12. What is a 100% stacked bar chart?
It is another name for a segmented bar chart in which every bar represents 100% of its group.
13. What is the difference between a bar chart and a segmented bar chart?
A basic bar chart displays one categorical variable. A segmented bar chart displays two categorical variables and their conditional distributions.
14. What is a mosaic plot?
A mosaic plot represents two categorical variables using rectangles whose widths and subdivisions show group proportions and category distributions.
15. Does association mean causation?
No. Association indicates a relationship between variables, but it does not establish that one variable causes changes in the other.
16. Why do conditional percentages matter when comparing groups?
They account for differences in group sizes and allow a fairer comparison of distributions.
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THE ONE-PAGE AP STATISTICS CHEAT SHEET
Frequency = number of observations in a category.
Relative frequency = category frequency ÷ grand total.
Percentage = relative frequency × 100.
Joint relative frequency = cell frequency ÷ grand total.
Marginal relative frequency = marginal total ÷ grand total.
Conditional relative frequency = cell frequency ÷ specified group total.
Conditional distribution = all conditional percentages for one group.
Two-way table = a summary of two categorical variables.
Bar chart = graphical display of one categorical variable.
Segmented bar chart = graphical comparison of conditional distributions.
Mosaic plot = graphical representation of two categorical variables using rectangle areas.
Association = differences in conditional distributions.
Association does not establish causation.
For examination answers, include context, numerical evidence and comparison.
Always check row totals, column totals and percentage totals.
The most useful habit is simple:
Identify the group. Choose its total. Divide the correct cell. Explain what the percentage means.
That is the foundation of accurate statistical interpretation.
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CURRICULUM AND STUDY COVERAGE
This guide supports the following areas of study:
AP Statistics (2026–27):
Unit 1: Exploring One-Variable Data and Collecting Data.
Topic 1.3: Tabular Representation and Summary Statistics for One Categorical Variable.
Topic 1.4: Graphical Representations for One Categorical Variable.
Unit 2: Probability, Random Variables, and Probability Distributions.
Topic 2.1: Tabular and Graphical Representations for the Distributions of Two Categorical Variables.
Other academic applications:
Introductory college statistics.
GCSE Mathematics and Statistics.
A-level Mathematics and Statistics.
IB Mathematics and data analysis.
High-school statistics.
Descriptive statistics and categorical data analysis.
The guide also develops skills useful for interpreting survey results, research data, business reports and statistical graphs.
The key is not memorizing a collection of formulas. It is understanding what each number represents, which group is being compared and why the denominator changes.
