HSS.ID.B.5Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.ID.B.5: Two-Way Frequency Tables and Relative Frequencies
In plain English: HSS.ID.B.5 is the Common Core statistics standard that asks students to organize data on two categorical variables in a two-way frequency table. Students compute and interpret joint, marginal and conditional relative frequencies in context, and compare conditional percents to decide whether the variables appear to be associated. It is usually taught in Algebra I.
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Common Core State Standards for Mathematics · Domain: Interpreting Categorical and Quantitative Data (ID) · Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables Also written as HSS-ID.B.5 or S-ID.5 · Official standard
Students learn to summarize data on two categorical variables, such as grade level and having a job, in a two-way frequency table. They build tables from raw lists and from verbal descriptions, and they check that the inner cells add up to the row totals, the column totals and the grand total.
The heart of the standard is interpretation. Students turn counts into joint, marginal and conditional relative frequencies and explain each one in the context of the data, paying close attention to the denominator. They then compare conditional relative frequencies across groups to recognize possible associations and trends, and they practice saying why an association in a table does not by itself show cause and effect.
Learning Objectives
By the end of this lesson, students will be able to:
Organize data on two categorical variables in a two-way frequency table with row and column totals
Compute joint, marginal and conditional relative frequencies and say which total each one uses
Interpret each kind of relative frequency in a sentence about the context of the data
Compare conditional relative frequencies to recognize a possible association or trend, and describe it without claiming cause and effect
Prior Knowledge Required
Students should already be comfortable with:
Displaying frequencies and relative frequencies of bivariate categorical data in a two-way table 8.SP.A.4
Understanding ratio and rate language, such as "for every" and "out of" 6.RP.A.1
Finding a percent of a quantity as a rate per 100 6.RP.A.3
Converting between fractions, decimals and percents
Write this invented list on the board. Each entry is one student: the grade (9 or 10) and whether the student brings lunch from home (B) or buys lunch (L).
Warm-Up Prompt
"9-B, 10-L, 9-B, 9-L, 10-L, 10-B, 9-B, 10-L, 9-L, 10-L, 9-B, 10-L. Organize these 12 students so that someone can answer, at a glance, how many 10th graders buy lunch. Then answer: is buying lunch more common in one grade?"
Let students invent their own organizers first; many make two separate lists or a tally chart. Then show the two-way table: rows 9th and 10th, columns Brings and Buys. The counts are 9th: 4 bring, 2 buy (6 students); 10th: 1 brings, 5 buy (6 students); totals 5 bring and 7 buy. Point out that every student is counted in exactly one inner cell, so the four cells add to 12. For the second question, 5 of 6 tenth graders buy lunch but only 2 of 6 ninth graders do, which is a first look at an association.
Direct Instruction20 minutes
Two-way frequency tables. When each individual is classified by two categorical variables, a two-way table shows the count for every combination (the inner cells) and the totals for each category (the margins). A relative frequency is a count divided by a total. Which total you divide by decides what the number means:
Joint relative frequency: an inner cell divided by the grand total. It describes a combination, as a share of everyone.
Marginal relative frequency: a row or column total divided by the grand total. It describes one variable by itself.
Conditional relative frequency: an inner cell divided by its row total or its column total. It describes one variable within a single group of the other, such as "of the seniors, what percent...".
Association: compare conditional relative frequencies across groups. If they are about the same in every group, the data show no association; if they differ noticeably, the variables appear associated. An association in data does not show that one variable causes the other.
Building the table
An invented survey of 200 students: 120 are ninth graders and 80 are twelfth graders. 72 students have a part-time job, and 48 of the job holders are twelfth graders. Complete the two-way table.
Equation: 9th: 24 job, 96 no job (120); 12th: 48 job, 32 no job (80); totals: 72 job, 128 no job, 200
Joint relative frequency
What percent of all 200 students are twelfth graders with a job? Ninth graders with a job?
Equation: 48 / 200 = 24%; 24 / 200 = 12%
Marginal relative frequency
What percent of the students have a job? What percent are twelfth graders?
Equation: 72 / 200 = 36%; 80 / 200 = 40%
Conditional relative frequency
Of the twelfth graders, what percent have a job? Of the ninth graders? Of the job holders, what percent are twelfth graders?
Do grade level and having a job appear to be associated in these data?
Equation: Yes: 60% of twelfth graders have a job compared with 20% of ninth graders; with no association both would be near the overall 36%
Use Diagram 2 during Examples 2-4: the same cell, 48, gives three different percents depending on the denominator, and students should say the group aloud ("of all students", "of the twelfth graders", "of the job holders") before dividing. Then use Diagram 1 for Example 5. Stress that 48 / 80 and 48 / 72 answer different questions: the first is about twelfth graders, the second about job holders. Close by asking for a reason other than grade itself that could explain the pattern (age, work permits, schedules), so students do not read the association as cause and effect.
Guided Practice15 minutes
Pairs work with this invented survey of 240 households in one town, classified by type of home and by pet. The table has three columns, but the ideas are the same.
Invented survey of 240 households: type of home and pet
Dog
Cat
No pet
Total
Apartment
18
30
72
120
House
54
30
36
120
Total
72
60
108
240
Ask, in this order, and have pairs name the type of relative frequency before computing:
What percent of all households live in a house and have a dog? (Joint: 54 / 240 = 22.5%)
What percent of all households have no pet? (Marginal: 108 / 240 = 45%)
Of the apartment households, what percent have a dog? Of the house households? (Conditional: 18 / 120 = 15% and 54 / 120 = 45%)
Of the dog owners, what percent live in a house? (Conditional: 54 / 72 = 75%)
Of the cat owners, what percent live in an apartment? (Conditional: 30 / 60 = 50%)
Then pairs build the full row-conditional table (each row adds to 100%): Apartment 15%, 25%, 60%; House 45%, 25%, 30%. Discuss: dog ownership and home type appear associated, but cat ownership is the same 25% in both rows. An association can show up in some categories and not in others.
Independent Practice10-15 minutes
This invented table gives joint relative frequencies for 500 commuters, classified by trip length and by how they usually travel.
Invented joint relative frequencies for 500 commuters
Car
Transit
Bike or walk
Total
Under 5 miles
14%
8%
18%
40%
5 miles or more
42%
16%
2%
60%
Total
56%
24%
20%
100%
Students (1) turn the table back into counts, (2) find the percent of short-trip commuters who bike or walk and the percent of long-trip commuters who bike or walk, and (3) write two sentences about the association. Answers: (1) 70, 40, 90 and 210, 80, 10, with 200 short-trip and 300 long-trip commuters. (2) 18% / 40% = 45% of short-trip commuters and 2% / 60% ≈ 3.3% of long-trip commuters bike or walk. (3) Trip length and biking or walking appear strongly associated. The key skill is that a conditional relative frequency can be found from joint relative frequencies by dividing by the right marginal percent.
Closure10 minutes
Exit ticket, from an invented survey of 60 dog owners: 20 live in apartments (16 with small dogs, 4 with large dogs) and 40 live in houses (14 with small dogs, 26 with large dogs). (1) Make the two-way table. (2) What percent of all 60 owners live in an apartment and have a small dog? (3) Of the house dwellers, what percent have a large dog? (4) Is dog size associated with home type? Answers: (2) 16 / 60 ≈ 26.7%, joint. (3) 26 / 40 = 65%, conditional. (4) Yes: 80% of apartment dwellers have small dogs but only 35% of house dwellers do.
Differentiation Strategies
For Struggling Students
Give a sentence frame for every relative frequency: "Of the ___ (group), ___% are ___." The group in the first blank is always the denominator
Have students highlight the denominator in the table in one color and the numerator in another before they divide
Start with 2-by-2 tables with small counts, such as the 12-student Warm-Up, before moving to 3-column tables
For Advanced Students
Give a table in which the association in each of two subgroups reverses when the subgroups are combined, and ask students to explain how that can happen
Ask students to design a table of 100 people with given marginal totals and no association at all, and explain why the inner cells are forced
Have students collect two categorical variables from a school survey, build the table in a spreadsheet and present a segmented bar chart with a written conclusion
Assessment Guidance
What to Look For
Check that inner cells add to the totals and that students can name the type of relative frequency before they compute it. Listen for the group in every interpretation: "of all students", "of the twelfth graders", "of the job holders". A common error is dividing by the grand total when the question says "of the ...", or reversing a conditional (using 48 / 72 when the question asks about twelfth graders). For associations, students should compare conditional percents across groups, not raw counts, and should avoid causal language.
02
Classroom Activities
3 Activities
1
Our Class in a Two-Way Table
20 minWhole class, then pairs
The class answers two yes-or-no questions, builds a two-way table from sticky notes, and computes all three kinds of relative frequency from its own data.
Procedure
Each student answers two questions on a sticky note: "Do you have at least one sibling?" and "Would you rather do homework in the morning or at night?" Students with a sibling use one color of sticky note and students without a sibling use the other
Draw a large 2-by-2 grid on the board. Students place their notes in the right cell, and the class counts the cells and totals
Pairs copy the table and compute one joint, two marginal and four conditional relative frequencies, writing each one in a sentence that starts "Of the ..."
Pairs build the row-conditional table and decide whether the two variables appear associated in this class
Discussion Questions
Why must the four inner cells add up to the number of students in the room?
Would you expect the same pattern in another class? What could make it different?
With only about 30 students, how big must a difference in conditional percents be before you believe it?
Modification for Distance Learning
Use a two-question online poll and share the results in a spreadsheet. Pairs build the table with a pivot table or with COUNTIFS and post their sentences in a shared document.
2
Which Denominator? Card Match
15 minGroups of 3-4
Groups match 6 statement cards to 6 fraction cards for one invented table, then label each match as joint, marginal or conditional.
The Table and the 12 Cards
Invented data on 150 library visitors: of 90 adults, 36 borrowed a book; of 60 teenagers, 42 borrowed a book (78 borrowed in all, 72 did not).
Statement cards: "the percent of all visitors who are adults that borrowed"; "the percent of all visitors who borrowed"; "of the adults, the percent who borrowed"; "of the teenagers, the percent who borrowed"; "of the borrowers, the percent who are teenagers"; "the percent of all visitors who are teenagers"
Key: in the order listed, joint 24%, marginal 52%, conditional 40%, conditional 70%, conditional about 53.8%, marginal 40%
Procedure
Groups build the full two-way table first, including the "did not borrow" column
Groups match each statement to a fraction, label its type, and convert it to a percent
Two of the answers are 40%. Groups explain in writing why those two statements mean different things
Finish with a claim about association: do teens and adults borrow at different rates?
Challenge Variation
Groups write three new statement cards for the "did not borrow" column, one of each type, and trade with another group to match.
3
Association Detective
20 minPairs
Pairs receive three invented tables, draw a segmented bar chart for each, and decide whether each shows an association, no association, or an association in a surprising direction.
The 3 Tables (invented)
Table 1, 200 drivers: of 80 who took a driving class, 8 had a crash in their first year; of 120 who did not, 30 had a crash
Table 2, 160 students: of 100 who play a sport, 35 wear glasses; of 60 who do not play a sport, 21 wear glasses
Table 3, 300 shoppers: of 180 who used a coupon, 126 spent over $50; of 120 who did not, 48 spent over $50
Procedure
For each table, pairs complete the 2-by-2 table and compute the conditional percent of the outcome in each group (Table 1: 10% and 25%; Table 2: 35% and 35%; Table 3: 70% and 40%)
Pairs draw a 100% segmented bar for each group on the same scale
Pairs write one sentence per table: association or not, and how strong
For Tables 1 and 3, pairs list one other variable that could explain the association
Discussion Questions
Which table shows no association? How can you tell from the bars?
Does Table 1 prove that the driving class prevents crashes? What else could differ between the two groups of drivers?
Why compare percents instead of counts such as 8 and 30?
Modification for Distance Learning
Pairs enter each table in a spreadsheet and insert a 100% stacked bar chart, then paste the three charts into a shared slide with their conclusions.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Segmented Bar Chart of Conditional Relative Frequencies
Each bar shows one grade from the invented survey in the worked examples, split into the percent with a part-time job and the percent without. Because each bar uses its own row total, bars of different group sizes can be compared directly. The grade bars (20% and 60%) are far apart and on opposite sides of the overall rate, which is what an association looks like.
Diagram 2: Joint, Marginal and Conditional Relative Frequencies in One Table
The same invented table of 200 students. A joint relative frequency divides an inner cell by the grand total, a marginal relative frequency divides a row or column total by the grand total, and a conditional relative frequency divides an inner cell by the total of the group it is conditioned on. The two conditional percents for the shaded cell differ because they describe different groups.
04
Homework Assignment
~30 min
HSS.ID.B.5 Homework: Two-Way Tables and Relative Frequencies
Directions: All data are invented. Show each table with its row and column totals. For every relative frequency, name its type (joint, marginal or conditional), show the fraction you used, round to one decimal place and write a sentence that starts with the group it describes.
Part 1: Building Two-Way Tables (Problems 1-2)
A cafeteria surveys 150 students about their favorite lunch, pizza or sandwich. There are 80 sophomores and 70 juniors. In all, 90 students chose pizza, and 35 of the juniors chose pizza. (a) Make the two-way frequency table with all totals. (b) How many sophomores chose a sandwich?
Twenty students report whether they take music lessons (M = yes, N = no) and whether they are in grade 9 or grade 10: 9M, 10N, 9N, 9M, 10M, 10N, 9N, 10N, 9M, 10N, 9M, 10M, 9N, 10N, 9M, 10N, 9M, 10N, 9N, 10M. (a) Make the two-way frequency table. (b) Make a table of joint relative frequencies (every cell divided by 20).
Part 2: Joint, Marginal and Conditional (Problems 3-4)
A survey of 250 adults asks how they watch TV. Of the 120 adults under 30, 95 mostly stream and 25 mostly use cable. Of the 130 adults aged 30 and over, 65 mostly stream and 65 mostly use cable. (a) Make the two-way table. (b) What percent of all adults are under 30 and mostly stream? (c) What percent of all adults mostly stream? (d) What percent of all adults are under 30?
Use the table from Problem 3. (a) Of the adults under 30, what percent mostly stream? (b) Of the adults aged 30 and over, what percent mostly stream? (c) Of the adults who mostly stream, what percent are under 30? (d) Explain in one or two sentences why the answers to (a) and (c) are different.
Part 3: Associations and Trends (Problems 5-6)
At one school, 180 students were asked whether they ride the bus and whether they were late to first period at least once this month. Of the 60 bus riders, 12 were late at least once. Of the 120 students who do not ride the bus, 24 were late at least once. (a) Make the two-way table. (b) Compute the percent of each group that was late. (c) Is riding the bus associated with being late in these data? Explain.
A town surveys 400 residents about a proposed new park. The table of joint relative frequencies is: lives within 1 mile of the site and supports, 10%; within 1 mile and opposes, 15%; farther and supports, 45%; farther and opposes, 30%. (a) Find the marginal relative frequencies and the counts for each cell. (b) Of the residents within 1 mile, what percent support the park? Of those farther away? (c) Describe the association, and give one reason a town council should be careful about how it uses this result.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Building Tables
All cells and totals correct and consistent
One cell or total wrong
Table missing or inconsistent
Relative Frequencies
Correct type, fraction and percent for every part
Right numerator with a wrong denominator once or twice
Types confused throughout
Interpretation in Context
Every sentence names the group and the category
Sentences given but the group is unclear
Numbers only, no interpretation
Association
Compares conditional percents and avoids causal claims
Compares counts or makes a causal claim
No conclusion
05
Quiz: 20 Questions
Interactive, with answers
Instructions
All data in this quiz are invented. Answer each question before you open its explanation. The score at the top counts your multiple-choice answers, and Reset quiz starts the whole set again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
An invented survey of 250 students: of the 150 who use a phone in bed, 27 sleep 8 or more hours a night and 123 sleep less; of the 100 who do not use a phone in bed, 58 sleep 8 or more hours and 42 sleep less. What percent of all 250 students use a phone in bed and sleep 8 or more hours?
Answer: B
This is a joint relative frequency: the inner cell divided by the grand total, 27 / 250 = 0.108 = 10.8%. Choice A treats the count 27 as a percent. Choice C is a different cell, the phone users who sleep less (123 / 250). Choice D divides by the 85 students who sleep 8 or more hours instead of by all 250.
Question 2 of 20 · Multiple Choice
Same survey. What percent of all 250 students sleep 8 or more hours a night?
Answer: A
This is a marginal relative frequency. The column total is 27 + 58 = 85 students, and 85 / 250 = 34%. Choice B is the percent within the no-phone group only. Choice C uses the count 85 as a percent.
Question 3 of 20 · Multiple Choice
Same survey. Of the students who use a phone in bed, what percent sleep 8 or more hours?
Answer: C
The words "of the students who use a phone in bed" set the denominator: 27 / 150 = 18%, a conditional relative frequency. Choice A reads the count 27 as a percent. Choice B divides by the 85 students who sleep 8 or more hours, which reverses the condition. Choice D is the share of all students who use a phone in bed (150 / 250).
Question 4 of 20 · Multiple Choice
Same survey. Of the students who sleep less than 8 hours, what percent use a phone in bed?
Answer: D
The condition is sleeping less than 8 hours, so divide by that column total, 123 + 42 = 165: 123 / 165 ≈ 0.745, about 74.5%. Choice A divides by the 150 phone users, which reverses the condition. Choice B divides by all 250 students, which gives a joint relative frequency. Choice C is the percent of that column who do not use a phone in bed (42 / 165).
Question 5 of 20 · Multiple Choice
Same survey. Which statement best describes the data?
Answer: B
Associations are judged by comparing conditional relative frequencies across groups: the percent who sleep 8 or more hours among phone users (27 of 150) is far below the percent among the others (58 of 100). Choice C claims cause and effect, which a survey table cannot show; for example, students who already sleep poorly may reach for their phones. Choice D compares a raw count, which depends on group sizes.
Question 6 of 20 · Multiple Choice
An invented café records 120 drinks. The joint relative frequencies are: small coffee 0.25, large coffee 0.35, small tea 0.25, large tea 0.15. What percent of the drinks are coffee?
Answer: C
The marginal relative frequency for coffee adds its row: 0.25 + 0.35 = 0.60, or 60%. Choice A is only the large-coffee cell, a joint relative frequency. Choice D assumes coffee and tea are equally common.
Question 7 of 20 · Multiple Choice
Same café. Of the tea drinks, what percent are large?
Answer: A
A conditional relative frequency can be found from joint relative frequencies. Tea makes up 0.25 + 0.15 = 0.40 of the drinks, so 0.15 / 0.40 = 0.375, or 37.5%. Choice B is the joint relative frequency of large tea among all drinks. Choice C divides 0.15 by the large-drink total 0.50, which gives the percent of large drinks that are tea. Choice D is the percent of teas that are small.
Question 8 of 20 · Multiple Choice
Same café. How many of the 120 drinks were large teas?
Answer: C
Multiply the joint relative frequency by the total: 0.15 × 120 = 18 drinks. Choice A reads 0.15 as a count of 15. Choice B is the number of small coffees (0.25 × 120 = 30), not large teas.
Question 9 of 20 · Multiple Choice
Same café. Is drink size associated with drink type?
Answer: B
Compare the conditional percent of large drinks within each type: large coffee divided by all coffee, and large tea divided by all tea. The two percents differ by about 20 percentage points, which suggests an association. Choice A looks at joint cells, not conditional percents. Choice D compares marginal totals, which say nothing about how size depends on type.
Question 10 of 20 · Multiple Choice
In a two-way table of invented data on 300 hikers (experienced or beginner, and whether they carried a map), which calculation is a conditional relative frequency?
Answer: A
A conditional relative frequency divides a cell by the total of the group it is conditioned on, here all beginners. Choice B divides a cell by the grand total, which is a joint relative frequency. Choices C and D divide a margin by the grand total, which are marginal relative frequencies.
Question 11 of 20 · Multiple Choice
An invented two-way table of 90 volunteers shows: 40 adults and 50 teens; 36 volunteers worked at the food bank; 12 of the food-bank volunteers were adults. How many teens did not work at the food bank?
Answer: D
Teens at the food bank are 36 - 12 = 24, so teens not at the food bank are 50 - 24 = 26. Choice C stops at the food-bank teens. Choice B is the number of adults not at the food bank (40 - 12 = 28).
Question 12 of 20 · Multiple Choice
Which invented result shows no association between the two variables?
Answer: A
In choice A, 20 / 50 = 40% of juniors and 30 / 75 = 40% of seniors have a license: the conditional percents are equal, so there is no association. Choice D has equal counts (30 and 30) but different percents, 60% and 40%, so it does show an association.
Question 13 of 20 · Multiple Choice
A news report on invented data says: "Of 400 teens surveyed, 22% were teens who volunteer and play a sport." What kind of relative frequency is 22%?
Answer: C
The 22% describes one combination (volunteers and plays a sport) as a share of all 400 teens, which makes it a joint relative frequency. Choice A would need a group such as "of the athletes" as the denominator. Choice B describes a single variable, not a combination.
Question 14 of 20 · Multiple Choice
For invented data on 200 shoppers, 30% of weekend shoppers used a self-checkout and 30% of weekday shoppers used a self-checkout. What can you conclude?
Answer: B
The conditional percents are the same in both groups, so the data show no association. Choice C does not follow: equal percents can come from groups of different sizes. Choice D confuses a percent with a count; 30% of 200 is 60 shoppers.
Question 15 of 20 · Short Answer
A gym surveys 120 members (invented). 70 members attend in the morning and 50 in the evening. In all, 48 members take group classes, and 30 of the class-takers attend in the evening. Complete the two-way table with all totals.
Morning: 18 take classes and 52 do not (70). Evening: 30 take classes and 20 do not (50). Totals: 48 take classes and 72 do not, 120 in all.
Question 16 of 20 · Short Answer
Invented data on 80 dog walkers: 44 walk before school and 36 after school; 20 of the before-school walkers and 24 of the after-school walkers use a leash app. (a) What percent of all walkers walk after school and use the app? (b) What percent of all walkers use the app?
Invented data on 200 seniors: of 125 who applied to college early, 100 were accepted to their first choice; of 75 who did not apply early, 30 were accepted to their first choice. (a) Of the early applicants, what percent were accepted to their first choice? (b) Of the students accepted to their first choice, what percent applied early? (c) Is there an association?
(a) 100 / 125 = 80%. (b) 100 / 130 ≈ 76.9%. (c) Yes: 80% of early applicants versus 30 / 75 = 40% of the others. This does not show that applying early causes acceptance; students who apply early may differ in other ways.
Question 18 of 20 · Short Answer
A segmented bar chart of invented data shows that 65% of students who eat breakfast at home feel ready for class, compared with 62% of students who do not. Describe what the chart shows about an association.
The two conditional percents are close (65% and 62%), so the data show little or no association between eating breakfast at home and feeling ready. A 3-point difference could easily be due to which students happened to be surveyed.
Question 19 of 20 · Short Answer
Invented joint relative frequencies for 250 people at a concert: 0.12 under 18 and wore earplugs; 0.28 under 18 and did not; 0.18 aged 18 or over and wore earplugs; 0.42 aged 18 or over and did not. (a) Give the count for each cell. (b) What percent of the people under 18 wore earplugs? (c) What percent of the people aged 18 or over wore earplugs?
(a) 30, 70, 45 and 105. (b) 0.12 / 0.40 = 30%. (c) 0.18 / 0.60 = 30%. The percents are equal, so age group and wearing earplugs are not associated in these data.
Question 20 of 20 · Short Answer
An invented survey of 300 students finds that 45% of students with a part-time job own a car but only 15% of students without a job own a car. A student concludes, "Getting a job makes students buy cars." Explain what the table does and does not show.
The table shows an association: the conditional percent of car owners is much higher among students with jobs. It does not show cause and effect. Owning a car may make it possible to get to a job, or a third variable such as age may explain both.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.ID.B.5 mean?
HSS.ID.B.5 means students can summarize data on two categorical variables in a two-way frequency table and interpret it. They compute joint, marginal and conditional relative frequencies, explain what each means in context, and compare conditional relative frequencies to spot possible associations.
Is HSS.ID.B.5 in Algebra 1?
Yes, in most course sequences. The Common Core appendix on course design places HSS.ID.B.5 in the traditional Algebra I course and in Mathematics I of the integrated pathway. It builds directly on the grade 8 standard 8.SP.A.4, and the conditional probability standards in HSS.CP use the same tables later.
What is the difference between joint, marginal and conditional relative frequency?
They use different denominators. A joint relative frequency divides one inner cell by the grand total, a marginal relative frequency divides a row or column total by the grand total, and a conditional relative frequency divides a cell by the total of one row or one column. The question wording, such as "of the seniors", tells you which total to use.
How do you know if there is an association in a two-way table?
Compare conditional relative frequencies across the groups. If the percent with a given outcome is about the same in every group, the data show no association; if the percents differ noticeably, the variables appear associated. Compare percents, not counts, because the groups can have different sizes.
Does an association in a two-way table mean one variable causes the other?
No. A two-way table from a survey or observational data can show that two variables are associated, but it cannot show cause and effect. Another variable may explain both, or the direction may be the reverse of what it seems. Causal claims need a randomized experiment, which is studied in HSS.IC.B.5.
What are common mistakes with two-way tables?
A frequent mistake is using the grand total as the denominator when the question asks about one group ("of the ..."). Others are reversing a conditional relative frequency, adding percents from different rows, and judging an association by comparing raw counts instead of percents.
Should the row percents or the column percents add to 100%?
It depends on which way you condition. In a table of row-conditional relative frequencies, each row adds to 100%; in a table of column-conditional relative frequencies, each column does. In a table of joint relative frequencies, only the whole table adds to 100%.
What graph goes with a two-way table?
A segmented (100% stacked) bar chart is the usual choice. Each bar is one group, split into the conditional percents of the other variable, so bars for groups of different sizes can be compared directly. Side-by-side bar charts of counts are harder to compare when group sizes differ.
How does HSS.ID.B.5 connect to probability?
Two-way tables are the main tool of the conditional probability standards. In HSS.CP.A.4, students use a two-way table as a sample space, and conditional relative frequencies become conditional probabilities such as P(job | senior). Equal conditional relative frequencies correspond to independence.
Is HSS.ID.B.5 on the SAT?
Questions that ask students to read a two-way table and compute a conditional percent belong to the Problem-Solving and Data Analysis domain of the digital SAT. The skills in this lesson, choosing the right denominator and interpreting the result, are the ones those questions test.
07
Related Standards
5 standards
These standards connect to HSS.ID.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.SP.A.4Prerequisite
Show patterns of association in categorical data with frequencies in a two-way table
