8.SP.A.4Common CoreMathStatistics and ProbabilityGrade 8
8.SP.A.4: Two-Way Tables and Relative Frequencies for Categorical Data
In plain English: 8.SP.A.4 is the Common Core grade 8 math standard that asks students to build and read two-way tables for two categorical variables, such as whether students have a curfew and whether they have chores. Students compute relative frequencies for rows or columns and compare them to decide whether the two variables seem to be associated. It extends scatter plot ideas to data in categories.
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Investigate patterns of association in bivariate data. Also written as 8.SP.4 · Official standard
So far in grade 8, students have looked for patterns of association (two variables that tend to change together) in scatter plots of measurements. This lesson finds the same kind of pattern in categorical data: answers that are categories instead of numbers, such as yes or no, or grade 6, 7 or 8. When two categorical questions are asked of the same people, the answers can be summarized in a two-way table, with one variable in the rows and the other in the columns. Each cell holds a frequency: the number of people in both categories.
Counts alone can mislead when groups have different sizes, so students also compute relative frequencies: a count divided by a total, written as a fraction, decimal or percent. A row relative frequency divides by the row total, and a column relative frequency divides by the column total. If the percents differ a lot from one row to another, the variables may be associated; if they are close, there is little evidence of association. The lesson works through the official example: do students who have a curfew on school nights also tend to have assigned chores? All data sets are invented for teaching.
Learning Objectives
By the end of this lesson, students will be able to:
Construct a two-way table from a list of answers to two categorical questions, including row, column and grand totals
Interpret the counts and totals in a two-way table in the words of the context
Compute row and column relative frequencies and choose the one that answers a given question
Use relative frequencies to decide whether two categorical variables show a possible association, and explain why an association does not prove a cause
Prior Knowledge Required
Students should already be comfortable with:
Describing patterns of association in scatter plots 8.SP.A.1
Writing a part-to-whole ratio as a percent 6.RP.A.3
Knowing that a statistical question is answered with data that vary 6.SP.A.1
Relative frequency as an estimate from data, used in probability 7.SP.C.6
"Two questions for everyone: Do you have a curfew on school nights? Do you have assigned chores at home? Write Y or N for each on a sticky note, like Y-N. Before we count: do you think students who have a curfew are more likely to have chores? Why?"
Collect the sticky notes on the board in four groups: Y-Y, Y-N, N-Y and N-N. Point out that each note carries two answers from the same student, which is what makes the data bivariate. Ask how the four groups could be arranged to answer the question, and keep the class data for Activity 1. For the rest of the lesson, use the invented data from two classes below.
Direct Instruction20 minutes
Build the table with the class, and define each part as it appears:
Two-way table: the categories of one variable label the rows and the categories of the other label the columns. Each person is counted in exactly one cell.
Totals: a row total counts everyone in that row's category, a column total counts everyone in that column's category, and the grand total counts everyone surveyed. The row totals and the column totals both add up to the grand total.
Relative frequency: a count divided by a total. Choose the total that matches the question: "Of the students with a curfew, what percent..." divides by the curfew row total.
Row relative frequencies: divide each cell by its row total, so each row adds to 100%. Column relative frequencies: divide each cell by its column total, so each column adds to 100%.
Looking for association: compare the percents between rows (or between columns). A large difference suggests the variables are associated; a small difference suggests little or no association.
Association is not cause: a pattern in a survey does not show that one variable causes the other. Families that set curfews may also set chores for another reason, such as having more rules in general.
Invented data: curfew and chores for 50 grade 8 students
Chores
No chores
Total
Curfew
20
6
26
No curfew
9
15
24
Total
29
21
50
The official example: curfew and chores (Diagrams 1 and 2)
Use the table above. Is there evidence that students who have a curfew also tend to have chores?
Equation: Row relative frequencies: of the 26 students with a curfew, 20 have chores: 20 ÷ 26 ≈ 76.9%. Of the 24 students without a curfew, 9 have chores: 9 ÷ 24 = 37.5%. About 77% compared with about 38% is a large difference, so yes: in these data, students with a curfew tend to have chores. That is an association, not proof that curfews cause chores.
Constructing a table from a list
Twenty students gave their grade (7 or 8) and their usual lunch (H = hot lunch, P = packed): 7H, 7P, 8H, 8H, 7H, 8P, 7P, 7H, 8P, 8H, 7H, 8P, 8P, 7H, 7P, 8H, 8P, 7H, 8P, 8P. Build the two-way table.
Equation: Tally each pair: 7H: 6, 7P: 3, 8H: 4, 8P: 7. Row totals: grade 7 has 9 students and grade 8 has 11. Column totals: 10 hot and 10 packed. Grand total 20, matching the number of students in the list.
Column relative frequencies for the same table
In the curfew table, what percent of the students with chores have a curfew? What percent of the students without chores have one?
Equation: Divide by column totals: 20 ÷ 29 ≈ 69.0% of students with chores have a curfew, and 6 ÷ 21 ≈ 28.6% of students without chores do. The columns tell the same story: the two variables go together.
Little or no association
A survey of 110 students asked their grade and whether they prefer print books or e-books (invented data). Grade 7: 36 print, 22 e-book. Grade 8: 29 print, 23 e-book. Is there an association?
Equation: Row relative frequencies for print: 36 ÷ 58 ≈ 62.1% in grade 7 and 29 ÷ 52 ≈ 55.8% in grade 8. The difference is about 6 percentage points (62.1 - 55.8, a difference between two percents), which is small for groups of this size, so there is little evidence that book preference is associated with grade.
Use Diagram 1 to name each part of the table and to show where each row percent comes from. Use Diagram 2 to show the same row percents as bars: when the bars for the two rows look very different, the variables are associated. The bottom bar shows all 50 students; the two row bars fall on opposite sides of it.
Guided Practice15 minutes
A middle school surveyed 40 students in each grade: "Do you have your own smartphone?" (invented data). Pairs complete the totals and answer the questions.
Invented data: smartphone ownership by grade
Smartphone
No smartphone
Total
Grade 6
22
18
40
Grade 7
29
11
40
Grade 8
34
6
40
Total
85
35
120
Guided practice questions with answers
Question
Answer
How many students were surveyed, and how many have a smartphone?
Is there evidence of an association between grade and owning a smartphone?
Yes: the percent rises from 55% to 85% as the grade goes up
What percent of the smartphone owners are in grade 8? Which total did you use?
34 ÷ 85 = 40%, using the smartphone column total
Why is comparing 22, 29 and 34 enough here, but not in general?
Each grade has 40 students, so the counts compare fairly; with different group sizes you must compare percents
Watch for students who divide by the grand total (120) when the question asks about one grade. Ask them to point to the group the question starts with: "Of the grade 6 students...".
Independent Practice15 minutes
Students work alone, then compare with a partner. A swim coach asked 80 students whether they went to a summer camp last year and whether they can swim 50 meters without stopping (invented data).
Invented data: summer camp and swimming
Can swim 50 m
Cannot
Total
Went to camp
24
6
30
Did not go
27
23
50
Total
51
29
80
Independent practice problems with answers
Problem
Answer
Fill in the totals. How many students cannot swim 50 m?
Camp row 30, no-camp row 50; 51 can swim and 29 cannot; grand total 80
What percent of the camp students can swim 50 m? Of the others?
24 ÷ 30 = 80% and 27 ÷ 50 = 54%
Is there an association? Explain with your percents.
Yes: 80% compared with 54% is a large difference
More non-camp students than camp students can swim 50 m (27 compared with 24). Does that mean camp students swim worse?
No: the groups have different sizes. As percents, camp students do better
Does this show that camp teaches swimming? Give another explanation.
No. Students who already swim well may be more likely to choose a camp
Closure5-10 minutes
Exit ticket: 50 students were asked whether they have a pet and whether they play a musical instrument (invented data). Pet: 14 play, 11 do not. No pet: 13 play, 12 do not. (1) What percent of pet owners play an instrument? (14 ÷ 25 = 56%.) (2) What percent of students without a pet play? (13 ÷ 25 = 52%.) (3) Is there evidence of an association? (Little: 56% and 52% are close.)
Differentiation Strategies
For Struggling Students
Give a blank two-way table with the row and column labels already filled in and the total cells shaded
Have students circle the group named after "of the" in each question before they divide, and write that total as the denominator
Use the sticky-note board from the warm-up: move the notes into the four cells so each count is something students can touch
For Advanced Students
Ask students to invent a 2-by-2 table of 60 students where the counts suggest one conclusion but the row percents suggest the opposite
Have students compute both row and column relative frequencies for the curfew table and explain why both show the same association
Extension (beyond this standard): in high school (HSS.CP.A.4), students use two-way tables to find conditional probabilities and to decide whether two events are independent
Assessment Guidance
What to Look For
A strong answer builds a table whose cells add to the row and column totals, names the total used for each relative frequency, and compares percents rather than raw counts when group sizes differ. The conclusion names both variables and says "associated" or "little evidence of association" with numbers, not "causes". Watch for students who divide by the grand total, who mix up row and column percents, who compare counts across groups of different sizes, and who call a difference of a few percentage points an association.
02
Classroom Activities
3 Activities
1
Our Class: Curfew and Chores
20 minWhole class, then pairs
The class answers the official question with its own data: is there evidence that students who have a curfew on school nights also tend to have assigned chores at home?
Procedure
Draw a large 2-by-2 grid on chart paper, with Curfew and No curfew as rows and Chores and No chores as columns
Each student places the warm-up sticky note in the matching cell
Pairs copy the counts, add the row, column and grand totals, and check that the grand total equals the number of students present
Pairs compute the percent of each row that has chores and write a two-sentence conclusion
Sample Class Data for Teachers (invented)
A class of 28: Curfew and chores 11, curfew and no chores 4, no curfew and chores 5, no curfew and no chores 8. Row percents with chores: 11 ÷ 15 ≈ 73.3% for curfew and 5 ÷ 13 ≈ 38.5% for no curfew, so this class shows an association.
Discussion Questions
Our class is small. How much would the percents change if one student moved to a different cell?
Does our class match the 50-student example? What might make a different class give different results?
Name a reason, other than the curfew itself, why students with curfews might also have chores.
Modification for Distance Learning
Students answer the two questions in an online form, and the teacher shares the four counts for pairs to analyze.
2
Association or Not? Card Sort
15 minPairs
Pairs get 6 table cards. For each, they compute row percents and sort the card into "association" or "little or no association", writing the two percents on the card.
The 6 Cards (invented data)
Card A, owns a bike (rows) and lives within 3 km of school (columns): yes-yes 28, yes-no 12; no-yes 10, no-no 30
Card B, has a younger sibling (rows) and likes spicy food (columns): yes-yes 17, yes-no 13; no-yes 19, no-no 16
Card C, plays video games weekly (rows) and wears glasses (columns): yes-yes 15, yes-no 35; no-yes 9, no-no 21
Card D, in the school garden club (rows) and eats vegetables daily (columns): yes-yes 18, yes-no 6; no-yes 21, no-no 35
Card E, early riser (rows) and does homework before school (columns): yes-yes 22, yes-no 18; no-yes 6, no-no 34
Card F, birthday in the first half of the year (rows) and has a library card (columns): yes-yes 26, yes-no 16; no-yes 25, no-no 17
Answer Key
Card A: 70% compared with 25%: association
Card B: about 56.7% compared with about 54.3%: little or no association
Card C: 30% compared with 30%: no association in these data
Card D: 75% compared with 37.5%: association
Card E: 55% compared with 15%: association
Card F: about 61.9% compared with about 59.5%: little or no association
Discussion Questions
Card C has more gamers than non-gamers who wear glasses (15 compared with 9). Why is that not evidence of an association?
For Card D, give two different explanations for the association.
3
Build It From the Cards
15 minGroups of 3
Each group gets 24 student cards. Each card shows one student's answers to two questions: "How do you get to school?" (S = school bus, N = not by bus) and "Where do you usually eat lunch?" (B = the cafeteria, L = the library or a club room). Groups sort the cards into a two-way table and look for an association.
Bus and cafeteria: 9. Bus and library: 7. Not by bus and cafeteria: 3. Not by bus and library: 5. Row totals: 16 bus riders and 8 others; column totals: 12 cafeteria and 12 library. Row percents in the cafeteria: 9 ÷ 16 = 56.25% for bus riders and 3 ÷ 8 = 37.5% for the others. That is a difference of about 19 percentage points in a small group, so it is a possible association that a larger survey should check.
Discussion Questions
How did your group check that no card was lost or counted twice?
With only 24 students, one card can change a percent by several points. How would you make the evidence stronger?
Challenge Variation
Groups add a third lunch category, "outside", by writing 6 new cards, and rebuild the table as a 2-by-3 table.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Parts of a Two-Way Table (Curfew and Chores)
The invented curfew and chores data for 50 students. Each cell count is in blue, and the shaded cells are totals. Below each count is its row relative frequency: for example, 20 of the 26 students with a curfew, about 76.9%, have chores.
Diagram 2: Row Relative Frequencies as Segmented Bars
Each bar is one row of the curfew table, drawn to scale as 100%. About 76.9% of students with a curfew have chores, compared with 37.5% of students without one: a large gap, which suggests an association. The bottom bar shows all 50 students (58% have chores).
04
Homework Assignment
~30 min
8.SP.A.4 Homework: Two-Way Tables and Association
Directions: All data are invented. Show each division you use for a relative frequency, and round percents to the nearest tenth. Write every conclusion in a full sentence that names both variables.
Part 1: Build and Read Two-Way Tables (Problems 1-3)
Twenty students answered two questions: their pet (D = dog, C = cat, N = no pet) and their home (A = apartment, H = house). Their answers were DA, DH, CA, NH, DH, CA, NA, DH, CH, NA, DH, CA, NH, DH, CA, NA, DA, CH, NH, DH. (a) Build a two-way table with pets as rows and homes as columns, with all totals. (b) How many students have a dog and live in a house? (c) How many students live in an apartment?
At a book fair, 90 students were asked their grade and whether they prefer fiction or nonfiction. There were 50 grade 7 students, 32 grade 7 students preferred fiction, and 55 students in all preferred fiction. (a) Complete the two-way table with all totals. (b) How many grade 8 students preferred nonfiction?
A school asked students in grades 6, 7 and 8 whether they read for fun every week.
Reads weekly
Does not
Total
Grade 6
28
12
40
Grade 7
24
21
45
Grade 8
19
26
45
Total
71
59
130
(a) What does the 24 in the table mean? (b) What does the total 71 mean? (c) What percent of all the students surveyed are grade 6 students who read weekly?
Part 2: Relative Frequencies and Association (Problems 4-6)
Students were asked whether they have a younger sibling and whether they have ever babysat.
Has babysat
Has not
Total
Younger sibling
21
15
36
No younger sibling
9
35
44
Total
30
50
80
(a) Find the row relative frequencies. (b) Is there evidence of an association? Explain with your percents.
Students were asked whether they play a musical instrument and whether they study with music playing.
Studies with music
In silence
Total
Plays an instrument
26
19
45
Does not
14
31
45
Total
40
50
90
(a) Find the column relative frequencies. (b) What percent of the students who study in silence play an instrument? (c) Describe any association in one sentence.
A town surveyed 150 students on whether they live within 2 km of school and whether they walk or bike to school.
Walks or bikes
Does not
Total
Within 2 km
42
18
60
Farther
12
78
90
Total
54
96
150
(a) Find the percent of each row that walks or bikes. (b) Describe the association. (c) The town says, "Living close to school makes students walk." Is that claim supported by the table alone? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Building Tables
Every cell and total correct, and totals check
One or two cells or totals wrong
Table missing or mostly wrong
Relative Frequencies
Correct total used for each, with the division shown
Correct method, one computing slip or wrong total once
Wrong totals throughout
Association
Compares percents and names both variables
Correct verdict with weak reasoning or counts only
No verdict or no reasoning
Cause and Context
Explains that association is not cause, with another possible reason
Mentions cause without a reason
Claims a cause from the table
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. All data sets are invented. Your score updates as you answer, and Reset quiz clears everything so you or your students can try 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
Which pair of survey questions gives bivariate categorical data?
Answer: B
Bivariate categorical data come from two questions with category answers asked of the same people, such as two yes-or-no questions. Choice A gives two measurements, which belong in a scatter plot. Choice C is one variable. Choice D is one categorical variable.
Question 2 of 20 · Multiple Choice
A survey of 75 students found that 40 play a sport, and 23 of the sport players also play an instrument. In a two-way table, how many sport players do not play an instrument?
Answer: B
The sport row total is 40, so 40 - 23 = 17 sport players do not play an instrument. Choice A adds 40 and 23. Choice C is the number of students who do not play a sport, 75 - 40. Choice D subtracts 23 from the grand total, 75, instead of from the row total.
Question 3 of 20 · Multiple Choice
Use this table:
In a club
Not in a club
Total
Rides the bus
14
22
36
Does not
10
14
24
Total
24
36
60
What does the 14 in the bottom row mean?
Answer: C
The bottom data row is students who do not ride the bus, and the second column is students not in a club, so 14 students are in neither group. Choice A uses the bus row instead. Choice B describes the 10 in the first column. Choice D confuses a cell with the grand total, 60.
Question 4 of 20 · Multiple Choice
Use the same table. What percent of the bus riders are in a club, to the nearest percent?
Answer: C
"Of the bus riders" means divide by the bus row total: 14 ÷ 36 ≈ 38.9%, about 39%. Choice A divides by the grand total: 14 ÷ 60 ≈ 23%. Choice B divides by the club column total: 14 ÷ 24 ≈ 58%. Choice D is the percent of bus riders not in a club, 22 ÷ 36.
Question 5 of 20 · Multiple Choice
In a survey, 65% of students who play a sport eat breakfast every day, compared with 30% of students who do not play a sport. What is the best conclusion?
Answer: A
A difference of 35 percentage points between the rows is large, so the two variables appear to be associated. Choices B and D claim a cause, which a survey alone cannot show. Choice C ignores the size of the difference.
Question 6 of 20 · Multiple Choice
In a survey, 48% of grade 7 students prefer summer break to winter break, compared with 52% of grade 8 students. What is the best conclusion?
Answer: B
A 4-point difference is small, so these data give little evidence of an association. Choice C treats any difference as an association. Choice D adds percents from different rows, which do not need to add to 100%: each row adds to 100% on its own.
Question 7 of 20 · Multiple Choice
What is a relative frequency in a two-way table?
Answer: A
A relative frequency compares a count with a total, such as 20 ÷ 26 ≈ 76.9%. Choice B is a frequency (a count), not a relative frequency. Choices C and D do not divide by a total.
Question 8 of 20 · Multiple Choice
Twelve students were asked "Do you play an instrument?" and "Do you sing in the choir?" (Y or N, in that order). Their answers were YY, YN, NN, YY, NY, NN, YY, YN, NN, NY, YY, NN. How many play an instrument but do not sing in the choir?
Answer: D
Only the YN answers count, and there are 2 of them. Choice A counts the YY answers. Choice B counts every instrument player (4 + 2 = 6). Choice C counts all the students who answered N to at least one question.
Question 9 of 20 · Multiple Choice
In a two-way table of pet ownership (rows) and grade (columns), the row total for "has a pet" is 38. What does 38 tell you?
Answer: A
A row total counts everyone in that row's category, across all columns, so 38 students have a pet. Choice B confuses a count with a percent. Choice C describes a single cell. Choice D describes column totals.
Question 10 of 20 · Multiple Choice
A two-way table shows whether students own a bike (rows) and whether they play a sport (columns). To find the percent of the students who play a sport that own a bike, what do you divide by?
Answer: C
The question starts with "of the students who play a sport", so the sport column total is the whole: divide the bike-and-sport cell by it. Choice A gives a row relative frequency, which answers a different question: what percent of bike owners play a sport. Choice B gives a percent of all students. Choice D is a single cell, not the total of a group.
Question 11 of 20 · Multiple Choice
Students chose a field trip: a science museum or the zoo.
Museum
Zoo
Total
Grade 7
18
22
40
Grade 8
27
13
40
Total
45
35
80
What percent of the students who chose the museum are in grade 8?
Answer: B
"Of the students who chose the museum" means divide by the museum column total: 27 ÷ 45 = 60%. Choice A divides by the grade 8 row total, 27 ÷ 40. Choice C divides by the grand total, 27 ÷ 80. Choice D is the percent of museum students in grade 7, 18 ÷ 45.
Question 12 of 20 · Multiple Choice
Use the same table. What percent of all 80 students are grade 7 students who chose the zoo?
Answer: D
Divide the cell by the grand total: 22 ÷ 80 = 27.5%. Choice A divides by the grade 7 row total, 22 ÷ 40. Choice B divides by the zoo column total, 22 ÷ 35 ≈ 62.9%. Choice C gives the count as if it were a percent.
Question 13 of 20 · Multiple Choice
A different class answered the curfew and chores questions.
Chores
No chores
Total
Curfew
8
4
12
No curfew
3
10
13
Total
11
14
25
Is there evidence that students with a curfew also tend to have chores?
Answer: D
Row percents: 8 ÷ 12 ≈ 66.7% of students with a curfew have chores, and 3 ÷ 13 ≈ 23.1% of the others do. The large difference is evidence of an association. Choice A looks at one count, not percents. Choice B is about group sizes, which the percents already take into account. Choice C uses the column total and does not compare the rows.
Question 14 of 20 · Multiple Choice
A survey shows that students who eat lunch in the cafeteria are more likely to be in a club. Which statement is best?
Answer: A
A two-way table can show that two variables go together, which is an association. Choices B and C claim a cause, but a survey cannot show which variable, if either, causes the other. Choice D contradicts the survey.
Question 15 of 20 · Short Answer
Of 60 students, 35 are in the band and 28 are in the choir. Twenty students are in both. Build a two-way table (band yes or no as rows, choir yes or no as columns) with all totals.
Band and choir: 20. Band, not choir: 35 - 20 = 15. Choir, not band: 28 - 20 = 8. Neither: 60 - 20 - 15 - 8 = 17. Row totals 35 and 25; column totals 28 and 32; grand total 60.
Question 16 of 20 · Short Answer
Seventy students were asked whether they ate breakfast and whether they felt tired in first period.
Tired
Not tired
Total
Ate breakfast
12
33
45
Skipped breakfast
16
9
25
Total
28
42
70
Find the percent of each row that felt tired, and describe any association.
Ate breakfast: 12 ÷ 45 ≈ 26.7% felt tired. Skipped breakfast: 16 ÷ 25 = 64% felt tired. The difference is large, so skipping breakfast is associated with feeling tired in first period in these data. The table alone does not show that skipping breakfast causes tiredness: for example, students who go to bed late may both skip breakfast and feel tired.
Question 17 of 20 · Short Answer
One hundred students were asked whether their family has a screen-time rule and whether they sleep 8 or more hours on school nights.
Sleeps 8+ h
Less than 8 h
Total
Rule
31
9
40
No rule
22
38
60
Total
53
47
100
(a) What percent of students with a rule sleep 8 or more hours? (b) What percent of students who sleep 8 or more hours have a rule? (c) Why are the two answers different?
(a) 31 ÷ 40 = 77.5%, a row relative frequency. (b) 31 ÷ 53 ≈ 58.5%, a column relative frequency. (c) They divide the same count by different totals: (a) looks only at the 40 students with a rule, and (b) looks only at the 53 students who sleep 8 or more hours.
Question 18 of 20 · Short Answer
Sixteen students gave how they get to school (B = bus, C = car, W = walk) and their lunch (H = hot, P = packed): BH, BP, CP, WH, BH, CP, WP, BH, CH, BP, WP, CP, BH, WH, CP, BP. Build a two-way table with the three ways of travel as rows and lunch as columns, with all totals.
Bus: hot 4, packed 3 (total 7). Car: hot 1, packed 4 (total 5). Walk: hot 2, packed 2 (total 4). Column totals: 7 hot and 9 packed; grand total 16, which matches the list.
Question 19 of 20 · Short Answer
A school asked students in each grade whether they stay for after-school clubs.
Stays
Does not
Total
Grade 6
12
28
40
Grade 7
18
27
45
Grade 8
28
22
50
Total
58
77
135
Find the percent of each grade that stays, and describe the association.
Grade 6: 12 ÷ 40 = 30%. Grade 7: 18 ÷ 45 = 40%. Grade 8: 28 ÷ 50 = 56%. The percent rises with each grade, so grade and staying for clubs are associated: older students in this school are more likely to stay.
Question 20 of 20 · Short Answer
School A has 300 students and 90 of them walk to school. School B has 120 students and 60 walk. A parent says, "More students walk at School A, so walking is more popular there." Build a two-way table (school by walks or not) and explain whether the parent is right.
Table: School A, walks 90 and does not 210 (total 300); School B, walks 60 and does not 60 (total 120); grand total 420. As row percents, 30% of School A walks and 50% of School B walks. The parent compared counts from schools of different sizes; walking is actually more common at School B.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.SP.A.4 mean?
8.SP.A.4 means students organize answers to two category questions in a two-way table and use relative frequencies to decide whether the two variables seem to be associated. The official example asks whether students who have a curfew on school nights also tend to have chores at home.
What is a two-way table?
A two-way table shows how many people fall into each combination of two categorical variables. One variable labels the rows and the other labels the columns. Each cell is a count, and the totals appear in the last row and column.
What is the difference between frequency and relative frequency?
A frequency is a count, such as 20 students. A relative frequency divides a count by a total, such as 20 out of 26, about 76.9%. Relative frequencies let you compare groups of different sizes fairly.
Should I use row or column relative frequencies?
Use the total of the group the question is about. "Of the students with a curfew" means divide by the curfew row total; "of the students with chores" means divide by the chores column total. To look for association, either choice works if you compare the percents across the groups you divided by.
How do you tell if there is an association in a two-way table?
Compare relative frequencies across the rows (or across the columns). If the percents are very different, the variables appear to be associated. If they are close, there is little evidence of association. Small groups can give different percents by chance, so be careful with small surveys.
Is 8.SP.A.4 about probability?
Not directly. 8.SP.A.4 is about describing data. The same tables return in high school probability (HSS.CP.A.4), where relative frequencies are read as conditional probabilities and used to test whether events are independent.
Does an association in a two-way table mean one thing causes the other?
No. A survey only shows that two variables go together. A third factor could explain both, or the cause could run the other way. To show a cause, you need a designed experiment, which students study later.
How is 8.SP.A.4 different from scatter plots in 8.SP.A.1?
Scatter plots show association between two measured, numerical variables. Two-way tables show association between two categorical variables. The question is the same in both: do the two variables tend to go together?
What mistakes do students make with two-way tables?
Common ones are dividing by the grand total instead of a row or column total, mixing up row and column percents, comparing counts from groups of different sizes, forgetting to check that the totals add up, and claiming that one variable causes the other.
How can parents help with 8.SP.A.4 at home?
Ask two yes-or-no questions of family and friends, such as "Do you drink coffee?" and "Are you a morning person?", and sort the answers into a table together. Then ask your child what percent of each group answered yes, and whether the two answers seem to go together.
07
Related Standards
5 standards
These standards connect to 8.SP.A.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.3Prerequisite
Use ratio and rate reasoning, including percents, to solve problems
