8th Grade Two-Way Tables and Relative Frequency Worksheet Solutions (PDF)
This two way tables answers for 8th Grade covers two-way tables and relative frequency (Common Core 8.SP.A.4), with a free PDF to print.
This answer key gives a full worked solution for each of the 20 problems on the two-way tables worksheet (Common Core 8.SP.A.4). Compare each step with your own work, not only the final answer, and note any step where your method differed.
Grade 8 · Common Core State Standards: 8.SP.A.4
Two way tables answers: exercises 1 to 10
Exercise 1 – Read a joint frequency
Read the cell in the Girls row and the Phone column.
38 girls
Exercise 2 – Marginal totals
Boys: \(34 + 6 = 40\). Phone: \(34 + 38 = 72\). Grand total: \(40 + 50 = 90\).
40, 72 and 90
Exercise 3 – Relative frequency of the total
\(\frac{38}{90} \approx 0.422\).
about 42.2%
Exercise 4 – Row relative frequency
Boys: \(\frac{34}{40} = 0.85\). Girls: \(\frac{38}{50} = 0.76\).
85% of boys and 76% of girls
Exercise 5 – Column relative frequency
\(\frac{34}{72} \approx 0.472\).
about 47.2%
Exercise 6 – Compare groups
85% and 76% are only 9 percentage points apart, so the association is weak.
Only a weak association
Exercise 7 – Missing cells
Sport row: \(30 – 18 = 12\). No sport, reads daily: \(32 – 18 = 14\), also \(50 – 36 = 14\).
12, 14 and 32
Exercise 8 – Sport and reading percents
Sport: \(\frac{18}{30} = 60\%\). No sport: \(\frac{14}{50} = 28\%\).
60% and 28%
Exercise 9 – Association?
60% versus 28% is a large difference of 32 percentage points.
Yes, there is an association
Exercise 10 – Of the daily readers
\(\frac{18}{32} = 0.5625\).
56.25%, about 56%
Exercises 11 and up
Exercise 11 – Three categories
7th: \(\frac{20}{50} = 40\%\). 8th: \(\frac{15}{60} = 25\%\).
40% and 25%
Exercise 12 – Largest category
Pizza total: \(25 + 35 = 60\). \(\frac{60}{110} \approx 54.5\%\).
Pizza, about 54.5%
Exercise 13 – Recycling
Maple: \(\frac{45}{50} = 90\%\). Oak: \(\frac{30}{50} = 60\%\).
90% and 60%
Exercise 14 – Recycling association
The rates differ by \(90 – 60 = 30\) percentage points.
Yes, a clear association (30 points)
Exercise 15 – Bake sale table
Morning total: 30. Afternoon total: 30. Muffins: 32. Cookies: 28.
Morning muffins: \(\frac{20}{30} \approx 66.7\%\).
about 66.7%
Exercise 16 – Build a table
Bike: 8; no bike: 4; walks: 4; does not walk: 8; total: 12
Exercise 17 – Reading a percent
There are 40 boys and 50 girls. Counts depend on the group size, so compare percents: 85% of boys and 76% of girls.
Compare relative frequencies, not counts
Exercise 18 – Sleep survey
Grade 7: \(\frac{22}{40} = 55\%\). Grade 8: \(\frac{15}{40} = 37.5\%\).
55% and 37.5%; Grade 7 is 17.5 points higher, a moderate difference
Exercise 19 – Cause and effect
No. The table shows an association only. Other factors, like how much free time a student has, could explain the pattern.
No, association is not causation
Exercise 20 – Interpret a percent
Sixty percent of the students who play a sport read daily, because \(\frac{18}{30} = 0.6\).
60% of athletes read daily
Key facts behind this two way tables answers
- Joint frequencies are the counts inside the table; marginal frequencies are the totals.
- Relative frequency = count ÷ total, usually as a percent.
- Row percents compare groups defined by the rows.
- If row percents differ a lot, the variables are associated.
- Association does not prove that one variable causes the other.
How to use this two way tables answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
All Grade 8 chapters: Grade 8 math.
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