Representations of Categorical Data:
A Practice Understanding Task
I. What’s your favorite color? When asked this question, the most popular
color named was blue.
Symbols
Key:
Male = M
Blue = B
2-way Table
Female = F
Not Blue = N
Blue
Not Blue
Total
Sample size = 200
Male
P(B) = 84/200
P(M) = 64/200
Female
P(F|B) = 48/84
Total
P(B|F) =
P(MB) =
Is color preference independent of gender? How
do you know?
P(MB) =
Venn Diagram
Tree Diagram
II.
Are you a lefty or a righty?
Symbols
Key:
Male = M
Lefty = L
2-way Table
Female = F
Righty = R
Lefty
Righty
Total
Sample size =
Male
P(L) =
P(M) =
Female
P(F) =
Total
P(L|F) =
Is “handedness” independent from gender?
How do you know?
P(L|M) =
In this sample are there equal proportions of males and
females who are left handed? Explain.
Venn Diagram
Tree Diagram
III.
Do you eat breakfast or not?
Symbols
Key:
Male = M
Eats Breakfast = E
2-way Table
Female = F
Doesn’t Eat Breakfast = D
Eats
Doesn’t
Total
Sample size =
Male
P(E) =
P(E|M) =
Female
P(EM)
685
Total
P(E|F) =
P(EF)
Is eating breakfast (or not) independent from
gender? How do you know?
Venn Diagram
Tree Diagram
IV.
The Humane Society likes to keep track of the
percent of people who are dog and cat owners.
Some people own only dogs, some only own
cats, and some own both. The Human Society
reports that 13% of households own both dogs
and cats, 33% own cats, 39% own dogs, and
41% don’t own either.
Symbols
2-way Table
Venn Diagram
Tree Diagram
Representations of Categorical Data: – Teacher Notes
A Practice Understanding Task
Purpose: To reinforce the connections between different representations. This task would follow
instruction on tree diagrams.
Core Standards Focus:
Related Standards:
Launch (Whole Class):
Minimal launch. Each part of this task is a puzzle with missing parts. You may use the first item as a
launch by letting the students work for 3-5 minutes on the table representation. Then, as a class, checking
for understanding and then deciding whether or not to do the Venn diagram and tree diagram as a guided
practice with whole-class or let it be a part of the small-group explore.
Explore (Small Group):
Participants complete the missing parts, with the goal of solidifying the connections between the multiple
representations that have been developed.
Discuss (Whole Class):
Allow participants to verify solutions and self correct.
1. Ask participants, what did you go to first to solve? (Most will say the table.) Have one
participant share the top half of the task sheet. Expose all the connections between the
probabilities and the tables. “Why did you put 200 here?” “Where did the 64 and the 136 come
from?” “What was the hardest piece of filling in the table.” Press for information, what did you
do and why did you do it? (Purposefully ignore independence for now.)
2. Reinforce any ideas that have surfaced during the day. For instance, finding P(Male and Blue)
using the table. Finding P(Male or Blue).
3. Why didn’t you go to the Venn diagram first? Ask participants to relate the Venn diagram to the
table (this is a difficult concept). Sketch one of the possible Venn diagrams directly on the table.
What are the other possible Venn diagrams.
4. Chose a participant’s tree diagram response with counts. Make connections, first between the 2way table and the tree diagram, then between the Venn diagram and the tree diagram. “Where is
the 64? Where is the 136?” Connect to the probabilities. Where is the 48/136? Where is the
48/84?” Could you build a tree diagram that made the other probabilities more visible? Where
is the 200 on the tree diagram?”
5. Repeat the discussion with a participant that used probabilities on their tree diagram.
6. Distribute the final 3 pages of the task and if there is time, allow participants to complete the
tasks.
7. Allow time to discuss independence on the dog & cat question. Why did you decide they were
independent?
8. How close do the probabilities need to be for us to say the categories are independent?
9. Go back to page 1, “Is color preference independent of gender? How do you know?”. Does this
make sense in real life?
10. Repeat independence discussion for pages 2 and 3.
11. Discuss that the purpose of many surveys and research is to establish either dependence or
independence between variables.