Two-way tables and scatterplots
Reading and constructing two-way frequency tables, describing association from a scatterplot, and telling association apart from causation
About three lessons of 45 to 60 minutes
Does one thing actually CAUSE the other, or do they just move together?
Fire departments notice that fires with more fire trucks sent to them also tend to have more damage. Does sending more trucks CAUSE more damage? Of course not, bigger fires cause both: more trucks are sent to bigger fires, and bigger fires cause more damage. The trucks and the damage are associated, but neither one causes the other directly.
This unit covers two ways of comparing two variables at once: a two-way table for two CATEGORIES (like owns a cat: yes/no, and owns a dog: yes/no), and a scatterplot for two NUMBERS (like hours of sunlight and plant height). Both let you describe how the variables relate, but describing an association is only the first step, the fire truck example shows why the next step, checking whether that association is trustworthy, matters just as much.
- Fire trucks sent vs damage causeda strong association, but fire SIZE causes both, not one causing the other
- Owning a cat AND a doga two-way table sorts every combination of two yes/no categories
- Hours of sunlight vs plant heighta scatterplot shows the pattern between two numerical measurements
- Ice cream sales and swimming pool visitsboth rise together in summer, driven by hot weather, not by each other
What students will be able to do
Students will construct and read a two-way table to find totals for combinations of two categories, describe the association shown by a scatterplot in terms of direction, strength and linearity, and explain why a strong association does not prove that one variable causes another.
- I can read a two-way table to find how many items fall into a given combination of two categories.
- I can find a row or column total from a two-way table's four combination counts.
- I can describe a scatterplot's association as positive, negative or none, and as strong, weak, or not linear.
- I can explain why a strong association between two variables does not prove that one causes the other.
Standards this unit teaches
- AC9M10ST04Australian Curriculum v9 (ACARA)Two-way tables
Construct two-way tables and discuss the possible relationship between categorical variables.
- AC9M10ST03Australian Curriculum v9 (ACARA)Scatterplots and association
Construct scatterplots and describe the association between two numerical variables by strength, direction and linearity.
- AC9M10ST05Australian Curriculum v9 (ACARA)Investigate bivariate data
Plan and conduct investigations with bivariate data, then evaluate and report findings, noting the limits of any inferences.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Year 8-9 compound & conditional probability teaching unitthe Year 8 two-way-table skill for combinations of 2 events; this unit extends the same table structure to categorical-variable relationships and numerical bivariate data
- Coordinates in the glossaryplotting an (x, y) point is exactly how each dot on a scatterplot is placed
- Year 7 statistics: mean, median, mode & range teaching unitsummarising a single numerical variable, before this unit compares two at once
Words to teach and display
- Two-way table
- a table that organises counts by two categories at once, such as owns a cat (yes/no) and owns a dog (yes/no)
- Bivariate data
- data that pairs up two different measurements or categories for the same item
- Association
- whether two variables tend to move together (positive), move oppositely (negative), or show no clear pattern
- Linearity
- how closely a scatterplot's points follow a straight line, rather than a curve or a random spread
- Causation
- when one variable directly makes another change; much stronger, and harder to prove, than an association
- Confounding variable
- a third factor that could explain why two variables appear associated, without one causing the other
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Two-way tables: organising two categories at once
ConcreteA two-way table sorts a group by two yes/no categories at the same time, giving four combination cells: both, first-only, second-only, and neither. Every item in the group falls into exactly one of the four cells, so the four counts always add up to the total.
For a survey of 40 students on pizza and tacos: 15 like both, 10 like pizza but not tacos, 8 like tacos but not pizza, and 7 like neither. Arranged in a table (rows: pizza / not pizza, columns: tacos / not tacos), each row and column total is found by adding across or down.
A survey of 40 students found: 15 like both pizza and tacos, 10 like pizza but not tacos, 8 like tacos but not pizza, and 7 like neither. How many students like tacos in total?
- Check the total: 15 + 10 + 8 + 7 = 40, matching the group size.
- 'Likes tacos in total' combines the 'both' cell and the 'tacos but not pizza' cell: 15 + 8.
- 15 + 8 = 23.
Answer: 23 students like tacos in total (out of 40 surveyed).
- Why must the four combination cells in a two-way table always add up to the total group size?
- Why is the 'both' cell needed to find a row or column total, not just the two 'only' cells?
2. Scatterplots: describing association
PictorialA scatterplot plots two numerical measurements for the same item as a single dot, with no connecting line. Its association is described by direction (positive, negative or none), strength (how tightly the points cluster around a trend), and linearity (how closely that trend follows a straight line).
If the dots trend upward from left to right, that is a positive association: as one quantity increases, the other tends to increase too. A negative association trends downward. Points scattered with no clear upward or downward drift show no association.
A scatterplot plots hours since sunrise (x) against outdoor temperature in degrees C (y): (0, 12), (2, 16), (4, 19), (6, 23), (8, 26). Describe the association, including its direction and strength.
- Read the trend in the y-values as x increases: 12, 16, 19, 23, 26, each one larger than the last.
- Since y consistently increases as x increases, the direction is positive.
- The points sit close to a straight upward line with no big jumps, so the association is strong and roughly linear.
Answer: A strong, positive, roughly linear association: temperature tends to rise steadily as hours since sunrise increase.
- What would this scatterplot look like if there were a strong NEGATIVE association instead?
- How can two scatterplots both show a positive association, but one be described as 'strong' and the other 'weak'?
3. Association is not causation: investigating bivariate data honestly
AbstractDescribing an association is only the first step. Before concluding that one variable causes another, check for a confounding variable, a third factor that could explain both, and remember that a scatterplot alone can never prove causation, no matter how strong the pattern looks.
Ice cream sales and swimming pool visits both rise together every summer. Hot weather is the confounding variable causing both; ice cream sales do not cause more swimming, and swimming does not cause more ice cream sales. A genuine investigation reports the association it found, and is honest about what it does, and does not, prove.
A study finds a strong positive association between the number of fire trucks sent to a fire and the amount of damage caused. A newspaper concludes that 'sending fire trucks causes more damage'. Explain the flaw in this conclusion, and identify a more likely explanation.
- Check whether the association proves causation: a strong association only shows the two variables tend to move together, not that one causes the other.
- Look for a confounding variable, a third factor that could explain both: the SIZE of the fire likely determines both how many trucks are sent and how much damage occurs.
- State the more likely explanation: larger fires cause both more trucks to be sent and more damage, rather than the trucks causing the damage.
Answer: The conclusion confuses association with causation. Fire size is a confounding variable explaining both the number of trucks and the amount of damage; sending more trucks does not cause more damage.
- Why does a strong association NOT prove that one variable causes the other?
- What is a confounding variable, and how did it explain the fire truck example?
Common misconceptions and how to address them
MisconceptionA strong association between two variables always means one causes the other.
Why it happens: The word 'association' sounds causal, and a clear pattern feels like proof on its own.
How to address it: Always ask whether a confounding variable could explain both, as in the fire truck and ice cream examples. Only a carefully controlled experiment, not just observed data, can properly support a causal claim.
MisconceptionA two-way table's 'both' cell can be found by adding the two 'only' totals together.
Why it happens: Students confuse the overlap cell with the two individual category totals.
How to address it: The 'both' cell is given directly by the data (or found by subtraction from a total), never by adding the two 'only' cells, since that would double count nothing, it would simply give the wrong number entirely. Point at the actual cell in the table each time.
MisconceptionIf a scatterplot's points are not in a perfectly straight line, there is no association.
Why it happens: Students expect real-world data to look as clean as a textbook diagram.
How to address it: Real data almost always scatters around a trend. Association is judged by the OVERALL pattern, its direction and how tightly the points cluster, not by requiring a perfectly straight line.
Guided practice (with answers)
1. A survey of 30 students found 12 play both chess and soccer, 6 play chess but not soccer, 5 play soccer but not chess, and 7 play neither. How many students play chess in total?
Answer: 18, because 12 (both) + 6 (chess only) = 18.
2. Using the same survey, how many students play neither sport?
Answer: 7, read directly from the survey description.
3. A scatterplot shows y consistently decreasing as x increases, following a fairly straight downward trend. Describe the association.
Answer: A strong, negative, roughly linear association.
4. A scatterplot shows points scattered with no clear upward or downward trend. Describe the association.
Answer: No association: the two variables show no consistent relationship.
5. Two variables have a strong positive association. Does this prove that one causes the other? Explain.
Answer: No. Association does not prove causation; a confounding variable could explain both, or the relationship could be coincidental.
Independent practice worksheets
Practise reading two-way tables and describing scatterplot association with computed, never-wrong answer keys.
Differentiation
- Provide a blank 2-by-2 table template with row and column headers already labelled, ready for the four counts to be filled in.
- Use a 'both / only A / only B / neither' chart with physical counters before moving to numbers.
- Start scatterplot description with only very strong, obvious positive or negative examples before introducing weak or no-association data.
- Provide a simple checklist for describing association: direction (up, down, or none), strength (tight or loose spread), shape (straight or curved).
- Introduce a three-category two-way table (a 3-by-3 table) for a richer breakdown.
- Ask students to invent a scenario with a strong association that is clearly NOT causal, explaining the likely confounding variable.
- Explore how the same data can look like a stronger or weaker association depending on the scale used on each axis.
- Investigate a real, published data set (such as sports or weather data) and construct a genuine two-way table or scatterplot from it.
Assessment: exit ticket
A three-question exit ticket sampling two-way tables, scatterplot association, and association versus causation.
1. A survey of 50 people found 20 own both a cat and a dog, 14 own a cat but not a dog, 9 own a dog but not a cat, and 7 own neither. How many people own a dog in total?
Answer: 29, because 20 (both) + 9 (dog only) = 29.
2. A scatterplot shows y decreasing steadily as x increases, with points sitting close to a straight line. Describe the association.
Answer: A strong, negative, roughly linear association.
3. Two towns show a strong positive association between the number of ice cream shops and the number of swimming pool visits each summer. Does opening more ice cream shops cause more swimming? Explain.
Answer: No; both are more likely explained by hot summer weather, a confounding variable that increases both ice cream sales and swimming, rather than a direct causal link between the two.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 two-way tables (section 1), Lesson 2 scatterplots and association (section 2), Lesson 3 association versus causation (section 3) plus mixed review and the exit ticket.
- This unit assumes comfort with plotting coordinate points (Grade 6-7 coordinate plane units) and basic addition and subtraction for table totals.
- Language to repeat: 'the both cell is one of the four cells, never an addition of totals'; 'association describes a pattern, causation explains WHY'.
- Curriculum note: AC9M10ST03, AC9M10ST04 and AC9M10ST05 (Australian Curriculum v9) are all Year 10 codes. AC9M8P02 (Probability strand) covers a two-way table for combinations of 2 events one level earlier, at Year 8, in a probability context rather than a categorical-relationship context, linked as prior knowledge above rather than repeated here.
- Present and print both work: use Present to build the two-way table and the scatterplot live with the class, then print the worksheet for independent practice.