Graphical Representations Between Two Quantitative Variables: Study Guide (Statistics 5.1)

From one variable to two

In Unit 1 you described one quantitative variable at a time: its shape, centre and variability. Many questions are about how two quantitative variables relate. Do warmer days bring more customers? Do older cars sell for less?

To answer these, you need bivariate quantitative data. This means that for each individual you record two quantitative values, as an ordered pair (x, y). Both values must come from the same individual. If you measured temperature on some days and bike hires on different days, you could not pair them up.

A scatterplot shows bivariate quantitative data. Each individual becomes one dot. Its horizontal position is its x-value, and its vertical position is its y-value.

Explanatory and response variables

Before drawing, decide which variable goes on which axis.

• The explanatory variable is the one you use to explain or predict the other. It goes on the x-axis (horizontal).
• The response variable is the one whose values you want to explain or predict. It goes on the y-axis (vertical).

Ask: “Which variable might help me predict the other?” A bike-hire manager wants to predict how many bikes will be hired from the weather forecast. So temperature is explanatory (x) and number of bikes hired is response (y).

Sometimes there is no natural choice. For example, the heights of pairs of sisters. Then either choice is fine, as long as you label the axes.

“Explanatory” does not mean “cause”. It only says which variable you are using to predict the other.

The data set used in this guide

A fictional bike-hire scheme in the town of Harbourton recorded the maximum temperature (°C) and the number of bikes hired on 12 days in spring. Each day is one individual.

Max temperature (°C), x	8	10	11	13	14	16	17	19	20	22	23	25
Bikes hired, y	52	61	70	74	83	90	88	104	109	118	40	131

Constructing a scatterplot

1. Choose the axes. Explanatory variable on the x-axis, response on the y-axis.
2. Label each axis with the variable name and its units, for example “Maximum temperature (°C)”.
3. Choose scales. Each axis needs equal steps between tick marks. The scale should cover the smallest to the largest value and spread the points across most of the graph. Here x runs from 8 to 25 °C and y from 40 to 131 bikes, so axes from 5 to 25 °C and 30 to 140 bikes work well. An axis does not have to start at zero.
4. Plot one dot per individual. For the first day, go across to 8 °C and up to 52 bikes.
5. Do not join the dots. A scatterplot is not a line graph. The days are separate individuals.

Describing a scatterplot: four features

A full description covers form, direction, strength and unusual features, always in context.

Feature	Question to ask	Words to use
Form	Do the points follow a straight-line pattern or a curve?	linear; non-linear (curved)
Direction	As x increases, does y tend to increase or decrease?	positive; negative; (no clear direction)
Strength	How closely do the points follow the pattern?	strong; moderate; weak
Unusual features	Are there points far from the pattern, or separate groups?	unusual point (outlier); clusters; gaps

Direction in detail. A positive association means that as the explanatory variable increases, the response variable tends to increase. A negative association means that as the explanatory variable increases, the response tends to decrease. The words “tends to” matter. Not every point has to follow the trend.

Strength in detail. Strength is about scatter, not steepness. If the points lie in a tight band around the pattern, the association is strong. If they are widely scattered around it, it is weak. A gentle slope can still be a strong association.

Unusual features in detail.

• An unusual point lies well away from the general pattern. It may have an unusual x-value, an unusual y-value, or a usual x and y that do not fit together (like the rainy day above).
• Clusters are separate groups of points with a gap between them. They often mean the individuals come from different groups, such as weekdays and weekends.

Direction only makes sense for a pattern that keeps going one way. A curve that rises then falls has no single direction, so describe each part.

Worked example 1: describing the bike-hire scatterplot

Question. Describe the association between maximum temperature and the number of bikes hired shown in Figure 1.

1. Form. Apart from one day, the points follow a straight-line pattern. The form is linear.
2. Direction. As temperature increases, the number of bikes hired tends to increase. The direction is positive.
3. Strength. The 11 typical days lie very close to a straight band. The association is strong.
4. Unusual features. The day at 23 °C with only 40 hires is far below the pattern. Days near 23 °C had well over 100 hires. The scheme’s records show it rained all that day, which could explain it. There are no clusters.

Answer in context. There is a strong, positive, linear association between maximum temperature and the number of bikes hired at Harbourton: warmer days tend to have more bikes hired. One day stands out: at 23 °C only 40 bikes were hired, far fewer than on other days with similar temperatures.

Check. Each feature has a word and a reason from the plot, and the sentence names both variables.

Worked example 2: a curved, negative association

Question. A fictional dealer listed ten used cars of the same model. Construct and describe a scatterplot of price against age.

Age (years), x	1	2	3	3	4	5	6	7	8	10
Price ($ thousands), y	25.0	21.0	18.0	18.6	15.6	13.8	12.4	11.5	10.9	10.2

Step 1: axes. The dealer uses age to predict price, so age (years) is on the x-axis and price ($ thousands) on the y-axis. Two cars are 3 years old, so there are two dots above x = 3.

Step 2: form. The price falls by $4.0 thousand from age 1 to age 2 and by $3.0 thousand from age 2 to age 3, but by only $0.6 thousand from age 7 to age 8. The rate of decrease changes, so the form is non-linear (curved). Over the first four years (age 1 to 5) the price falls by $11.2 thousand; over the last four (age 6 to 10) by only $2.2 thousand.

Step 3: direction. As age increases, price tends to decrease. The direction is negative.

Step 4: strength. The points lie very close to a smooth curve with little scatter. The association is strong.

Step 5: unusual features. No point lies away from the curve, and there are no clusters.

Answer in context. There is a strong, negative, non-linear association between age and price for these cars: older cars tend to cost less, but the price drops quickly in the first few years and then levels off.

Check. “Non-linear” is justified by the changing rate of decrease, not just by the fact that the price changes.

Worked example 3: using a scatterplot to judge claims

Question. Use the Harbourton data to judge each claim.

(a) The manager says: “On warmer days we tend to hire out more bikes.” (b) A staff member says: “Whenever the maximum is 20 °C or more, we hire out at least 100 bikes.”

(a) Supported. The scatterplot shows a strong positive linear association. Days below 15 °C had between 52 and 83 hires. Six of the seven days above 15 °C had more than 85 hires. So warmer days did tend to have more hires.

(b) Not supported. Four days reached 20 °C or more. Three had at least 100 hires (109, 118 and 131), but the 23 °C day had only 40. One counterexample is enough to show “whenever” is false. Temperature is not the only thing that matters: rain matters too.

Careful with cause. The plot supports an association. It does not prove that warm weather alone causes more hires. Warm days in spring may also be weekends or school holidays. You will return to this in Topic 5.2.

Common misconceptions

• “It is linear because I can draw a line through it.” You can draw a straight line through any set of points. Form is linear only if the points follow a straight-line pattern, with a roughly constant rate of change.
• “Steep means strong.” Steepness is about how fast y changes. Strength is about how closely the points follow the pattern.
• “Positive means good.” Positive only means that both variables tend to increase together. A positive association between hours of screen use and eye strain is not “good”.
• “Every point must follow the trend.” An association describes a tendency. Write “tends to”, not “always”.
• Putting the response on the x-axis. The variable you want to predict goes on the y-axis.
• Joining the dots. The individuals are separate; a scatterplot has no line through the points.
• Describing without context. “Strong, positive, linear” is incomplete. Name both variables and say what the direction means for these individuals.
• Ignoring unusual features. One unusual point or a cluster can change the story. Mention it and, if you can, suggest a reason.

Where this leads

Next you will measure the strength and direction of a linear association with one number, the correlation coefficient r, in Topic 5.2: Correlation. Later in Unit 5 you will fit a line to a linear pattern and use it to predict. Try the practice questions now, then use the revision notes and the checklist to consolidate.