Question

Association. Suppose you were to collect data for each pair of variables. You want to make a scatterplot. Which variable would you use as the explanatory variable and which as the response variable? Why? What would you expect to see in the scatterplot? Discuss the likely direction, form, and strength. a) T-shirts at a store: price each, number sold b) Scuba diving: depth, water pressure c) Scuba diving: depth, visibility d) All elementary school students: weight, score on a reading test

Short answer

a) Explanatory: price, Response: number sold. Expect a negative trend. b) Explanatory: depth, Response: water pressure. Expect a positive trend. c) Explanatory: depth, Response: visibility. Expect a negative trend. d) Challenging to establish; expect weak or no pattern.

Step-by-step solution

Understanding Explanatory vs Response Variables

Explanatory variables are the ones that you believe might cause a change in another variable, which is the response variable. The explanatory variable is plotted on the x-axis, while the response variable is plotted on the y-axis. Let's evaluate which variable is the cause and which is the effect for each pair.

Scenario a - T-shirts at a store

For the pair 'price each' and 'number sold': The price of a T-shirt (explanatory variable) likely affects the number sold (response variable). Typically, a higher price might result in fewer sales, indicating a negative association. In the scatterplot, you might see a downward trend showing this negative relationship.

Scenario b - Scuba diving depth and water pressure

For 'depth' and 'water pressure': As you dive deeper (explanatory variable), water pressure increases (response variable). This is a direct positive relationship, and you would expect to see a linear, upward trend in the scatterplot, reflecting a strong positive association.

Scenario c - Scuba diving depth and visibility

For 'depth' and 'visibility': As you dive deeper (explanatory variable), visibility often decreases (response variable). Thus, you'd expect a negative association. The scatterplot would likely show a downward trend as depth increases, indicating this relationship.

Scenario d - School students' weight and reading test score

For 'weight' and 'score on reading test': It's challenging to establish a causal relationship. Generally, weight is not an explanatory variable for reading scores as it is less likely to cause changes in academic performance. Expect the scatterplot to show little to no consistent pattern, suggesting weak or no association.

Key concepts

Explanatory Variable

When working with pairs of variables in data analysis, particularly in scatterplots, identifying which is the explanatory variable is a crucial step. The explanatory variable is the one that might influence or explain changes in the other variable, known as the response variable. Think of it as the supposed cause in a cause-and-effect relationship. It is plotted along the x-axis of a scatterplot.
To better understand this, consider the example of the price of T-shirts and the number of shirts sold. Here, the price serves as the explanatory variable because variations in price can lead to changes in sales volume.
However, remember that an explanatory variable does not always imply causation but suggests a likely influence over the response variable.

Response Variable

The response variable is the variable that is of primary interest, and it's what you measure the changes in, based on the influence of the explanatory variable. It appears on the y-axis of a scatterplot and is often thought of as the effect of a study.
For instance, in a scuba diving scenario, if we're studying the depth of dive as it affects water pressure, the water pressure becomes the response variable.

• It changes in response to diving deeper.
• How much water pressure increases can be measured as we change the depth.

This relationship forms the backbone of analyzing data through scatterplots.

Data Association

Data association illustrates the relationship between two variables. It inspects how one variable changes concerning another, observed through trends in a scatterplot. There are mainly three types of associations: positive, negative, and no association.
A positive association occurs when increases in the explanatory variable lead to increases in the response variable, depicted as an upward trend. For instance, water pressure increasing with diving depth is a positive association.
On the other hand, a negative association has the response variable decrease as the explanatory variable increases, seen in the negative trend of T-shirt sales reducing as price increases.
Occasionally, variables may showcase no association, like student weight with reading scores, where no clear pattern emerges.

Scatterplot Trends

Scatterplots visually represent how two variables relate to each other. The key trends one might notice include direction, form, and strength.
**Direction** refers to whether the relationship is positive or negative. A positive trend rises from left to right, while a negative one falls.
**Form** involves the overall shape of the line, such as linear or non-linear. A linear trend shows consistent change, like the depth-pressure relationship in scuba diving, which is linear and upward.
**Strength** gauges how closely points follow the pattern. A strong association means most points are near the trend line, indicative of a reliable relationship, such as in water pressure's relation to depth.

• Strong, positive: closely packed upward line.
• Strong, negative: closely packed downward line.
• Weak association: scatter widely spread out.

Understanding these trends helps in making meaningful interpretations of data within a scatterplot.