Scatterplots (2 of 5)

Learning OUTCOMES

  • Use a scatterplot to display the relationship between two quantitative variables. Describe the overall pattern (form, direction, and strength) and striking deviations from the pattern.

Interpreting the Scatterplot

How do we describe the relationship between two quantitative variables using a scatterplot? We describe the overall pattern and deviations from that pattern.

This is the same way we described the distribution of one quantitative variable using a dotplot or a histogram in Summarizing Data Graphically and Numerically. To describe the overall pattern of the distribution of one quantitative variable, we describe the shape, center, and spread. We also describe deviations from the pattern (outliers).

Flowchart of graphing the distribution of 2 quantitative variables in a scatterplot, which includes overall patterns and derivations from the patterns

Similarly, in a scatterplot, we describe the overall pattern with descriptions of direction, form, and strength. Deviations from the pattern are still called outliers.

The direction of the relationship can be positive, negative, or neither:

Scatterplot showing positive relationship. Dots on scatterplot are making an upwards trajectory.Scatterplot showing negative relationship. Scatterplots are making a downwards trajectoryScatterplot showing neither positive nor negative relationship. Dots on the scatterplot form a U shape.

A positive (or increasing) relationship means that an increase in one of the variables is associated with an increase in the other.A negative (or decreasing) relationship means that an increase in one of the variables is associated with a decrease in the other.  Note that not all relationships can be classified as either positive or negative.

The form of the relationship is its general shape. To identify the form, describe the shape of the data in the scatterplot. In practice, forms that we commonly use have mathematical equations. We look at a few of these equations in this course. For now, we simply describe the shape of the pattern in the scatterplot. Here are a couple of forms that are quite common:

Linear form: The data points appear scattered about a line. We use a line to summarize the pattern in the data. We study the equation for a line in this module.

Scatterplot with linear form, where the dots are all clustered around a line

Curvilinear form: The data points appear scattered about a smooth curve. We use a curve to summarize the pattern in the data. We study some specific types of curvilinear forms with their equations in Modules 4 and 12.

Scatterplot with curvilinear form, where the dots follow a single curved line

The strength of the relationship is a description of how closely the data follow the form of the relationship. Let’s look, for example, at the following two scatterplots displaying positive, linear relationships:

Scatterplot showing strongly related data, where the dots all cluster densely around a single lineScatterplot showing weakly related data, where dots are only loosely clustered around a line

In the left scatterplot, the data points closely follow the linear pattern. This is an example of a strong linear relationship. In the right scatterplot, the data points also follow a linear pattern, but the points are not as close to the line. The data is more scattered about the line. This is an example of a weaker linear relationship. Labeling a relationship as strong or weak is not very precise. We develop a more precise way to measure the strength of a relationship shortly.

Outliers are points that deviate from the pattern of the relationship. In the scatterplot below, there is one outlier.

Scatterplot showing outlier, where one single red dot stands alone from a grouping of black dots

Try It

A series of 6 scatterplots, each with 10-12 points. Scatterplot 1: points are arranged in a U shape. Scatterplot 2: points are in a curve starting at the origin and increasing exponentially upwards. Scatterplot 3: points start at the origin and increase upward linearly. Scatterplot 4: points are fairly scattered, but display an overall downward trend from left to right. Scatterplot 5: points are arranged in a bell curve shape. Scatterplot 6: points are fairly scattered, but display an overall increasing upward trend.

Match each scatterplot with the correct description.

Descriptions:

A: X = month (January = 1), Y = rainfall (inches) in Napa, CA in 2010 (Hint: Napa has rain in the winter months and months with little to no rainfall in summer. Rainfall decreases from January to June, with no rainfall for several months in summer. It begins to rain again in October, and rainfall increases through the winter months.)

B: X = month (January = 1), Y = average temperature in Boston, MA, in 2010 (Hint: Boston has cold winters and hot summers. Temperature increases from January to mid-summer, peaks, then decreases through the fall.)

C: X = year (in five-year increments from 1970), Y = Medicare costs (in $) (Hint: the yearly increase in Medicare costs has gotten bigger and bigger over time.)

D: X = average temperature in Boston MA (°F), Y = average temperature in Boston MA (°C) each month in 2010 (Hint: you know that there is a mathematical formula relating °F and °C.)

E: X = chest girth (cm), Y = shoulder girth (cm) for a sample of men (Hint: men with larger chest girth tend to have larger shoulder girth).

F: X = engine displacement (liters), Y = city miles per gallon for a sample of cars (Hint: engine displacement is roughly a measure of engine size. Large engines use more gas. If more gas is needed, the cars get fewer miles.)

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