Investigating Relationships between Variables: Interpreting Scatterplots

Examining Relationships Is there a relationship between the amount of fat and the number of calories in a burger??? We can investigate this question by using techniques designed to look at the relationships among variables. Although there are many of these, for now we will concentrate on those used to look at relationships between two quantitative variables

The Data Fast food is often considered unhealthy because much fast food is high in both fat and calories. But are the two related?? Here are the fat and calorie contents of several brands of burgers. Fat Calories 19 410 31 580 34 590 35 570 39 640 680 43 660

Creating a Picture Of course, we still want to follow the first rule of statistics---Draw a picture. The plot we use to show relationships is called a scatterplot.

Fat & Calories We think that the amount of fat in a burger will help to explain the number of calories, so we call this variable our “explanatory” variable and we plot it on the x-axis. “Calories” is our response variable, and is graphed along the vertical or y-axis.

Interpreting Scatterplots When looking at the relationships between 2 quantitative variables, we want to consider any overall pattern that appears. Three aspects help to guide our description: Strength Type (form) Direction

Let’s start with strength--- To help determine the strength of a relationship, think about drawing an oval around the data. Create, if you will, a “data cloud” Interpretation: A diagonal oval either increasing or decreasing indicates a relationship The tighter the oval---the stronger the relationship A horizontal oval indicates no relationship A circle would indicate a weak or no relationship

The second element Type (form)— The type of relationship that exists can be described (at least for now) as either linear or nonlinear. In other words, if we can summarize the data with a line (not a curve) through the middle of the data, we would consider the relationship to be linear. If a curve would do a better job, then the relationship would be categorized as non-linear. (we’ll look at non-linear relationships later) We notice that a line, rather than a curve appears to summarize the data. (we’ll learn some additional ways to check this aspect later)

The last characteristic we use: Direction: To determine the direction of the relationship, we think about the how each variable is changing with respect to the other. There are several ways to think about this. A relationship is positive when above average values of one variable occur with above average values of the other variable. We can also see the positive relationship in the scatterplot very quickly As one variable is increasing, the other variable also increases

The scatterplot at the right shows horsepower ratings for several models of vehicles and the corresponding gas mileage. The scatterplot indicates that the relationship between these variables is negative Again we can “see” the negative relationship between the variables in the scatterplot. As one variable is increasing, the value of the second variable is decreasing. A negative relationship occurs when above average values of one variable happen with below average values of the second variable.

Outliers We know that an outlier is a piece of data that does not seem to fit the overall pattern So what does that look like in bivariate data? Consider the burger that has 19grams of fat and 410 calories This piece of data may need to be looked at more closely since it is so far away from the rest of the data

Putting it all Together Now that we know the individual elements for investigating the relationship between two quantitative variables, let’s see if we can summarize the association (another word for relationship) between the amount of fat and the calorie content in fast food burgers.

There appears to be a strong There appears to be a strong* relationship between the amount of fat in a burger and the calories in that burger. As the fat increases, so does the calories, indicating a positive association between these variables. *Note: we will learn some numeric guidelines on strength in the next lesson about correlation.

Additional Resources The Practice of Statistics: 1st Edition: YMM—pg 106-111 2nd Edition: YMS—pg 120-140 The Basic Practice of Statistics—Moore Pg 79-88 Against All Odds Video Video #8---Describing Relationships