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# Describing and Displaying Bivariate Data Study Guide

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## Introduction to Describing and Displaying Bivariate Data

Often, two or more numerical measurements are taken on each observational unit. For example, the weight and length of each fish taken from a lake may be recorded. In studies of wetlands, the phosphorus and nitrogen concentrations at several sites might be observed. In another study, the goal was to determine whether a person's ear grows throughout life. For each person in the sample, age and the length of the left ear were recorded. In these studies, we are interested in not only the values of each of the variables assumed, but also in how they relate to each other. In this lesson, graphical displays for bivariate data and a measure of the relationship of two variables will be explored.

## Scatter Plots

Suppose that more than one, say two, numerical values are recorded for each unit in the study. Sometimes, both of the variables are responses.At other times, we are interested in how the response variable, or dependent variable, relates to the explanatory variable, or independent or predictor variable. In the latter case,we may want to predict the value of the response variable for a specific value of the explanatory variable.More than one explanatory variable may exist for each response variable.

When working with univariate data, we saw that plots (pie charts, bar charts, dotplots, stem-and-leaf plots, histograms, and boxplots) aided our understanding of the data. Although we could construct these types of graphs for each variable and gain a better understanding of each variable, such graphs would not aid our understanding of how the two variables are related. A scatter plot is an effective graph for gaining insight into bivariate data. A scatter plot is a graph in which each observation (pair of numbers) is represented by a point in a rectangular coordinate system. The horizontal axis is identified with the x-axis, and the axis is scaled to cover the range of values of X. The vertical axis is identified with the y-axis, and the axis is scaled to cover the range of values of Y. If both an explanatory and a response variable exist, the x-axis is used for the explanatory variable, and the y-axis is used for the response variable. The point representing the observation (x,y) is placed at the intersection of the vertical line through x on the x-axis and the horizontal line through y on the y-axis. Figure 8.1 shows the point representing the observation (2.5,3) with the corresponding vertical and horizontal lines. These reference lines are not usually included in the plot; they are here only for illustration. All data points would be plotted using the same approach.

#### Example

A group of students wanted to know whether there was a relationship in the height from which a ball was dropped and its rebound height. Using a basketball, they dropped the ball from each of 11 heights three times and measured how high it rebounded. Both the height from which the ball was dropped and the height of the rebound were measured, in inches, from the bottom of the ball. The data are given in Table 8.1.

1. Specify whether drop height and rebound height are explanatory or response variables.
2. Create a scatter plot of the data and describe the relationship between drop height and rebound height.

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