4-3 Scatter Plots and Lines of Fit

Property

Bivariate data consists of pairs of values for two different variables. A scatter plot is a graph that displays these pairs of data as points $(x, y)$ on a coordinate plane to show the relationship between the two variables.

• $x$-axis: Represents the independent variable (the input or explanatory variable).
• $y$-axis: Represents the dependent variable (the output or response variable).
• Axis Break: A symbol used on an axis to indicate that a range of values starting from 0 has been skipped to avoid empty space.

Examples

• For data comparing "Hours Studied" and "Test Score", "Hours Studied" is the independent variable ($x$-axis) and "Test Score" is the dependent variable ($y$-axis). A single data point could be $(3, 85)$, representing 3 hours of study and a score of 85.
• If a dataset of weights ranges from 120 lbs to 160 lbs, place an axis break between 0 and 120 on the axis, then use consistent increments of 10.

Explanation

To construct a scatter plot, first determine which variable is independent ($x$-axis) and which is dependent ($y$-axis). Next, choose a scale with consistent increments that covers the entire range of your data. If data values start far from zero, an axis break skips the empty space, making the pattern in the data much easier to see. Finally, plot each observation as an ordered pair $(x, y)$.

Property

Correlation describes the relationship between two variables in a scatter plot: Positive correlation occurs when both variables increase together (points trend upward from left to right), Negative correlation occurs when one variable increases while the other decreases (points trend downward from left to right), and No correlation occurs when there is no clear linear pattern between the variables (points appear randomly scattered).

Examples

Property

Correlation means two variables show a pattern of change together in a scatter plot — as one variable increases or decreases, the other tends to as well. Causation means that a change in one variable directly causes a change in the other.

A strong correlation (positive or negative) between two variables does not necessarily mean one variable causes the other to change.

Property

A trend line (or line of best fit/regression line) is a straight line drawn on a scatter plot that models the relationship between two quantitative variables. It provides a one-dimensional summary of a bivariate (2-dimensional) data set by showing the general direction of the data.

To draw it, use the primary method of 'eye-balling' to find the line that minimizes the distance between each data point and that line. A line of best fit will have about the same number of points above and below it and may or may not pass through any of the data points.

Examples

• If the number of hours studied increases and test scores also tend to increase, a trend line would have a positive slope, showing a positive association.
• A scatter plot shows hours spent practicing piano versus number of mistakes made in a performance. The points trend downwards, so an 'eyeballed' line with a negative slope is drawn to show that more practice is associated with fewer mistakes.
• Data is collected on daily temperature and the number of bottles of water sold at a park. The points on the scatter plot go up and to the right. An 'eyeballed' line with a positive slope summarizes this positive association.