Scatter plots and trend lines Knowledge-Based Study

Study notes

1. What Is a Scatter Plot?

   A scatter plot is a two-dimensional graph used to display the relationship between two quantitative variables collected from the same set of individuals or subjects. Data displayed in this manner is called bivariate data.

   Each observation is represented as an ordered pair $(x, y)$ plotted on a coordinate grid:

   • Explanatory variable (Independent variable, $x$): Placed along the horizontal axis. This variable is examined to see if it explains changes in the other variable.
   • Response variable (Dependent variable, $y$): Placed along the vertical axis. This variable measures the outcome or response.

   By plotting all data pairs as distinct points, we can visually examine patterns, groupings, and overall trends that are difficult to detect in raw tables.

   Interpreting Scatter Plots

   When analyzing a scatter plot, we examine three primary characteristics of the association between the variables: direction, form, and strength.

   1. Direction

   • Positive Association: As $x$ increases, $y$ tends to increase. The points trend upward from left to right.
   • Negative Association: As $x$ increases, $y$ tends to decrease. The points trend downward from left to right.
   • No Association: Changes in $x$ do not correspond to any predictable increase or decrease in $y$.

   2. Form

   • Linear Form: The data points cluster along a roughly straight path.
   • Nonlinear (Curvilinear) Form: The points follow a curved pattern, such as a parabola or an exponential curve.

   3. Strength

   Strength describes how closely the points adhere to the underlying pattern. In a strong linear association, the points lie very close to an imaginary straight line. In a weak association, the points are widely scattered while still showing a general direction.

   Points that fall substantially outside the overall cluster of data are called outliers.

   Modeling Linear Relationships

   When bivariate data exhibits a linear association, we can summarize the pattern by modeling it with a trend line, also referred to as a line of best fit.

   A well-placed trend line possesses several key properties:

   • It follows the general direction and slope of the plotted points.
   • It passes through the approximate center of the data cluster, typically leaving roughly equal numbers of points above and below the line.
   • It minimizes the vertical distances between the actual data points and the line.
   • It does not need to connect the first and last points, nor does it need to pass through any specific data point.

   Algebraically, a linear trend line is expressed in slope-intercept form:

   $$y = mx + b$$

   Here, $m$ represents the rate of change (the predicted change in $y$ for each unit increase in $x$), and $b$ represents the $y$-intercept (the predicted value of $y$ when $x = 0$).

   Using Trend Lines for Estimation

   Once an equation for a trend line is determined, it can be used to estimate unknown values by substituting a specific value of $x$ or $y$ into the linear model.

   Interpolation

   Interpolation is the process of predicting a response value for an $x$-value that falls within the range of the observed data. Because the data supports the pattern across this domain, predictions made by interpolation are generally reliable.

   Extrapolation

   Extrapolation is the process of predicting a value for an $x$-value that lies outside the range of the observed data (either well below the minimum or well above the maximum observed $x$).

   Caution: Extrapolation is inherently risky because there is no guarantee that the linear trend continues indefinitely beyond the recorded values.

   A Critical Statistical Distinction

   A fundamental principle in statistics is that association does not imply causation.

   Observing a strong positive or negative trend line between variable $x$ and variable $y$ confirms that the two variables change together in a predictable pattern. However, this observation alone does not prove that changes in $x$ directly cause changes in $y$.

   Consider common explanations for observed correlations:

   • Confounding variables (Lurking variables): A third, unmeasured variable may influence both $x$ and $y$ simultaneously, producing an apparent relationship.
   • Coincidence: Two completely unrelated quantities may happen to change concurrently over time.
   • Reverse causation: Variable $y$ may actually be causing the observed changes in variable $x$.

   To establish a true cause-and-effect relationship, researchers must conduct controlled experiments that isolate the variables of interest, rather than relying solely on observational scatter plot data.