Lesson 2: Associations in Numerical Data

Property

A scatter plot is a graph in the coordinate plane of the set of all $(x, y)$ ordered pairs of bivariate data.
Consistent with the usual convention, we place the independent variable $X$ on the horizontal axis and the dependent variable $Y$ on the vertical axis.
The investigator wishes to quantify the relationship between two random variables, $X$ and $Y$, which describe an entire population.

Examples

• A researcher plots the number of hours a person sleeps ($X$) against their reaction time in milliseconds ($Y$) to see if more sleep is associated with faster reactions.
• To study car safety, an engineer creates a scatter plot with a car's speed in miles per hour ($X$) and its braking distance in feet ($Y$).
• A coffee shop owner tracks the daily temperature ($X$) and the number of hot coffees sold ($Y$) on a scatter plot to understand customer habits.

Explanation

Scatter plots are visual tools that help us see if a relationship exists between two sets of data. By plotting points for two variables, like height and weight, we can look for patterns or trends that might not be obvious from numbers alone.

Property

Four types of relationships can be identified in scatter plots:
Positive Linear - points form a pattern where as $x$ increases, $y$ increases along an approximate straight line;
Negative Linear - points form a pattern where as $x$ increases, $y$ decreases along an approximate straight line;
Nonlinear - points form a curved pattern (parabolic, exponential, etc.);
No Relationship - points show no discernible pattern or trend.

Examples

Property

Once a line of best fit is drawn, you can calculate its algebraic equation. First, locate two points that sit exactly on the line of fit to find the slope. Then, use the point-slope form to write the full equation.

Examples

• Given points $(2, 5)$ and $(6, 9)$ on a line of best fit, find its equation.
• First, find the slope: $m = \frac{9-5}{6-2} = \frac{4}{4} = 1$.
• Then, use the point-slope form with point $(2, 5)$: $y - 5 = 1(x - 2)$, which simplifies to $y = x + 3$.

Explanation

Even though a scatter plot is constructed by converting two-variable data into ordered pairs $(x, y)$ and plotting them, the line of fit may or may not pass through any of the actual data points.