Study Notes on Scatter Plots

Definition: A scatter plot is a graphical representation where dots are scattered across a two-dimensional space to represent individual data points from a population or a sample.
Purpose: Scatter plots are used to illustrate the relationship between two quantitative variables.

Elements to Include:
- Title: A descriptive title of what the scatter plot is illustrating.
- Axes Labels: Label the x-axis and y-axis with the respective variables being measured.
- x-axis (horizontal): Represents the independent or explanatory variable.
- y-axis (vertical): Represents the dependent or response variable.
- Data Points: Each dot on the plot corresponds to an individual case measured by two variables.
Example: In a scatter plot examining cars, you might have:
- Variable 1: Highway miles per gallon (mpg)
- Variable 2: City miles per gallon (mpg)
- Each dot represents a different car with specific mpg values plotted on the axes.
Constructing the Axes:
- Ensure even increments for hash marks on both axes to avoid misleading representations.
- For instance, if the lowest value on an axis is particularly low, use a lightning bolt symbol to indicate that part of the scale is compressed or adjusted.

Explanatory Variable: Placed on the x-axis; it is the independent variable that explains changes in the response variable.
- Example: Age can be an explanatory variable when plotting against height, as age generally explains height changes in children.
Response Variable: Placed on the y-axis; it is the dependent variable that responds to changes in the explanatory variable.

Correlation: A measure of how closely two variables are related, typically assessed in terms of a linear relationship.
- Correlation Coefficient (r): A numeric value between -1 and 1 that indicates the strength and direction of the linear relationship.
- Positive Correlation (0 < r ≤ 1): The variables increase or decrease together; r closer to 1 indicates a stronger relationship.
- Negative Correlation (-1 ≤ r < 0): One variable increases while the other decreases; r closer to -1 indicates a stronger inverse relationship.
- No Correlation (r ≈ 0): Indicates no discernible linear relationship between the variables.
Identifying Relationships via Scatter Plots:
- Dots that fit closely along a straight line indicate strong correlation.
- An approximation of the correlation coefficients for different examples:
- Highway vs. City MPG: Strong positive correlation, approximately $r ext{ ≈ } 0.9$.
- Driver's Age vs. Maximum Read Distance: Negative correlation, approximately $r ext{ ≈ } -0.75$.
- Unrelated Data (e.g., new chicken eggs vs. ice cream sold): r ≈ 0, indicating no correlation.

Regression Line: A special line that fits the data points; it can be used to predict values of the response variable given a value of the explanatory variable.
Equation of the Line: Often represented as: $y = b<em>0 + b</em>1 x$
- $b_0$: Y-intercept of the line.
- $b_1$: Slope of the line (indicates the rate of change of y with respect to x).
Example of Usage:
- If you wanted to determine the maximum distance you can read a sign at age 55, you would locate the corresponding age on the x-axis, then use the least squares regression line to find the predicted distance on the y-axis.
- If $b1 = 5$ and $b0 = 3$, a weight of 12 for a creature could yield:
- $y = 3 + 5(12) = 63$
- This means the predicted height (or other measurement) of the creature is 63 units.

Scatter plots are vital tools in statistics for visualizing relationships between variables.
Understanding and utilizing correlation coefficients and regression lines allows for deeper insights and predictive analytics in data analysis.