AP Statistics 9.1 Introducing Statistics: Do Those Points Align? Study Notes

Introducing Statistics: Do Those Points Align?

In statistics, a scatter plot displays the relationship between two quantitative variables. To interpret it, we often compare the observed points to a theoretical line, such as a regression line or a line predicted by a model. The way the points vary around this line helps us decide whether the relationship is random (expected natural scatter) or non-random (indicating the model is not a good fit).

Key Questions Suggested by Variation in Scatter Plots:

• Strength of alignment: Do the points closely follow a straight line, or are they widely scattered?
• Form of relationship: Is the relationship linear, or does the scatter show curvature (suggesting a quadratic or exponential trend)?
• Outliers and influential points: Are there unusual points far away from the general trend that could distort the fit?
• Consistency of spread (residuals): Do the points spread out equally along the line (homoscedasticity), or does the spread increase/decrease (heteroscedasticity)?
• Direction: Is the trend positive (both variables increase together) or negative (one increases as the other decreases)?

Why this matters:

Recognizing these patterns helps us decide:

• Whether a linear regression model is appropriate.
• Whether a different functional form (quadratic, exponential, logarithmic) might better explain the data.
• Whether certain data points should be investigated (possible errors, special cases, or important influential observations).

Example 1: Random Variation (Linear is Appropriate)
A researcher studies the relationship between hours studied (x) and exam score (y).

• The scatter plot shows an upward linear trend.
• Points cluster close to a straight line with small random deviations.

• Question suggested: Is a simple linear regression model suitable for predicting exam scores from study hours?

• Answer: Yes, random scatter around the line suggests linear regression is appropriate.

Example 2: Non-Random Variation (Curvature Present)
A biologist studies the relationship between temperature (x) and plant growth rate (y).

• The scatter plot shows a curved pattern: growth increases with temperature up to a peak, then declines.
• This is not random scatter but systematic curvature.

• Question suggested: Should a quadratic (curved) model be used instead of a straight line?

• Answer: Yes, a quadratic model would fit better than a linear one.

Example 3: Outlier / Influential Point
An economist studies the relationship between years of education (x) and income (y).

• Most data points follow a positive linear trend, but one point (a billionaire dropout) is far above the line.

• Question suggested: Does this point overly influence the slope of the line?

• Answer: Yes, it is an influential point that could distort the regression; it should be carefully examined.

Example 4: Unequal Spread (Heteroscedasticity)
An engineer studies the relationship between machine age (x) and repair cost (y).

• The scatter plot shows that for small values of x (new machines), the points are close to the line, but for older machines, the points spread out much more.

• Question suggested: Does the model assume constant variance in errors?

• Answer: No, the spread increases with age, so heteroscedasticity is present; another model may be required.