Chapter 4: Describing the Relation between Two Variables – Scatter Diagrams and Correlation

Chapter 4: Describing the Relation between Two Variables

4.1 Scatter Diagrams and Correlation

This section explores how to visually and numerically describe the relationship between two quantitative variables. Key concepts include scatter diagrams, correlation coefficients, and the distinction between correlation and causation.

4.1.1 Draw and Interpret Scatter Diagrams

• Response Variable: The variable whose value can be explained by the value of the explanatory or predictor variable.

• Scatter Diagram: A graph that displays the relationship between two quantitative variables measured on the same individual. Each point represents an individual, with the explanatory variable on the horizontal axis and the response variable on the vertical axis.

Example: The relationship between club-head speed (mph) and distance (yards) a golf ball travels is shown below:

Club-head Speed (mph)	Distance (yards)
100	257
102	264
103	274
104	272
105	275
99	258

As club-head speed increases, the distance the ball travels also increases, indicating a positive association.

• Scatter diagrams help distinguish between linear, nonlinear, and no relation between variables.

• Linear relations can be positive (upward slant) or negative (downward slant).

4.1.1 Types of Association

• Positively Associated: Above-average values of one variable are associated with above-average values of the other. As one increases, so does the other.

• Negatively Associated: Above-average values of one variable are associated with below-average values of the other. As one increases, the other decreases.

4.1.2 Properties of the Linear Correlation Coefficient

The linear correlation coefficient (Pearson product moment correlation coefficient) measures the strength and direction of the linear relationship between two quantitative variables.

• Population correlation coefficient: $\rho$ (rho)

• Sample correlation coefficient: $r$

Formula for sample correlation coefficient:

$r = \frac{\sum \left( \frac{x_i - \bar{x}}{s_x} \cdot \frac{y_i - \bar{y}}{s_y} \right)}{n-1}$ where $\bar{x}$ and $\bar{y}$ are sample means, $s_x$ and $s_y$ are sample standard deviations, and $n$ is sample size.

Alternative computational formula:

$r = \frac{\sum x_i y_i - \frac{\sum x_i \sum y_i}{n}}{\sqrt{\left( \sum x_i^2 - \frac{(\sum x_i)^2}{n} \right) \left( \sum y_i^2 - \frac{(\sum y_i)^2}{n} \right)}}$

Properties of the Linear Correlation Coefficient

• The value of $r$ is always between -1 and 1, inclusive: $-1 \leq r \leq 1$.

• If $r = +1$, there is a perfect positive linear relation.

• If $r = -1$, there is a perfect negative linear relation.

• The closer $r$ is to +1, the stronger the positive association.

• The closer $r$ is to -1, the stronger the negative association.

• If $r$ is close to 0, there is little or no evidence of a linear relation (but possibly a nonlinear relation).

• $r$ is unitless; the units of $x$ and $y$ do not affect its value.

• $r$ is not resistant; outliers can significantly affect its value.

Visual Examples: Scatter diagrams can show perfect, strong, moderate, or weak positive/negative linear relations, or no linear relation.

4.1.3 Compute and Interpret the Linear Correlation Coefficient

To compute $r$ by hand:

1. Calculate sample means and standard deviations for $x$ and $y$.

2. Standardize each value: $\frac{x_i - \bar{x}}{s_x}$ and $\frac{y_i - \bar{y}}{s_y}$.

3. Multiply standardized values for each pair and sum.

4. Divide by $n-1$ to obtain $r$.

Example: For the data set:

x	y
1	18
3	13
6	9
7	4

Calculated $r = -0.946$, indicating a strong negative association.

Technology (e.g., Excel, calculators) can also compute $r$ efficiently.

4.1.4 Determine Whether a Linear Relation Exists Between Two Variables

To test for a linear relation:

1. Determine the absolute value of $r$.

2. Find the critical value for the given sample size (from a table).

3. If $|r|$ is greater than the critical value, a linear relation exists; otherwise, it does not.

Interpretation: If $r$ is positive and greater than the critical value, the association is positive. If $r$ is negative and less than the negative critical value, the association is negative.

4.1.5 Explain the Difference Between Correlation and Causation

• Correlation does not imply causation. Observational data cannot establish a causal relationship.

• A lurking variable is related to both the explanatory and response variable and may confound the observed association.

Example: As air-conditioning bills increase, so does the crime rate. The lurking variable is air temperature, which affects both variables.

Bone Mineral Density Study: Researchers found a negative correlation between cola consumption and bone mineral density in women, but could not conclude causation due to possible lurking variables (e.g., age, BMI, smoking, calcium intake).

Number of Colas per Week	Bone Mineral Density (g/cm2)
0	0.892
1	0.882
2	0.881
3	0.884
4	0.876
5	0.875
6	0.867
7	0.862

Researchers must be careful to state association, not causation, when lurking variables may be present.