Quantitative and Qualitative Data Analysis: Scatter Diagrams and Pearson Correlation

Qualitative Data Relationship Analysis

• Categorical data analysis using bar graphs and side-by-side bar graphs:
  • Side-by-side bar graphs evaluate potential relationships between two qualitative variables (e.g., gender and work location inside or outside campus).
  • If the distribution pattern across categories is identical between groups (e.g., males and females showing identical proportions working inside vs. outside campus), there is no relationship between the variables.
  • If distribution patterns differ between groups (e.g., females exhibiting a higher tendency to work outside campus compared to males), a relationship exists where one qualitative variable depends on the other.

Quantitative Data and Scatter Diagrams

• A scatter diagram (or scatter plot) displays two quantitative variables plotted as paired coordinates $(x, y)$ collected from the same set of individuals or items.
• Graph Structure:
  • Horizontal Axis ($x$): Explanatory variable, predictor variable, or independent variable.
  • Vertical Axis ($y$): Response variable or dependent variable.
  • Data Points: Displayed as unconnected dots scattered across the coordinate plane.
• Variable Roles:
  • Response (Dependent) Variable ($y$): The variable being investigated or predicted; its value depends upon the explanatory variable.
  • Explanatory (Predictor / Independent) Variable ($x$): The variable that explains or predicts changes in the response variable.
• Sentence Identification Test:
  • Determine variable roles by placing them in the sentence: "Response depends upon Explanatory."
  • Rock Drilling Study Example:
  • Research Objective: Determine whether the time required to drill $5\,\text{feet}$ into rock depends on the depth at which drilling begins.
  • Sentence Test: "Time to drill depends on starting depth" makes common sense. "Starting depth depends on time to drill" does not.
  • Explanatory Variable ($x$): Depth at which drilling starts.
  • Response Variable ($y$): Time required to drill $5\,\text{feet}$.
  • Used Car Dealership Example:
  • Research Objective: Determine the relationship between used car mileage and price.
  • Sentence Test: "Price depends upon mileage" makes sense. "Mileage depends upon price" does not.
  • Explanatory Variable ($x$): Mileage.
  • Response Variable ($y$): Price.

Plotting Scatter Diagrams and Axis Scaling

• Step-by-Step Construction Rules:
  • Label both axes with variable names and appropriate units.
  • Select consistent scale intervals.
  • Axis Break ("Cut the Line"): When quantitative values start significantly above zero, place an axis break symbol on the scale to avoid compressing data points at the top of the grid.
• Rock Drilling Dataset Plotting Parameters ($12$ Paired Data Points):
  • Depth ($x$) range: $35$ to $190$ (horizontal scale marked up to $200$).
  • Time ($y$) range: $5.88$ to $7.97$ (vertical scale broken to cover $5.5$ or $5.9$ up to $8.0$).
  • Complete Paired Coordinate Data Set:
  1. $(35, 5.88)$
  2. $(50, 5.99)$
  3. $(75, 6.74)$
  4. $(95, 6.10)$
  5. $(120, 7.47)$
  6. $(130, 6.93)$
  7. $(145, 6.42)$
  8. $(155, 7.97)$
  9. $(160, 7.92)$
  10. $(175, 7.62)$
  11. $(185, 6.89)$
  12. $(190, 7.90)$
  • Final Rendering: Helper/grid lines used during coordinate plotting must be erased to leave a clean scatter diagram of unconnected dots.

Classifying Relationships in Scatter Diagrams

• Classification Types for Bivariate Relationships:
  • Linear Relationship: Points follow a straight-line trend.
  • Positive Linear Relationship: As $x$ increases, $y$ increases (line with positive slope).
  • Negative Linear Relationship: As $x$ increases, $y$ decreases (line with negative slope).
  • Nonlinear Relationship: Points display a distinct curved pattern (e.g., quadratic/parabolic equation, degree-3 polynomial curve).
  • No Relationship: Dots are scattered randomly across the plane without any discernible pattern.
• Analytical Decisions Based on Diagram Classifications:
  • No Relationship: Discard the data regarding that explanatory variable and evaluate a different predictor (e.g., test car age instead of mileage if mileage yields no pattern).
  • Linear Relationship: Conduct linear regression analysis to determine the line of best fit and use it to predict response values.
  • Nonlinear Relationship: Retain the data. Perform a mathematical transformation on a variable (e.g., replacing explanatory values $x$ with $\sqrt{x}$ for quadratic patterns) to linearize the relationship before performing linear regression.

Pearson Correlation Coefficient ($r$)

• Definition: A quantitative measure of the strength and direction of a linear relationship between two quantitative variables.
• Notation:
  • Sample correlation coefficient: $r$
  • Population correlation coefficient: $\rho$ (Greek letter rho)
• Mathematical Foundation:
  • Evaluates deviations of $x$ values from their mean ($\bar{x}$) and $y$ values from their mean ($\bar{y}$):     $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$
  • Evaluates deviations in horizontal and vertical directions simultaneously.
• Key Properties of $r$:
  • Bounded Range: $-1 \le r \le 1$
  • $r = 1$: Perfect positive linear relationship (all points lie exactly on a line with a positive slope).
  • $r = -1$: Perfect negative linear relationship (all points lie exactly on a line with a negative slope).
  • Unitless: $r$ has no units of measurement.
  • Sensitivity: Like standard deviation, $r$ is non-resistant to extreme values (outliers).
• Numerical Interpretation Thresholds:
  • $-0.3 \le r \le 0.3$: Weak or no linear relationship.
  • $0.3 < |r| \le 0.7$ (i.e., $0.3$ to $0.7$ or $-0.7$ to $-0.3$): Moderate linear relationship.
  • $|r| > 0.7$ (i.e., $> 0.7$ or $< -0.7$): Strong linear relationship.
• Correlation Values across Diagram Types:
  • $r = 1.0$: Perfect positive linear.
  • $r \approx 0.9$: Strong positive linear.
  • $r \approx 0.4$: Moderate positive linear.
  • $r = -1.0$: Perfect negative linear.
  • $r \approx -0.9$: Strong negative linear.
  • $r \approx -0.4$: Moderate negative linear.
  • $r \approx 0$: No linear relationship (can represent complete random scatter or a strong nonlinear relationship such as a parabola).
• Fundamental Sequence of Analysis:
  • Always inspect the scatter diagram first before interpreting $r$.
  • An $r$ value near zero confirms only the absence of a linear relationship; it does not rule out strong nonlinear relationships.

Texas Instruments TI-83/84 Calculator Instructions

• Step 1: Input paired quantitative data into statistical lists via STAT -> Edit.
  • Preserve strict paired order across lists (e.g., starting depth $35$ in L3 must align with drilling time $5.88$ in L4).
• Step 2: Enable Diagnostics (required if $r$ is missing from regression output):
  • Press 2nd then 0 to enter CATALOG.
  • Scroll down to DiagnosticOn.
  • Press ENTER, then press ENTER again until Done appears on screen.
• Step 3: Compute Linear Regression and Correlation Coefficient:
  • Press STAT, navigate to CALC menu.
  • Select option 4: LinReg(ax+b).
  • Specify Xlist (e.g., L3) and Ylist (e.g., L4).
  • Select Calculate and press ENTER.
• Output for Drilling Dataset:
  • Calculated correlation coefficient: $r = 0.7728$
  • Verbal Interpretation: Strong positive linear relationship.

Correlation versus Causation

• Fundamental Principle: High correlation does not imply causation.
• Empirical Example of Spurious Correlation:
  • Correlation between the percentage of female population with a bachelor's degree and the percentage of births to unmarried mothers by year (1990s onward) yields $r = 0.94$
  • Explanation: Earning a bachelor's degree does not cause higher unmarried birth rates. Both variables increased concurrently over time due to external social trends.
• Observational Data Caution: Correlation in observational data cannot prove direct causation. Domain context and logic must evaluate relationships beyond raw numerical calculations.