Quantitative Variables, Skewed Distributions, and Longitudinal Data Analysis

Quantitative Variables and Distribution Shapes

• Definition of Quantitative Variable:

  • A quantitative variable is a numerical measurement associated with individual people or objects within a study or dataset.
  • It differs from aggregate counts across categories (e.g., tracking the total count of people born in each month is a categorical grouping count, whereas measuring a specific numerical trait for each individual person defines a quantitative variable).

• Common Shapes of Distributions:

  • Symmetric distributions: Data is centered evenly around a middle value with equal tails on both sides.
  • Right-skewed distributions: Data exhibits a concentration of observations at lower values with a long tail extending toward higher extreme values on the right side of the graph.
  • Symmetric and right-skewed distributions represent the two most common distribution shapes encountered in statistical analysis.

Examples and Properties of Right-Skewed Distributions

• General Characteristics of Money-Based Variables:

  • Metrics involving monetary values serve as quintessential examples of right-skewed distributions due to lower structural bounds and unbounded upper potential.

• Income Distributions:

  • Typical Range: The vast majority of the population earns average incomes falling between $20,000$ (for non-students) and $80,000$.
  • Moderate Skew: Higher earners, such as doctors and specialized lawyers, earn incomes around $600,000$.
  • Extreme Skew: Multi-million dollar incomes earned by elite professional athletes and movie stars form the extreme right-hand tail of the distribution.

• Housing Prices:

  • Baseline Values: Average home prices typically sit below $1,000,000$, varying by specific local neighborhood conditions.
  • Right-Tail Extrema: Upper boundary values are virtually unlimited. Examples include ultra-luxury real estate purchases in Los Angeles, such as an $80,000,000$ home acquired by LeBron James or Jeff Bezos and his wife.

• Automobile Prices:

  • Baseline Values: The average standard new car price centers around $50,000$.
  • Right-Tail Extrema: Luxury and high-performance exotic vehicles (e.g., Mercedes-Benz, BMW, Maserati, and Ferrari) push the right tail out significantly.

Time Plots and Longitudinal Trends

• Concept of Time Plots:

  • Time plots evaluate how specific statistical measures or metrics evolve sequentially over established time intervals.
  • Applications include tracking changes in naming trends across generations or monitoring demographic shifts in course enrollment across academic terms (e.g., tracking the percentage of freshmen enrolled in Math 131 each semester).

• Academic Tuition Comparisons:

  • Educational costs tracked over time show that private university tuition consistently remains higher and more expensive relative to public university tuition.

Case Study: Analyzing Longitudinal Cancer Mortality Rates

• Dataset Overview:
  • A temporal study tracking death rates in the United States over a continuous $50\text{-year}$ period.
  • Measurement Unit: Standardized as the cancer death rate per $1,000$ people.
  • Primary Finding: Over the observed $50\text{-year}$ timeline, the cancer death rate per $1,000$ individuals consistently increased despite ongoing technological and medical advancements.

Questions & Discussion

• Question: Why are cancer death rates per $1,000$ people increasing over time even as medical technology improves and overall human life expectancy increases?

• Student Hypothesis 1 (Andrew):

  • Proposed Mechanism: The trend is driven by overall population growth combined with rising cancer incidence.
  • Evaluation & Corrective Analysis: Standardizing data as a rate per $1,000$ people mathematically controls for total population size. An increase in absolute population size alone does not cause the per-thousand rate to rise.

• Student Hypothesis 2 (Katie):

  • Proposed Mechanism: Modern agricultural and food processing modifications (e.g., genetically modified foods and less healthy processed dietary habits) increase cancer risk.
  • Evaluation & Corrective Analysis: Lifestyle alterations and processed foods do contribute to overall cancer incidence, but environmental and dietary changes alone are not the primary statistical driver of the long-term trend.

• Student Hypothesis 3 (James):

  • Proposed Mechanism: Improvements in diagnostic technology allow medical professionals to accurately identify and log deaths caused by cancer that were previously attributed to other unclassified causes.
  • Evaluation & Corrective Analysis: Enhanced diagnostic precision plays a valid historical role in clarifying mortality data, but it remains a secondary factor.

• Primary Scientific Explanation:

  • Differential Mortality Reduction: Medical advancements over the last $50$ years successfully cured, treated, or managed major non-cancer diseases—most notably cardiovascular conditions—while direct cures for advanced cancers remain elusive.
  • Cardiovascular Interventions: Historically (e.g., $50$ years ago), individuals with unmanaged high cholesterol or high blood pressure frequently suffered fatal strokes or heart attacks at younger ages (e.g., age $52$).
  • Preventive Care Impact: Modern medicine introduced routine diagnostic testing (cholesterol screens, blood pressure monitoring) and pharmaceutical therapies (anticoagulants/blood thinners, antihypertensives), substantially reducing sudden cardiovascular mortality.
  • Demographic Outcome: Because individuals no longer die at early ages from cardiovascular diseases, a substantially larger proportion of the population survives into advanced age brackets where cancer risk is naturally highest. This demographic shift elevates the aggregate cancer death rate per $1,000$ individuals.