Day 13

Math 216: Statistical Thinking

Bastola

Introduction to Assessing Normality

  • Goal: Verify if data comes from a normal distribution, required for many parametric tests (e.g., t-tests, ANOVA).
  • Key Context: Ensures validity of statistical inferences.
  • Primary Methods:
    1. Visual checks (plots)
    2. Empirical rule alignment
    3. IQR/SD ratio

Visual Assessment Tools

  • Histogram/Density Plot:
    • Look for symmetric bell-shaped curve.
    • Check tails: abrupt drops may indicate non-normality.
  • Q-Q Plot:
    • Data points should align with 45° reference line.
    • Curves or outliers = potential skew/kurtosis.

Numerical Assessment Methods

  • Empirical Rule:
    • \(\pm1\sigma\): ≈68%, \(\pm2\sigma\): ≈95%, \(\pm3\sigma\): ≈99.7%.
    • Compare calculated %s to these benchmarks.
  • IQR/SD Ratio:
    • For normality: \(\frac{\text{IQR}}{s} \approx 1.3\).
    • Compute via \(\frac{Q3 - Q1}{s}\) and compare.

Case Study: EPA ratings


The dataset consists of EPA gas mileage ratings for 100 cars. Each value represents the miles per gallon (MPG) that a particular car achieves under standardized testing conditions. This dataset can help understand how statistical methods are applied in real-world scenarios like assessing the fuel efficiency of vehicles.

EPA Gas Mileage Ratings for 100 Cars (miles per gallon)

col1 col2 col3 col4 col5 col6 col7 col8 col9 col10
36.3 41.0 36.9 37.1 44.9 36.8 30.0 37.2 42.1 36.7
32.7 37.3 41.2 36.6 32.9 36.5 33.2 37.4 37.5 33.6
40.5 36.5 37.6 33.9 40.2 36.4 37.7 37.7 40.0 34.2
36.2 37.9 36.0 37.9 35.9 38.2 38.3 35.7 35.6 35.1
38.5 39.0 35.5 34.8 38.6 39.4 35.3 34.4 38.8 39.7
36.3 36.8 32.5 36.4 40.5 36.6 36.1 38.2 38.4 39.3
41.0 31.8 37.3 33.1 37.0 37.6 37.0 38.7 39.0 35.8
37.0 37.2 40.7 37.4 37.1 37.8 35.9 35.6 36.7 34.5
37.1 40.3 36.7 37.0 33.9 40.1 38.0 35.2 34.8 39.5
39.9 36.9 32.9 33.8 39.8 34.0 36.8 35.0 38.1 36.9 
# store 100 mpgs in an object called 'mpg'
mpg <- c(
  36.3, 41.0, 36.9, 37.1, 44.9, 36.8, 30.0, 37.2, 42.1, 36.7,
  32.7, 37.3, 41.2, 36.6, 32.9, 36.5, 33.2, 37.4, 37.5, 33.6,
  40.5, 36.5, 37.6, 33.9, 40.2, 36.4, 37.7, 37.7, 40.0, 34.2,
  36.2, 37.9, 36.0, 37.9, 35.9, 38.2, 38.3, 35.7, 35.6, 35.1,
  38.5, 39.0, 35.5, 34.8, 38.6, 39.4, 35.3, 34.4, 38.8, 39.7,
  36.3, 36.8, 32.5, 36.4, 40.5, 36.6, 36.1, 38.2, 38.4, 39.3,
  41.0, 31.8, 37.3, 33.1, 37.0, 37.6, 37.0, 38.7, 39.0, 35.8,
  37.0, 37.2, 40.7, 37.4, 37.1, 37.8, 35.9, 35.6, 36.7, 34.5,
  37.1, 40.3, 36.7, 37.0, 33.9, 40.1, 38.0, 35.2, 34.8, 39.5,
  39.9, 36.9, 32.9, 33.8, 39.8, 34.0, 36.8, 35.0, 38.1, 36.9
)

mpg_data <- data.frame(mpg = mpg)

Visualization Approach

Normality Assessment

IQR SD Ratio Expected Difference
2.65 2.42 1.1 1.3 -0.2

Key Findings from EPA Case Study

  1. Visual Evidence:

    • Histogram shows right skew (confirmed by boxplot outlier at 44.9 MPG)
    • Q-Q plot shows systematic deviation from reference line in upper tail

  2. Empirical Rule Compliance:

    • 72% within μ±1σ (expected 68%) - Close
    • 95% within μ±2σ (exact match) - Good
    • 100% within μ±3σ (expected 99.7%) - Borderline

  3. IQR/SD Ratio:

    • Observed 1.1 vs Expected 1.3 - Suggests lighter tails

Implications for Future Analysis