Chapter 2: Numerical Descriptors

Chapter Objectives

• Describing distribution with numbers:
  • Measure of center: Mean and Median
  • Measure of spread: Quartiles and Standard Deviation
  • Five-number summary and Box plots (a.k.a. Box and Whisker plots)
  • Interquartile range (IQR) and Outliers
  • Dealing with outliers
  • Choosing among summary statistics
  • Organizing statistical problems

Measure of Center

Mean (Arithmetic Average)

• Calculation: Sum of values / Number of values (n)
• Symbol: x̄ (x-bar)
• Formula: x̄ = (Σ xi) / n where xi are the values and n is the number of values
• Nature: Sensitive to skewing and outliers

Median

• Definition: Midpoint of a distribution
• Calculation:
  • Sort values from smallest to largest
  • Median location: (n + 1) / 2
  • If n is odd: Median is the center value
  • If n is even: Median is the mean of the two center values
• Nature: Resistant to skewing and outliers
• Median and Mean should be close if the distribution is symmetric

Study Calculation Example:

Group Sizes in Bars

• n Value: 500 groups
• Median: Group size 2 (most frequent)
• Mean: Slightly larger than 2

Measure of Spread

Five-Number Summary

• Components: Minimum, Q1 (First Quartile), Median, Q3 (Third Quartile), Maximum
• Quartiles:
  • Q1: Median of values below the overall median
  • Q3: Median of values above the overall median
• Example Calculation:
  • Median: n + 1 / 2
  • Q1: Median of lower half
  • Q3: Median of upper half

Standard Deviation (SD)

• Definition: Measure of spread concerning how far each observation is from the mean
• Calculation:
  • Variance (s²): s² = Σ (xi - x̄)² / (n - 1)
  • Standard Deviation (s): s = √s²
• Example Calculation: Metabolic rates of 7 men
  • Sample mean (x̄) = 1600
  • Variance (s²) = 35811.7
  • Standard Deviation (s) = 189.2

Box and Whisker Plot

• Components: Box (Q1 to Q3), Median line, Whiskers (min and max), and Outliers marked as asterisks
• IQR: Q3 - Q1
• Finding Outliers:
  • Lower bound: Q1 - 1.5 * IQR
  • Upper bound: Q3 + 1.5 * IQR
• Example Calculation:
  • Q2 (Median): 3.4
  • Q1: 2.2
  • Q3: 4.35
  • IQR: 2.15
  • Upper Outlier: 7.9 (outlier)
  • Whisker max adjusted to 5.6

Dealing with Outliers

• Types of Outliers:
  • Human error (recording/experiment)
  • Legitimate but wild observations
• Action: Assess whether to exclude based on interest in typical vs. all individuals; never exclude to force data fit

Choosing Summary Statistics

• Mean + SD: Use for fairly symmetrical distributions
• Median + Five-Number Summary: Use for skewed distributions or with outliers
• Example: Phytopigment concentrations—highly right-skewed data, likely Median preferred

Organizing Statistical Problems

1. State: Practical question in real-world context
2. Plan: Specific statistical operations needed
3. Solve: Make graphs and calculations
4. Conclude: Practical conclusion in real-world context

End of Chapter 2