L4- Distribution of Observations

Learning Objectives

• By the end of this lecture you should be able to:
  • Describe and identify several statistical distributions (Normal, t, \chi^2, Binomial, Bernoulli).
  • Apply reference ranges to the Normal distribution when appropriate.
  • Calculate the probability that an observation lies between two values using the standard Normal distribution.

Core Definitions & Concepts

• Population vs Sample
  • Population: Entire group of interest; possesses a fixed, but usually unknown, numerical characteristic (parameter).
  • Sample: Subset drawn from the population, used to estimate population characteristics.
• Parameter vs Statistic
  • Parameter (Greek letters): Numerical characteristic of a population (e.g., population mean cholesterol of all Asian males in Nguyen).
  • Statistic (Latin letters): Numerical summary derived from a sample (e.g., mean cholesterol of a sample of 50 Asian males in Melbourne) used to infer the parameter.

Notational Conventions

• Mean
  • Population mean: \mu
  • Sample mean: \bar{x} (spoken “x-bar”).
• Standard Deviation
  • Population SD: \sigma
  • Sample SD: S or SD
  • Excel command: STDEV(range) computes the sample standard deviation.
• Proportion
  • Population proportion: \pi
  • Sample proportion: p

Overview of Distributions Covered

• Bernoulli
  • Used for a single trial (sample size n=1) with two outcomes.
  • Rarely applied in practice; not covered further in the course.
• Binomial
  • Two mutually exclusive outcomes; fixed number of trials n; constant probability \pi.
  • Concerned with the count of “successes”.
  • Example question: “In 10 events, what is P(X=5)?”
• Normal (Gaussian) Distribution – Focus of this lecture.
• t Distribution – Will be addressed in later sessions.
• Chi-Square (\chi^2) Distribution – Will be addressed in later sessions.

The Normal (Gaussian) Distribution

• Also called the Gaussion distribution.
• Applies to continuous data.
• Characteristics:
  • Shape: Symmetric, bell-shaped curve.
  • Parameters:
  • Central location: \mu
  • Spread: \sigma
• Visual: Mean at center, curve spreads outward in units of \sigma.
• Importance: Foundation for many statistical methods and probability calculations.

Summary Statistics Revisited

• Provide concise, essential information about data distribution.
• Two classes:
  • Measures of Central Tendency: Mean, median, mode.
  • Measures of Dispersion: Standard deviation, inter-quartile range (IQR), range.
• Interpretation of dispersion (illustrated by two curves):
  • Narrow curve ⇒ small variability.
  • Wide curve ⇒ large variability.

Shapes of Distributions

• Symmetric / Normal: Mean = Median = Mode.
• Negatively (Left) Skewed:
  • Tail extends to smaller values.
  • Outliers pull mean left: \text{Mean} < \text{Median} < \text{Mode}.
• Positively (Right) Skewed:
  • Tail extends to larger values.
  • Outliers pull mean right: \text{Mode} < \text{Median} < \text{Mean}.
• Terminology:
  • Symmetric ⇒ normal
  • Asymmetric ⇒ skewed

Choosing Appropriate Statistics to Report

• If data ~ Normal ⇒ report mean ± SD.
• If data skewed ⇒ report median & IQR (mean is distorted by outliers).

Checking for Normality

1. Graphical Methods
   • Histogram
   • Box-plot
   • Look for symmetry, bell shape, absence of long tails.
2. Numerical Comparison
   • Compare mean and median:
     • Close ⇒ likely symmetric/normal.
     • Far apart ⇒ likely skewed.

Graphical & Numerical Examples

• Weight Histogram
  • Visually symmetric; mean = median = 79 ⇒ weight ~ Normal.
• Age Distribution (Left-Skewed)
  • Most values > 70; tail toward younger ages.
  • Mean < Median ⇒ left skew.
• Post-Procedure Length of Stay (Right-Skewed)
  • Many short stays, few very long stays.
  • Mean > Median ⇒ right skew.

Clinical Data Example – Cardiac Surgery Database

• Variables: Pre-operative creatinine (mmol/L) & dialysis status.
• Overall Creatinine
  • Histogram & box-plot show long right tail (outliers up to <2 mmol/L).
• Grouped by Dialysis Status
  • No Dialysis:
  • Mean =0.10, Median =0.09 (similar) ⇒ approximate normality despite long tail.
  • Yes Dialysis:
  • Mean =0.462, Median =0.426 (mean significantly higher) ⇒ strong right skew.
• Statistical Reporting Decision
  • Because Yes Dialysis group is skewed, compare groups using median & IQR rather than mean & SD.

Additional Example – Body Mass Index (BMI)

• Histogram nearly symmetric; a few outliers up to 68 BMI.
• Mean ≈ Median (red lines overlap).
• Conclusion: BMI data ~ Normal; mean ± SD appropriate.

Practical Implications & Next Steps

• Always examine distribution shape before selecting summary statistics or formal tests.
• Reference ranges (e.g., \mu \pm 2\sigma) valid only under approximate normality.
• Standard Normal table or software allows probability calculations once data are standardized.
• Future lectures will build on these ideas to cover t and \chi^2 distributions, and to perform inferential tests based on Normal theory.