Sample and Population Dilemma

Overview

• Discusses the relationship between samples and populations in statistics.
• Focus on three properties: height, age, and IQ of a population of 50,000 people.

Population Distribution

• Height Distribution:
  • Two groups identified:
    • Group 1: Average height ~178 cm
    • Group 2: Average height ~160 cm
• Box Plot: Shows population distribution versus sample distribution.

Sampling Process

• Sample Size Impact:
  • Default sample size is 6.
  • As sample size changes, representation of the population varies.
  • Example of 1 sample being a good approximation of the population despite small size.
• Percentiles:
  • Indicates how many values are smaller than a given value.
  • Comparison of population and sample percentiles.
• Variation:
  • Smaller sample sizes often lead to less variability than the entire population.

Effects of Sample Size

• Small Sample Sizes (e.g., size = 3):
  • Higher variability and potential for extreme results.
  • Example: Sample shifted to the left.
• Larger Sample Sizes (e.g., size = 30):
  • Better representation of the population.
  • Points closer to theoretical lines and boxplots align closely with population boxplots.

Importance of Random Sampling

• Samples should be completely random for better representation of the population.
• Regardless of distribution type (e.g., uniform or normal), larger sample sizes improve approximation.

Case Studies

• Age Distribution:
  • Uniform distribution, leading to straight line in percentiles.
  • Samples generally close to population.
• IQ Distribution:
  • Normal distribution with extremes available.
  • Boxplot rule: Approximately 99.2% of points within range, 0.8% as outliers based on large population.

Summary

• Smaller sample sizes yield increased uncertainty and less representative results.
• Experimenting with sample sizes can lead to better understanding of population representation.