Introduction and Prerequisites

• Josh Starmer presenting StatQuest on statistics fundamentals
• Focus: estimating mean, variance, standard deviation
• Prerequisites: histograms, statistical distributions, normal distribution, population parameters
• Example: counting mRNA transcripts from gene X in liver cells (alternative examples: green apples in grocery stores, green t-shirts in clothing stores)

Sample Data Setup

• 5 liver cells measured with mRNA transcript counts: 3, 13, 19, 24, 29
• Population: 240 billion liver cells total
• Goal: estimate population parameters from small sample

Population Parameters (When All Data Available)

• Population Mean (μ):
  • Formula: sum of all measurements / number of measurements
  • Result: 24 mRNA transcripts (from all 240 billion measurements)
  • Not an estimate - actual population mean
• Population Variance:
  • Formula: Σ(X - μ)² / N
  • Steps: subtract population mean from each measurement, square differences, sum, divide by N
  • Squaring prevents negative differences from canceling positive ones
  • Result: 100 (units: mRNA transcripts squared)
• Population Standard Deviation:
  • Formula: √(population variance)
  • Result: √100 = 10 mRNA transcripts
  • Can be plotted on graph (same units as original data)
  • Shows mean ± standard deviation range

Sample-Based Estimation (Practical Application)

• Estimated Population Mean (x̄):
  • Same formula as population mean but with sample data
  • Result: 17.6 mRNA transcripts
  • Different from population mean (24) but should converge with more data
• Estimated Population Variance:
  • Formula: Σ(X - x̄)² / (n-1)
  • Key differences from population variance:
    • Uses sample mean (x̄) instead of population mean (μ)
    • Divides by (n-1) instead of n
  • n-1 compensates for using sample mean vs population mean
  • Prevents consistent underestimation of variance
  • Sample calculations: differences from mean (17.6), squared, summed, divided by 4
  • Result: 101.8
• Estimated Standard Deviation:
  • Formula: √(estimated variance)
  • Result: √101.8 = 10.1 mRNA transcripts

Comparison and Accuracy

• Estimated parameters: mean = 17.6, standard deviation = 10.1
• True parameters: mean = 20, standard deviation = 10
• Reasonable accuracy with only 5 measurements vs 240 billion
• More data would improve accuracy and confidence

Software Implementation Notes

• Most software packages estimate variance/standard deviation by default
• Microsoft Excel exception:
  • VAR.P: calculates population variance
  • VAR.S: estimates population variance
  • Should almost always use VAR.S for sample data

Key Formulas Summary

• Population mean: Σx / N
• Sample mean: Σx / n
• Population variance: Σ(X - μ)² / N
• Sample variance: Σ(X - x̄)² / (n-1)
• Standard deviation: √variance (for both population and sample)