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Mastering One-Way ANOVA Calculations

Apr 29, 2025

Lecture on Calculating One-Way ANOVA by Hand

Introduction

  • Purpose: Learn to calculate One-Way ANOVA (Analysis of Variance) by hand.
  • Complexity: Involves numerous steps and formulas.
  • Goal: Understand the process to get the correct answer every time.

Key Concepts

F Ratio/Test Statistic

  • Definition: ANOVA uses the F ratio (F test statistic), different from Z or T tests.
  • Formula: F ratio is a variance divided by a variance (analysis of variances).
    • Numerator: Variance due to group differences.
    • Denominator: Variance within groups (error/random chance).

Interpretation

  • Small Variance Between Groups: Small F statistic, non-significant p-value.
  • Large Variance Between Groups: Large F statistic, significant p-value.

Calculating F Ratio

  • Formula: F = MS between / MS within
    • MS: Mean Sum of Squares.
    • MS Between: Variability between groups.
    • MS Within: Variability within groups.

Mean Sum of Squares (MS)

  • MS Between: SS between / DF between.
  • MS Within: SS within / DF within.

Degrees of Freedom (DF)

  • Between (DF): K - 1, where K = number of groups.
  • Within (DF): n total - K.

Sums of Squares (SS)

  • Complexity: 95% of the work is in calculating the sums of squares.

Calculating Within Groups

  1. Sample Size (n): For each group.
  2. Sum (ΣX): Sum of all values in each group.
  3. Mean (XÌ„): Calculate for interpretation (not used in F directly).
  4. ΣX²/n: Square ΣX, divide by n.
  5. Σx²: Sum of squared values within each group.

Calculating Across Groups

  1. n Total (nT): Sum of all group sample sizes.
  2. ΣΣX: Sum of ΣX for each group.
  3. ΣΣX²/nT: Square ΣΣX total, divide by nT.
  4. Σ(ΣX²/n): Sum ΣX²/n for all groups.
  5. ΣΣx²: Sum Σx² for all groups.

Summing Formulas

  • SS Between: Σ(ΣX²/n) - ΣΣX²/n total.
  • SS Within: ΣΣx² - Σ(ΣX²/n).

Final Calculation Steps

  1. Calculate MS: Use SS and DF.
  2. Calculate F: MS between/MS within.
  3. Effect Size: Optional, use SS between / SS total.

Example Problem

  • Given Data: Three groups with squared values.
  • Steps:
    1. Calculate group-specific stats (n, ΣX, X̄, ΣX²/n, Σx²).
    2. Sum across groups for nT, ΣΣX, and others.
    3. Compute SS, MS, and finally F.

Conclusion

  • Result: F test statistic calculated.
  • Additional: Calculate effect size (η²) if needed.
  • Practice: Key to mastering the process is repeated practice.

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