EasyDeepLearn

Why divide by n-1 in the sample variance?

easy

Answer

  • Sample variance uses estimated mean X̄ instead of true μ.
  • Since X̄ is closer to the data than μ (it minimizes SSD by definition), Σ(Xi  −  X)2{\Sigma}(X_{i}\; - \;X)^{2} is systematically smaller than Σ(Xi  −  μ)2{\Sigma}(X_{i}\; - \;{\mu})^{2}.
  • Dividing by n-1 (not n) corrects this bias → unbiased estimator of σ2{\sigma}^{2}.
  • Formally: one degree of freedom is used to estimate the mean, leaving n-1 for variance.
  • Same reason ANOVA / regression use df corrections.
Check yourself — multiple choice
  • Convention
  • X̄ is closer to data than μ → Σ(X  −  X)2{\Sigma}(X\; - \;X)^{2} biased low; dividing by n-1 corrects to unbiased estimator (1 df used for mean)
  • Wrong choice
  • Only for small n

n-1: Bessel correction — X̄ absorbs one df; makes sample variance unbiased.

#estimation#variance

Practise Statistics Fundamentals

215 interview questions in this topic.

Read the lessonFlashcardsTake the quiz

Related questions