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Statistics Fundamentals · section 3 of 15

Expectation & variance

3 interview questions on expectation & variance, each answered in full. Free to read, no account needed.

Why do we use standard deviation instead of variance in interpretation?

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  • Variance has squared units (dollars2,  meters2)(\mathrm{dollars}^{2}, \;\mathrm{meters}^{2}), which are unintuitive.
  • Standard deviation is the square root of variance and shares the units of the data, so it can be plotted alongside the mean and interpreted directly.
  • Statistical theory tends to use variance because it decomposes nicely (sum of independent variances), while reporting typically uses SD.
#descriptive#variancePermalink & quiz →

Skewness and kurtosis — what do they measure?

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  • Skewness  =  E[((X  −  μ)/σ)3]\mathrm{Skewness}\; = \;E[((X\; - \;{\mu}) / {\sigma})^{3}] — asymmetry of the distribution.
  • Positive: long right tail (log-normal, income).
  • Negative: long left tail.
  • Kurtosis  =  E[((X  −  μ)/σ)4]\mathrm{Kurtosis}\; = \;E[((X\; - \;{\mu}) / {\sigma})^{4}] — 'tailedness'.
  • Normal has kurtosis 3 (excess kurtosis 0).
  • Higher (leptokurtic): heavier tails, more outliers (t, Laplace, financial returns).
  • Lower (platykurtic): lighter tails than normal.
  • Both give quick diagnostic beyond mean/variance.
#moments#descriptivePermalink & quiz →

Why divide by n-1 in the sample variance?

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  • 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.
#estimation#variancePermalink & quiz →

Practise Statistics Fundamentals

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