Controls
Std only: three Normals with σ=1,2,3 (same μ=0).
Skew only: same μ & σ (standardized) via Normal mixtures (right-skew, left-skew).
Kurtosis only: symmetric Normal mixtures: heavy tails (rare wide component), light/flat (two nearby narrow components), all standardized.
Solid = mean μ, dashed = median. Shaded center μ±σ; shaded tails |x−μ|≥2σ.
Equations (population → sample)
\[\textbf{Mean}\quad \mu=\mathbb{E}[X],\quad \hat{\mu}=\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i\]
\[\textbf{Variance / Std}\quad \sigma^2=\mathbb{E}[(X-\mu)^2],\quad s^2=\frac{1}{n-1}\sum (x_i-\bar{x})^2,\ s=\sqrt{s^2}\]
\[\textbf{Skewness}\quad \gamma_1=\frac{\mathbb{E}[(X-\mu)^3]}{\sigma^3}\approx\frac{\frac{1}{n}\sum (x_i-\bar{x})^3}{\left(\frac{1}{n}\sum (x_i-\bar{x})^2\right)^{3/2}}\]
\[\textbf{Excess kurtosis}\quad \gamma_2=\frac{\mathbb{E}[(X-\mu)^4]}{\sigma^4}-3\approx\frac{\frac{1}{n}\sum (x_i-\bar{x})^4}{\left(\frac{1}{n}\sum (x_i-\bar{x})^2\right)^2}-3\]
\[\textbf{Jarque–Bera}\quad \mathrm{JB}=\frac{n}{6}\left(\gamma_1^2+\frac{1}{4}\gamma_2^2\right)\]
Excel quick-start (copy/paste)
Mean
=AVERAGE(A2:A1001)Std dev
=STDEV.S(A2:A1001)Variance
=VAR.S(A2:A1001)Skewness
=SKEW(A2:A1001)Kurtosis (excess)
=KURT(A2:A1001)Center mass (≤1σ)
=COUNTIF(ABS(A2:A1001-AVERAGE(A2:A1001)),"<="&STDEV.S(A2:A1001))/ROWS(A2:A1001)Tail mass (≥2σ)
=COUNTIF(ABS(A2:A1001-AVERAGE(A2:A1001)),">="&2*STDEV.S(A2:A1001))/ROWS(A2:A1001)Notes (student life / engineering intuition)
- Same mean, different σ: Two classes can have the same average exam score, but one has bigger spread ⇒ more risk of very low and very high scores.
- Same μ & σ, different skew: Projects with the same average cost and spread can still differ—right-skew means rare but painful overruns (like “one part failed and delayed shipping”).
- Same μ, σ, skew≈0, different kurt: Heavy tails = more outliers (e.g., extreme queue times when one machine goes down). Light/flat tails = fewer extremes but more “shoulders.”
- Always check skew and kurtosis before assuming “Normal.” The shaded tails make outlier risk obvious at a glance.