Bowley's Skewness, Kurtosis and Moments

Lesson 14 of STA301 explores advanced distributional parameters: Bowley's quartile skewness, statistical Moments, and Kurtosis. Arthur Bowley developed a relative measure of skewness based entirely on quartiles: Sk_B = [(Q3 - Median) - (Median - Q1)] / (Q3 - Q1) = (Q3 + Q1 - 2*Q2) / (Q3 - Q1). Ranging strictly between -1 and +1, Bowley's coefficient is valuable when dealing with open-ended distributions or extreme outliers where the arithmetic mean cannot be reliably computed. The lecture then introduces Statistical Moments, which provide a unified mathematical framework for describing central location, dispersion, skewness, and kurtosis. Raw moments about zero (m'_r = sum(X^r) / n) and moments about an arbitrary origin A (m'_r = sum(X - A)^r / n) are contrasted with Central Moments about the mean (m_r = sum(X - X-bar)^r / n). Fundamental relationships are established: the first central moment m1 is always zero, while the second central moment m2 equals sample variance s^2. Binomial expansions link raw and central moments: m2 = m'_2 - (m'_1)^2, m3 = m'_3 - 3*m'_2*m'_1 + 2*(m'_1)^3, and m4 = m'_4 - 4*m'_3*m'_1 + 6*m'_2*(m'_1)^2 - 3*(m'_1)^4. For grouped continuous data, Sheppard's Corrections are applied to correct for interval grouping errors: corrected m2 = m2 - h^2 / 12 and corrected m4 = m4 - h^2*m2 / 2 + 7*h^4 / 240. The lecture concludes with Kurtosis, measuring the degree of peakedness or flatness of a unimodal frequency curve relative to the standard normal distribution. Karl Pearson's Moment Ratios are derived: beta1 = (m3)^2 / (m2)^3 measures skewness, and beta2 = m4 / (m2)^2 measures kurtosis. A distribution is Mesokurtic if beta2 = 3 (normal curve), Leptokurtic if beta2 > 3 (sharper central peak, heavier tails), and Platykurtic if beta2 < 3 (flatter peak, lighter tails). In Virtual University midterm assessments, moment derivations, Sheppard's adjustments, and beta2 classifications represent core computational exam questions. Consulting sta301 handouts and practicing solved MCQs by Moaaz guarantees academic mastery.