Statistics

Section: Statistics and Descriptive Analysis Estimated study time: 45 minutes Content: Descriptive statistics summarize the key characteristics of a dataset, providing the foundation for quantitative analysis in finance. The two primary dimensions are measures of central tendency (what is typical?) and measures of dispersion (how spread out is the data?). Measures of central tendency include the arithmetic mean, geometric mean, weighted mean, harmonic mean, median, and mode. For investment returns, the choice of mean matters significantly. The arithmetic mean is the simple average and is appropriate for estimating the expected return in a single future period. The geometric mean (G = [(1+R1)(1+R2)…(1+Rn)]^(1/n) – 1) is the compound annual growth rate and reflects the actual long-run wealth accumulation of a multi-period investment. When returns are volatile, the geometric mean is always less than or equal to the arithmetic mean. Measures of dispersion quantify the spread around the central tendency. Variance is the average squared deviation from the mean; standard deviation is its square root, expressed in the same units as the original data. For a sample, variance uses (n–1) in the denominator (Bessel's correction) to produce an unbiased estimate. The coefficient of variation (CV = standard deviation / mean) normalizes dispersion, allowing comparison across assets with different means. A fund with a 15% mean return and 10% standard deviation has a CV of 0.67, while one with a 5% mean and 6% standard deviation has a CV of 1.20 — the second fund has more dispersion per unit of return. Range and mean absolute deviation (MAD) are simpler dispersion measures but less commonly used in finance. Skewness and kurtosis describe the shape of a distribution. A symmetric distribution has zero…