Learning Module 3
Statistical Measures
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Calculate, interpret, and evaluate measures of central tendency and location to address an investment problem.
Calculate, interpret, and evaluate measures of dispersion to address an investment problem.
Interpret and evaluate measures of skewness and kurtosis to address an investment problem.
Interpret correlation between two variables to address an investment problem.
1 · Measures of Central Tendency & Location
ARITHMETIC MEAN
Sum of all observations divided by n. Most common measure. Sensitive to outliers. Best for estimating expected future single-period returns.
MEDIAN
Middle value when sorted. For even n: average of two middle values. Not influenced by outliers. Best for skewed distributions.
MODE
Most frequently occurring value. Can be unimodal, bimodal, trimodal, or none. Only measure usable with nominal/categorical data.
GEOMETRIC MEAN
Compound rate of growth. Always ≤ arithmetic mean (equal only if all values identical). Best for multi-period return performance.
TRIMMED MEAN
Arithmetic mean computed after removing a stated % from both the top and bottom of the distribution. Reduces outlier effect.
WINSORIZED MEAN
Replaces extreme values (top and bottom %) with the nearest non-extreme value, then computes the mean. Retains sample size.
X̄ = (X₁ + X₂ + … + Xₙ) / n = Σᵢ Xᵢ / n
Population mean: μ = ΣXᵢ / N (divide by N, not N-1)
Sample mean: X̄ = ΣXᵢ / n
Quantiles divide the distribution into equal parts:
Quartiles → 4 parts (Q1=25%, Q2=50%=median, Q3=75%)
Quintiles → 5 parts (each 20%)
Deciles → 10 parts (each 10%)
Percentiles → 100 parts (each 1%)
Interquartile Range (IQR) = Q3 − Q1
Box plot: box = IQR; whiskers = upper/lower fences = Q3 ± 1.5×IQR
Observations beyond fences = outliers (plotted as individual points)
Order matters for skewed data: In a positively skeweddistribution: Mode < Median < Mean. In a negatively skeweddistribution: Mean < Median < Mode. The mean is always pulled toward the tail. For investment analysis, skewed returns make the median more representative than the mean.
2 · Measures of Dispersion
Range = Maximum value − Minimum value
Simplest measure. Uses only 2 data points. Highly sensitive to outliers.
Provides no information about the shape of the distribution.
Mean Absolute Deviation — average of absolute deviations from mean:
MAD = Σᵢ |Xᵢ − X̄| / n
Uses all observations (unlike range). Avoids negative deviations cancelling.
Less mathematically tractable than variance (no squaring).
Average of squared deviations, divided by n−1 (degrees of freedom):
s² = Σᵢ(Xᵢ − X̄)² / (n − 1)
Divide by n−1 (not n) to get an unbiased estimator of population variance σ².
Units = squared units of the original data (e.g., %²).
Square root of variance — same units as original data:
s = √[ Σᵢ(Xᵢ − X̄)² / (n − 1) ]
Most widely used dispersion measure. Expressed in same units as returns (%),
making it directly interpretable alongside the mean.
Focuses only on downside risk — returns below a target B:
s_Target = √[ Σ(Xᵢ − B)² / (n − 1) ] for all Xᵢ ≤ B
B = minimum acceptable return (e.g., 0%, risk-free rate, or benchmark).
Only observations below the target enter the calculation.
Preferred by investors who care more about losses than gains.
Risk per unit of return — scale-free, allows comparison across assets:
CV = s / X̄
Lower CV = more efficient (less risk per unit of return).
Useful when assets have different means and/or different units.
Only meaningful when X̄ > 0.
Measure | Formula | Best Use | Limitation |
|---|---|---|---|
Range | Max − Min | Quick overview | Only 2 data points |
MAD | Σ|Xᵢ−X̄|/n | All observations equally | Not mathematically tractable |
Variance (s²) | Σ(Xᵢ−X̄)²/(n−1) | Statistical theory | Squared units |
Std Dev (s) | √s² | Most common, same units | Treats upside = downside risk |
Target Semidev | √Σ(Xᵢ−B)²/(n−1), Xᵢ≤B | Downside risk focus | Ignores upside observations |
CV | s/X̄ | Cross-asset comparison | Meaningless if X̄ ≤ 0 |
3 · Skewness & Kurtosis — Shape of the Distribution
Mean > Median > Mode. Long right tail. Frequent small losses, rare large gains. Mean pulled upward by extreme positive values.
Mean = Median = Mode. Normal distribution has zero skewness. Gains and losses are mirror images.
Mean < Median < Mode. Long left tail. Frequent small gains, rare large losses. Relevant for strategies with crash risk.
Average cubed deviation from the mean (standardised by s³):
Skewness ≈ (1/n) × Σᵢ(Xᵢ − X̄)³ / s³
Skewness = 0 → symmetric distribution
Skewness > 0 → right-skewed (positive); tail on the right
Skewness < 0 → left-skewed (negative); tail on the left
Investment implication: positive skewness is preferred (limited loss,
unlimited gain). Many equity strategies have negative skew.
Fat-tailed / excess kurtosis > 0. More frequent extreme outcomes than normal. Most equity return distributions.
Normal distribution. Excess kurtosis = 0. Benchmark for comparison. Tails as expected under normality.
Thin-tailed / excess kurtosis < 0. Fewer extreme outcomes than normal. Less tail risk than a normal distribution.
Kurtosis relative to the normal distribution (kurtosis = 3):
K_E ≈ (1/n) × Σᵢ(Xᵢ − X̄)⁴ / s⁴ − 3
K_E > 0 → fat-tailed (leptokurtic) — more extreme events than normal
K_E = 0 → mesokurtic (normal distribution)
K_E < 0 → thin-tailed (platykurtic) — fewer extreme events than normal
Most equity return series exhibit positive excess kurtosis (fat tails).
Ignoring fat tails underestimates the probability of extreme losses.
Investment preference: investors generally prefer positive skewness (limited downside, large upside potential) and lower kurtosis (fewer extreme shocks).
Normal distribution baseline: skewness = 0, kurtosis = 3, excess kurtosis = 0. Financial returns almost universally deviate from this — fatter tails and some skew.
Combined effect: a distribution with negative skew AND positive excess kurtosis is particularly dangerous — frequent small gains but occasional large losses, with more extreme outcomes than normal.
4 · Correlation Between Two Variables
Measures how two variables move together (jointly):
s_XY = Σᵢ(Xᵢ − X̄)(Yᵢ − Ȳ) / (n − 1)
Positive s_XY → X and Y tend to move in the same direction
Negative s_XY → X and Y tend to move in opposite directions
Units = product of units of X and Y (difficult to interpret in isolation)
Standardized covariance — scale-free, always between −1 and +1:
r_XY = s_XY / (s_X × s_Y)
r = +1 → perfect positive linear relationship
r = 0 → no linear relationship (may still be nonlinear)
r = −1 → perfect negative (inverse) linear relationship
Correlation does NOT imply causation. Two variables may be correlated
due to a third variable or pure chance (spurious correlation).
Correlation Range | Interpretation | Portfolio Implication |
|---|---|---|
r close to +1 | Strong positive — assets move together | Little diversification benefit |
0 < r < +1 | Positive — partial co-movement | Some diversification benefit |
r = 0 | No linear relationship | Maximum diversification of linear risk |
−1 < r < 0 | Negative — tend to move opposite | Strong diversification benefit |
r close to −1 | Strong inverse relationship | Near-perfect hedge possible |
Scatter plots: tight clustering around a line signals high correlation; loose/scattered points signal low correlation. Non-linear patterns can exist with low linear correlation.
Spurious correlation: occurs when (1) correlation reflects chance, (2) both variables are driven by a third hidden variable, or (3) the calculation mixes variables via a common divisor. Never build investment strategies on spurious correlations.
Outliers: a single extreme observation can dramatically change the correlation coefficient. Always visualise data with scatter plots before relying on a numerical correlation.
Correlation ≠ causation: even a high correlation of 0.90 does not mean one variable causes the other. Anscombe's Quartet shows four datasets with identical means, standard deviations, and correlations but completely different distributions.