01Descriptive Statistics

02Correlation and Regression

Correlation
Correlation and Causation
Ordinary Least Squares Regression
Multiple Linear Regression
Beta and Hedging Applications

03Sampling and Inference

Populations and Samples
Standard Errors
Confidence Intervals
Hypothesis Testing
p-Values and Statistical Decisions
Type I and Type II Errors

04Bayesian Inference

Priors, Likelihoods, and Posteriors
Bayesian Updating and Decision-Making

05Time Series Fundamentals

Autocorrelation
Stationarity and Regime Change
Autoregressive and Moving Average Models

06Statistical Properties of Market Data

Heavy-Tailed Distributions
Signal, Noise, and Overfitting
Value-at-Risk and Expected Shortfall

07Estimation

Estimating Probabilities and Parameters
Maximum Likelihood Estimation
Estimation Error and Uncertainty
The Bootstrap

← Statistics I: Foundations

Measures of Dispersion

9 min read

Two datasets with the same centre can be entirely different objects, and dispersion measures quantify the difference: how spread out the values are. The variance family dominates for its algebraic properties; the quantile family for its robustness; and knowing both, plus when each misleads, completes the descriptive toolkit.

Variance and standard deviation of data

The average squared distance from the mean, and its root, restoring the data's units. (The estimation module explains the n - 1 variant used when data are a sample from something larger.)

The standard deviation inherits everything the probability courses established: it is the natural scale for deviations, roughly two thirds of well-behaved data sit within one of the mean, and it feeds every downstream formula, standard errors, correlations, regression slopes. Its cost is sensitivity: squaring amplifies outliers, so one wild value can dominate the entire measure, exactly as it dominated the mean.

Quantile-based spread

Range and interquartile range

The range is max minus min: simple and maximally fragile, one value each end. The interquartile range (IQR) is the spread of the middle half, 75th percentile minus 25th: unmoved by anything in the outer quarters.

Worked example: the same data, four spreads

Data: 10, 12, 13, 15, 20. Mean 14; deviations .

Replace the 20 with 90 and the range explodes to 80, jumps to about 31, and the IQR stays 3. Reporting a robust pair (median, IQR) alongside the classical pair (mean, s) costs one line and reveals whether the tails are running the summary.

Relative spread

The coefficient of variation

Spread as a fraction of the mean, unit-free: the honest way to compare variability across datasets measured on different scales, valid when the mean is positive and meaningfully non-zero.

A standard deviation of 5 is enormous for quantities averaging 10 and negligible for quantities averaging 10,000; dividing by the mean puts both on one scale. The same normalising instinct produced the z-score in the probability course, , which locates a single observation within its dataset: spread measures are also the units in which "unusual" is defined.

Quick check

For the data 2, 4, 6, 8, 10 what is the population variance?

Quick check

Dataset A: mean 20, s = 4. Dataset B: mean 200, s = 30. Which has larger relative spread? Answer with its CV as a decimal.

Practise this

From Variance to Sigma →The Spread of a Pair →Variance of a Constant →

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