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 Central Tendency

9 min read

Statistics begins where probability ends: with data rather than distributions, and the first question about any dataset is where its centre lies. Three standard answers, mean, median, and mode, agree for symmetric data and diverge informatively otherwise, and choosing among them is the first modelling decision most analyses make.

Mean, median, mode

The mean is the arithmetic average, . The median is the middle value when the data are sorted (the average of the two middle values for even n). The mode is the most frequent value, or the peak of a distribution.

The mean is the balance point: deviations from it sum to exactly zero, and it minimises the sum of SQUARED deviations, which ties it to variance and to least squares later in this course. The median minimises the sum of ABSOLUTE deviations and is a positional measure: half the data on each side. The mode answers a different question entirely, what is typical to SEE, and can sit far from both.

Robustness: how the measures fail

Worked example: one large value moves one measure

Incomes 30, 35, 40, 45, 50 (thousands): mean 40, median 40. Replace the top value with 500:

The mean followed the outlier; the median did not move at all. Neither behaviour is wrong: the mean answers "total divided by count" (relevant when totals matter), the median answers "the typical unit" (relevant when representativeness matters). The divergence itself is information: it announces asymmetry.

A right-skewed distribution and its three centres

Density

With a long right tail the three measures separate in a fixed order: mode, then median, then mean, the mean dragged furthest toward the tail. Left skew reverses the order.

Weighted means and which average

When observations carry different importance, sizes, durations, counts, the weighted mean replaces the plain one, and most practical averages are secretly weighted: an average computed over groups must weight by group size, or it answers a question about GROUPS rather than individuals. The ratios lesson's harmonic mean is the reminder that even the right weighting depends on what is held fixed. The recurring error is averaging averages without weights, which silently changes the question.

Quick check

What is the median of the data 12, 5, 9, 21, 14?

Quick check

Group A (40 people) averages 60; group B (10 people) averages 85. What is the overall mean? Answer as a decimal.

Practise this

Mean of the First Ten →Means Add Up →

Next lesson →