Descriptive Statistics

Overview

Descriptive statistics summarize a dataset with a few meaningful numbers. They answer three fundamental questions: Where is the center? How much do values vary? What is the shape? Measures of center (mean, median, mode) identify the typical value. Measures of spread (variance, standard deviation, IQR, range) quantify how dispersed observations are. Measures of shape (skewness, kurtosis) describe asymmetry and tail heaviness. Together, these three pillars give you a complete picture of a distribution. The choice of summary measure depends on the data's distribution shape and the presence of outliers — using the wrong one can be deeply misleading.

Key Concepts

Measures of Center

The mean uses every data point but is sensitive to extreme values. It minimizes the sum of squared deviations: . It is the natural measure for symmetric distributions without outliers.

The median is the 50th percentile — the value separating the lower half from the upper half. It is robust to outliers and skewed distributions. When the mean is significantly different from the median, the distribution is skewed.

The mode is the most frequently occurring value. A dataset can have zero, one, or multiple modes. It is especially useful for categorical data where means are undefined.

Weighted and Specialized Means

Use when observations have different importance: combining group means with different sample sizes, weighting by reliability, or applying time decay in time series.

The geometric mean is appropriate for multiplicative processes: growth rates, compound returns, and ratios. It satisfies (AM-GM inequality) and requires positive values.

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. It is useful for rates (speed = distance/time) and is the basis of the F1-score: .

The Means Inequality

For any dataset with positive values: (harmonic ≤ geometric ≤ arithmetic), with equality only when all values are identical.

Measures of Spread

The denominator corrects bias because is computed from the same data, using one degree of freedom. This ensures .

For approximately normal distributions, about 68% of data falls within , 95% within , and 99.7% within (the 68-95-99.7 rule).

The IQR measures the spread of the middle 50% of data. It is robust to outliers. The standard outlier rule: values outside are outliers.

For normal data, is a consistent estimator of the standard deviation, more robust than when outliers are present.

Shape Measures

Right-skewed (positive): Mean > Median > Mode. Common in income data. Left-skewed (negative): Mean < Median < Mode. Common in test scores with ceiling effects.

The subtraction makes the normal distribution have zero excess kurtosis. High kurtosis means more probability in the tails and center, leading to more outliers.

Quick Example

Measure	Formula	Sensitive to Outliers?	Best For
Mean		Yes	Symmetric, no outliers
Median	Middle value when sorted	No	Skewed data, outliers
Mode	Most frequent value	No	Categorical data
Geometric Mean		Less	Growth rates, ratios
Harmonic Mean		Less	Rates, F1-score

Key Takeaways

Deep Dive

For detailed explanations, worked examples, and advanced theory, explore the dedicated statistics lessons:

Measures of Center

Measures of Central Tendency — Mean, median, mode compared with when each is appropriate and Python computation
Arithmetic Mean Deep Dive — Balance point, minimization of squared deviations, and properties
Median — Robust center, computation methods, and relationship to quartiles
Mode — Most frequent value, multimodality, and categorical data analysis
Weighted Mean — Assigning different importance to observations with examples
Geometric Mean — Multiplicative processes, compound growth, and AM-GM inequality
Harmonic Mean — Rates, ratios, F1-score connection, and when to use

Measures of Spread

Range and IQR — Robust spread measures, quartile computation, and outlier detection
Variance — Bessel's correction, computational formulas, population vs. sample, and properties
Standard Deviation — Interpreting spread in original units, the empirical rule

Measures of Shape

Skewness — Fisher-Pearson coefficient, right vs. left skew, and implications for the mean
Kurtosis — Excess kurtosis, tail behavior, leptokurtic vs. platykurtic distributions

Descriptive Statistics