Descriptive Statistics
Overview
Descriptive statistics summarize and describe the main features of a dataset without drawing inferences beyond the data itself. They provide simple summaries about the sample and the measures.
Measures of Center
| Property | Statement |
|---|---|
| Mean | \(\bar{{x}}=\frac{{1}}{{n}}\sum_{{i=1}}^n x_i\) — arithmetic average |
| Median | Middle value when data are ordered; robust to outliers |
| Mode | Most frequently occurring value |
Measures of Spread
| Property | Statement |
|---|---|
| Range | Max − Min |
| Variance | \(s^2=\frac{{1}}{{n-1}}\sum(x_i-\bar{{x}})^2\) |
| Std Dev | \(s=\sqrt{{s^2}}\) |
| IQR | Q3 − Q1 |
Shape of Distributions
Skewness measures asymmetry; kurtosis measures tail heaviness. A symmetric, bell-shaped distribution has skewness ≈ 0.
Examples
Example 1. Find the mean and standard deviation of {{2,4,4,4,5,5,7,9}}.
Solution. Mean = 5. Variance = \(\frac{{(2-5)^2+3(4-5)^2+2(5-5)^2+(7-5)^2+(9-5)^2}}{{7}}=4\). SD = 2.
Deep Dive: Descriptive Statistics
This section builds durable understanding of descriptive statistics in statistics through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using descriptive statistics and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.