Exploratory Data Analysis & Descriptive Statistics

Understand variance, kurtosis, skewness, and IQR outlier detection across continuous and discrete random variables.

1. Measures of Central Tendency & Dispersion

For a sample dataset \(X = \{x_1, x_2, \dots, x_n\}\), the sample variance \(s^2\) and standard deviation \(s\) use Bessel's correction (\(n - 1\)) to eliminate sample bias:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, \quad s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \]

2. Skewness & Kurtosis (Higher-Order Moments)

Skewness (3rd Moment): Measures distributional asymmetry: \(\gamma_1 = \mathbb{E}\left[\left(\frac{X - \mu}{\sigma}\right)^3\right]\). Positive skew indicates a heavy right tail.
Kurtosis (4th Moment): Measures tail heaviness and outlier propensity: \(\text{Kurt} = \mathbb{E}\left[\left(\frac{X - \mu}{\sigma}\right)^4\right]\). Leptokurtic distributions (excess kurtosis > 0) feature fat tails.

3. Interquartile Range (IQR) Outlier Rule

John Tukey's robust outlier detection rule identifies points outside \([Q_1 - 1.5 \cdot \text{IQR}, \, Q_3 + 1.5 \cdot \text{IQR}]\), where \(\text{IQR} = Q_3 - Q_1\).

🎯 Module Mastery Certification Quiz

+100 XP

Why does sample variance divide by (n - 1) instead of n (Bessel's correction)?

To correct for the fact that sample mean underestimates the true population variance.

To prevent division by zero when n = 0.

To convert the variance into a percentage.

To make the calculation run faster on CPUs.