Introductory Statistics

Exploratory Data Analysis and Five-Number Summaries

Box and Whisker Plot

A graphical summary of data based on the five-number summary: Minimum, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum, identifying outliers via $1.5 \times \text{IQR}$.

Mathematical Expectation

The weighted average of all possible values of a random variable $X$, where weights are the respective probabilities: $E(X) = \sum x_i P(x_i)$.

Skewness and Kurtosis

Skewness measures the asymmetry of a distribution around its mean; kurtosis measures the peakedness and fatness of tails relative to a normal distribution (mesokurtic $\beta_2 = 3$).

Financial Data Distribution, Skewness, and Kurtosis Dynamics

Five-Number Summary and Outlier Fences

1. Quartiles Computation:
   • Interquartile Range (IQR): $\text{IQR} = Q_3 - Q_1$.
   • Lower Inner Fence: $\text{LIF} = Q_1 - 1.5 \times \text{IQR}$.
   • Upper Inner Fence: $\text{UIF} = Q_3 + 1.5 \times \text{IQR}$.
   • Any observation falling beyond LIF or UIF is classified as an outlier.
2. Karl Pearson’s Coefficient of Skewness ($S_k$):

   Swipe to see the full formula↔

   $$S_k = \frac{\bar{X} - \text{Mode}}{s} = \frac{3(\bar{X} - \text{Median})}{s}$$
   • If $S_k > 0$: Positively skewed (right tail is longer; Mean $>$ Median $>$ Mode).
   • If $S_k < 0$: Negatively skewed (left tail is longer; Mode $>$ Median $>$ Mean).

Financial Data Visualization and Correlation Analysis

• Diagrams and Graphs in Banking and Finance:
  • Time Series Line Graphs: Displaying daily equity index price fluctuations, interest rate spreads, and foreign exchange reserves over time.
  • Histograms & Frequency Polygons: Displaying return distributions, assessing portfolio skewness and fat-tailed kurtosis.
  • Scatter Plots: Plotting asset returns against market benchmark returns to visualize portfolio beta ($\beta$).
• Bivariate Correlation Analysis:
  • Karl Pearson’s Coefficient of Correlation ($r$): Quantifies linear association between continuous financial variables (e.g., inflation rate and interest rates):

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    $$r = \frac{N \sum XY - \sum X \sum Y}{\sqrt{[N \sum X^2 - (\sum X)^2][N \sum Y^2 - (\sum Y)^2]}}$$
  • Spearman’s Rank Correlation ($\rho$): Evaluates monotonic associations for non-parametric or ordinal ranked data (e.g., credit rating tier vs bond yield rank):

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    $$\rho = 1 - \frac{6 \sum d_i^2}{N(N^2 - 1)}$$

Comparative Analysis of Dispersion and Central Tendency Measures

Five-Number Summary, IQR, and Outlier Detection Computations

Applied Problem: Five-Number Summary and Outlier Detection

Given the quarterly returns (%) of 11 banking stocks: $4, 7, 8, 9, 10, 12, 14, 15, 18, 22, 38$.

1. Determine the five-number summary ($Min, Q_1, Q_2, Q_3, Max$).
2. Compute the Interquartile Range (IQR) and identify any outliers.

Solution:

• $\text{Minimum} = 4$
• $Q_1 = \text{Value of } \frac{n+1}{4} = \text{3rd item} = 8$
• $Q_2 \text{ (Median)} = \text{Value of } \frac{n+1}{2} = \text{6th item} = 12$
• $Q_3 = \text{Value of } \frac{3(n+1)}{4} = \text{9th item} = 18$
• $\text{Maximum} = 38$

Since $38 > 33$, the stock return of $38\%$ is an outlier.

Key analytical nuance: they define IQR as (Q3 - Q1)/2 (which is the quartile deviation, not IQR) or neglect to sort the data before finding quartile positions.

Detailed Mathematical Framework: Measures of Central Tendency and Dispersion

In statistical data modeling, summarizing sample distributions requires rigorous calculation of location and dispersion parameters:

• Arithmetic Mean: $\bar{X} = \frac{\sum_{i=1}^n X_i}{n}$ (for ungrouped data) and $\bar{X} = \frac{\sum f X}{N}$ (for frequency distributions). Sensitive to extreme values and outliers.
• Median ($M_d$): Value of the middle observation when data is arranged in ascending order. Positional measure: $\text{Position} = \frac{n+1}{2}$. Unaffected by extreme outliers.
• Sample Variance ($s^2$) and Standard Deviation ($s$):

  Swipe to see the full formula↔

  $$s^2 = \frac{\sum_{i=1}^n (X_i - \bar{X})^2}{n - 1}, \quad s = \sqrt{s^2}$$
  Division by $n-1$ (Bessel’s correction) provides an unbiased estimator of the true population variance $\sigma^2$.
• Coefficient of Variation (CV):

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  $$\text{CV} = \frac{s}{\bar{X}} \times 100\%$$
  Used in financial analysis to compare relative risk and volatility between securities with different price levels or units of measurement. The distribution with a lower CV is more consistent and stable.

Chebyshev’s Theorem vs. The Empirical Rule

1. Chebyshev’s Theorem: Applies to any probability distribution regardless of shape. For any constant $k > 1$, the proportion of values that lie within $k$ standard deviations of the mean is at least:

   Swipe to see the full formula↔

   $$P(|X - \mu| < k\sigma) \ge 1 - \frac{1}{k^2}$$
   For $k = 2$, at least $75\%$ ($1 - 1/4$) of values lie within $\mu \pm 2\sigma$. For $k = 3$, at least $88.89\%$ ($1 - 1/9$) lie within $\mu \pm 3\sigma$.
2. Empirical Rule (68-95-99.7 Rule): Applies strictly to symmetric, bell-shaped Normal Distributions:
   • Approximately $68.27\%$ of observations lie within $\mu \pm 1\sigma$.
   • Approximately $95.45\%$ of observations lie within $\mu \pm 2\sigma$.
   • Approximately $99.73\%$ of observations lie within $\mu \pm 3\sigma$.