Statistics Cheat Sheet - CDS 2027 - Paper 3: Elementary Mathematics

1. Descriptive Statistics

Descriptive statistics forms a crucial pillar of the CDS Elementary Mathematics syllabus. It deals with summarizing, organizing, and measuring key features of a collection of data.

Measures of Central Tendency

Central tendency identifies a single central value as representative of an entire distribution:

• Sample Mean: \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\)
• Population Mean: \(\mu = \frac{1}{N} \sum_{i=1}^{N} x_i\)
• Median: The middle value when data points are arranged in ascending or descending order.
• Mode: The value that appears most frequently in a dataset.

Measures of Dispersion

Dispersion quantifies the degree of spread or variability within a dataset:

• Sample Variance: \(s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\)
• Population Variance: \(\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2\)
• Standard Deviation: \(s = \sqrt{s^2}\), \(\sigma = \sqrt{\sigma^2}\)
• Interquartile Range: \(\text{IQR} = Q_3 - Q_1\)
• Coefficient of Variation: \(\text{CV} = \frac{s}{\bar{x}} \times 100\%\)

2. Probability Theory

Probability quantifies the likelihood of occurrence of random events, serving as the foundation for statistical inference.

Axioms and Basic Rules

• Fundamental bounds: \(0 \le P(A) \le 1\), with sample space probability \(P(\Omega) = 1\)
• Complement Rule: \(P(A^c) = 1 - P(A)\)
• Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
• Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
• Independence Condition: Events \(A\) and \(B\) are independent if and only if \(P(A \cap B) = P(A)P(B)\)

Bayes' Theorem

Bayes' theorem updates prior probabilities based on new evidence or observed events:

3. Random Variables

A random variable maps outcomes of a random process to numerical values, quantified through mathematical expectation and variance.

Expectation and Variance

• Discrete Random Variable Expectation: \(E[X] = \sum x P(X=x)\)
• Continuous Random Variable Expectation: \(E[X] = \int_{-\infty}^{\infty} x f(x) dx\)
• Variance Identity: \(\text{Var}(X) = E[X^2] - (E[X])^2\)

Linear Transformations

• \(E[aX + b] = a E[X] + b\)
• \(\text{Var}(aX + b) = a^2 \text{Var}(X)\)
• \(\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y) \pm 2\text{Cov}(X,Y)\)

4. CDS Exam Pattern & PYQ Analysis

In UPSC CDS Paper 3 (Elementary Mathematics), Statistics consistently accounts for approximately 8% to 12% of total questions. Key focal points include calculations of mean, median, and mode for grouped vs. ungrouped data, alongside standard deviation properties under linear transformations.

Mastering basic probability principles, conditional probability, and simple applications of Bayes' Theorem ensures speed and accuracy under strict exam conditions.

5. Frequently Asked Questions (People Also Ask)