NDA cheat sheet for Statistics and Probability! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Statistics and Probability for NDA

Mastering Statistics and Probability is crucial for securing top marks in the NDA Mathematics paper. This comprehensive reference guide covers key formulas, measure definitions, and foundational probability laws tailored to the National Defence Academy entrance syllabus.

1. Measures of Central Tendency

Measures of central tendency summarize a dataset through a single representative value.

• Mean (Arithmetic Average): For individual observations, \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i\). For grouped data, \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\).
• Median: The middle value of an ordered dataset. If \(n\) is odd, position is \(\frac{n+1}{2}\); if \(n\) is even, it is the average of \(\frac{n}{2}\) and \(\frac{n}{2}+1\).
• Mode: The value that appears most frequently. Empirical relationship: \(\text{Mode} = 3(\text{Median}) - 2(\text{Mean})\).

2. Measures of Dispersion

Dispersion quantifies the degree of spread or variability within a distribution.

• Variance (\(\sigma^2\)): Calculated as \(\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2 = \frac{1}{n}\sum x_i^2 - (\bar{x})^2\).
• Standard Deviation (\(\sigma\)): The positive square root of variance, \(\sigma = \sqrt{\sigma^2}\).
• Coefficient of Variation (CV): Measures relative dispersion: \(\text{CV} = \frac{\sigma}{\bar{x}} \times 100\).

3. Fundamentals of Probability

Probability evaluates the likelihood of occurrence of random events.

• Classical Definition: \(P(A) = \frac{n(A)}{n(S)}\), where \(n(A)\) is favorable outcomes and \(n(S)\) is total sample space outcomes.
• Addition Theorem: For any two events \(A\) and \(B\), \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). If mutually exclusive, \(P(A \cap B) = 0\).
• Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), provided \(P(B) > 0\).
• Independent Events: Events \(A\) and \(B\) are independent if and only if \(P(A \cap B) = P(A) \cdot P(B)\).
• Binomial Distribution: Probability of exactly \(k\) successes in \(n\) independent trials is \(P(X=k) = \binom{n}{k}p^k q^{n-k}\), with mean \(\mu = np\) and variance \(\sigma^2 = npq\).