CDS 2027 - Paper 3: Elementary Mathematics cheat sheet for Statistics! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Statistics for CDS 2027

Statistics is a fundamental section of the Elementary Mathematics paper for CDS 2027. It deals with the collection, classification, presentation, and mathematical analysis of numerical data.

1. Measures of Central Tendency

Central tendency identifies a single central value representing an entire dataset.

• Arithmetic Mean: For ungrouped data \(x_1, x_2, \dots, x_n\), the mean is given by: \[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\] For grouped frequency data with frequencies \(f_i\): \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\]
• Median: The middle value when data is arranged in ascending order.
  • If \(n\) is odd, Median = \(\bigl(\frac{n+1}{2}\bigr)^{\text{th}}\text{ term}\).
  • If \(n\) is even, Median = \(\frac{1}{2}\bigl(\bigl(\frac{n}{2}\bigr)^{\text{th}} + \bigl(\frac{n}{2}+1\bigr)^{\text{th}}\bigr)\text{ terms}\).
• Mode: The value that occurs with maximum frequency in a dataset.
• Empirical Relationship: \[\text{Mode} = 3(\text{Median}) - 2(\text{Mean})\]

2. Measures of Dispersion

Dispersion quantifies the scatter or spread of data around a central value.

• Range: \[\text{Range} = \text{Maximum Value} - \text{Minimum Value}\]
• Mean Absolute Deviation (MAD): \[\text{MAD} = \frac{\sum |x_i - \bar{x}|}{n}\]
• Variance (\(\sigma^2\)): \[\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - (\bar{x})^2\]
• Standard Deviation (\(\sigma\)): \[\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}\]
• Coefficient of Variation (CV): \[\text{CV} = \frac{\sigma}{\bar{x}} \times 100\]

3. Frequency Distributions & Graphical Representation

• Histograms & Frequency Polygons: Used to represent continuous class intervals.
• Ogives (Cumulative Frequency Curves): The point of intersection between a 'less than' ogive and a 'more than' ogive gives the median value of the data.