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

Chapter Summary: Statistics (CDS Elementary Mathematics)

Statistics deals with the collection, presentation, analysis, and interpretation of numerical data. For the CDS examination, mastery of measures of central tendency, measures of dispersion, and graphical representations is essential.

1. Key Concepts & Definitions

• Data Types: Primary (raw data collected directly) and Secondary (data collected from secondary sources).
• Class Mark: The midpoint of a class interval, given by \(\text{Class Mark} = \frac{\text{Upper Limit} + \text{Lower Limit}}{2}\).
• Cumulative Frequency: The running total of frequencies up to a given class boundary (Less-than or More-than type).

2. Measures of Central Tendency

• Arithmetic Mean (\(\bar{x}\)):
  Direct Method: \[\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\] Assumed Mean Method: \[\bar{x} = a + \frac{\sum f_i d_i}{N}\] where \(d_i = x_i - a\) and \(N = \sum f_i\).
• Median: The middle value of an ordered dataset.
  For grouped continuous data: \[\text{Median} = l + \biggl(\frac{\frac{N}{2} - c.f.}{f}\biggr) h\] where \(l\) is the lower boundary of the median class, \(c.f.\) is the cumulative frequency of the preceding class, \(f\) is the class frequency, and \(h\) is class width.
• Mode: The value occurring with the highest frequency.
  For grouped data: \[\text{Mode} = l + \biggl(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\biggr) h\] where \(f_1\) is the frequency of the modal class, \(f_0\) is the preceding frequency, and \(f_2\) is the succeeding frequency.
• Empirical Relationship: \[\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}\]

3. Graphical Representation & Dispersion

• Histograms & Ogives: The median of a grouped frequency distribution is determined graphically by the x-coordinate of the point of intersection of less-than and more-than ogives.
• Variance (\(\sigma^2\)) & Standard Deviation (\(\sigma\)): \[\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}\] \[\sigma = \sqrt{\frac{\sum f_i x_i^2}{N} - (\bar{x})^2}\]