Statistics

• Introduction
  • Meaning of Statistics
  • Collection, presentation, analysis, and interpretation of numerical data
• Mean of Grouped Data
  • Direct Method
    • Formula: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
    • $x_i$ = Class mark = $\frac{\text{Lower limit} + \text{Upper limit}}{2}$
  • Assumed Mean Method
    • Formula: $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$
    • $d_i = x_i - a$ ($a$ = Assumed mean)
  • Step-Deviation Method
    • Formula: $\bar{x} = a + \left(\frac{\sum f_i u_i}{\sum f_i}\right) \times h$
    • $u_i = \frac{x_i - a}{h}$ ($h$ = Class size)
• Mode of Grouped Data
  • Definition: Value that appears most frequently
  • Modal Class: Class interval with the maximum frequency
  • Formula: $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$
    • $l$ = Lower limit of modal class
    • $h$ = Size of class interval
    • $f_1$ = Frequency of modal class
    • $f_0$ = Frequency of preceding class
    • $f_2$ = Frequency of succeeding class
• Median of Grouped Data
  • Definition: Middle-most value of the data
  • Median Class: Class interval with cumulative frequency just greater than $\frac{N}{2}$
  • Formula: $\text{Median} = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$
    • $l$ = Lower limit of median class
    • $N$ = Sum of frequencies ($\sum f_i$)
    • $cf$ = Cumulative frequency of preceding class
    • $f$ = Frequency of median class
    • $h$ = Class size
• Graphical Representation of Cumulative Frequency Distribution
  • Ogives (Cumulative Frequency Curves)
    • LessThan Type Ogive
      • Plot upper class limits against less than cumulative frequencies
      • Shape: Rising curve
    • MoreThan Type Ogive
      • Plot lower class limits against more than cumulative frequencies
      • Shape: Falling curve
    • Median Determination
      • X-coordinate of the intersection point of both ogives gives the median