Learning Objectives
- Find the mean of grouped data using direct, assumed mean, and step-deviation methods
- Find the mode of grouped data
- Find the median of grouped data
- Represent cumulative frequency distribution graphically (ogive)
Key Concepts
Mean of Grouped Data
Direct Method: Mean = Σfᵢxᵢ / Σfᵢ, where xᵢ = class mark (midpoint), fᵢ = frequency.
Assumed Mean Method: Mean = a + (Σfᵢdᵢ / Σfᵢ), where dᵢ = xᵢ - a and a is the assumed mean.
Step-Deviation Method: Mean = a + (Σfᵢuᵢ / Σfᵢ) × h, where uᵢ = (xᵢ - a)/h and h is the class size.
Mode of Grouped Data
The class with the highest frequency is the modal class.
Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
where l = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency of class before modal class, f₂ = frequency of class after modal class, h = class size.
Median of Grouped Data
Median = l + [(n/2 - cf) / f] × h
where l = lower limit of median class, n = total frequency, cf = cumulative frequency of class before the median class, f = frequency of the median class, h = class size.
The median class is found by locating the class whose cumulative frequency is just greater than or equal to n/2.
Empirical Relationship
3 Median = Mode + 2 Mean (approximate relationship)
Ogive (Cumulative Frequency Curve)
Two types: Less-than ogive (rising curve) and More-than ogive (falling curve). The x-coordinate of the point where the two ogives intersect gives the median.
Summary
Statistics deals with organising, analysing, and interpreting grouped data. Mean, mode, and median are measures of central tendency for grouped data, each computed using specific formulas. Ogives provide graphical representation of cumulative frequency distributions and can be used to find the median.
Important Terms
- Class Mark
- The midpoint of a class interval: (upper limit + lower limit) / 2
- Modal Class
- The class interval with the highest frequency
- Median Class
- The class interval in which the median lies
- Cumulative Frequency
- The running total of frequencies up to a particular class
- Ogive
- A graph of cumulative frequency plotted against class boundaries
Quick Revision
- Mean (direct) = Σfᵢxᵢ / Σfᵢ
- Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h
- Median = l + [(n/2 - cf)/f] × h
- 3 Median ≈ Mode + 2 Mean
- The median can be found graphically as the intersection point of the two ogives