Core Concepts & Formulas
1. Methods to Find Mean (x̄)
Direct Method:
x̄ = ∑fᵢxᵢ / ∑fᵢ
Assumed Mean Method:
x̄ = a + (∑fᵢdᵢ / ∑fᵢ)
Where dᵢ = xᵢ - a
Step-Deviation Method:
x̄ = a + h × (∑fᵢuᵢ / ∑fᵢ)
Where uᵢ = (xᵢ - a) / h
2. Median of Grouped Data
Median = l + [ ((n / 2) - cf) / f ] × h
3. Mode of Grouped Data
Mode = l + [ (f₁ - f₀) / (2f₁ - f₀ - f₂) ] × h
4. Empirical Relationship
Mode = 3(Median) - 2(Mean)
Points to Remember
Empirical Relationship: A key formula connecting all three measures of central tendency: Mode = 3 Median - 2 Mean.
Class Mark: For grouped data, the mid-point of a class interval (class mark xᵢ) is calculated as (Upper Class Limit + Lower Class Limit) / 2.
Important Practice Questions
- Find the mean of a given frequency distribution using the step-deviation method.
- Calculate the median for a grouped frequency distribution table.
- Find a missing frequency in a given data set when either the mean or median is provided.
Quick Revision
Empirical Formula
Mode = 3(Median) - 2(Mean)
Class Mark
xᵢ = (Upper limit + Lower limit) / 2
Modal Class
The class interval with the highest frequency.
Median Class
The class interval whose cumulative frequency is just greater than or equal to n/2.