Statistics – Measures of Central Tendency & Dispersion

Introduction

• Statistics deals with collection, presentation, analysis and interpretation of data.
• Main objective: to represent the entire data using a single representative value.

Measures of Central Tendency

• Arithmetic Mean (A.M.)
• Geometric Mean (G.M.)
• Harmonic Mean (H.M.)
• Median
• Mode

Arithmetic Mean (A.M.)

Individual Series

• x̄ = (x₁ + x₂ + … + xₙ) / n
• x̄ = (1/n) Σxᵢ

Discrete Series

• x̄ = (Σfᵢxᵢ) / (Σfᵢ)
• Total frequency: N = Σfᵢ

Continuous Series

• Use mid-values of class intervals
• x̄ = (Σfᵢxᵢ) / N

Properties of Arithmetic Mean

• Σ(xᵢ − x̄) = 0
• If each observation is increased/decreased by k, mean changes by k
• If each observation is multiplied/divided by k, mean also changes by same factor
• Sum of squared deviations is minimum about mean

Step Deviation Method

• x̄ = A + h (Σfᵢdᵢ / N)
• dᵢ = (xᵢ − A)/h

Combined Mean

• x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

Weighted Arithmetic Mean

• x̄_w = (Σwᵢxᵢ) / (Σwᵢ)

Geometric Mean (G.M.)

• Individual series: G.M. = (x₁x₂…xₙ)^1/n
• Discrete/continuous series: G.M. = (x₁^f₁ x₂^f₂ … xₙ^fₙ)^1/N

Harmonic Mean (H.M.)

• Individual series: H.M. = n / Σ(1/xᵢ)
• Discrete/continuous series: H.M. = N / Σ(fᵢ/xᵢ)
• Relation (for xᵢ > 0): A.M. ≥ G.M. ≥ H.M.

Median

Individual Series

• Arrange data in ascending/descending order
• Odd n: Median = (n+1)/2^th item
• Even n: Median = average of n/2^th and (n/2 + 1)^th items

Discrete Series

• Find cumulative frequency
• Median corresponds to c.f. just greater than N/2

Continuous Series

• Median = l + [ (N/2 − C) / f ] × h
• l = lower limit of median class
• C = cumulative frequency before median class
• f = frequency of median class
• h = class width

Mode

Continuous Series

• Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
• f₁ = frequency of modal class
• f₀ = frequency of preceding class
• f₂ = frequency of succeeding class

Empirical Relation

• Mode = 3 Median − 2 Mean (when mode is ill-defined)

Measures of Dispersion

• Range
• Mean Deviation
• Quartile Deviation
• Standard Deviation

Range

• Range = Maximum value − Minimum value

Mean Deviation (M.D.)

• Individual series: M.D. = (1/n) Σ|xᵢ − A|
• Discrete/continuous series: M.D. = (1/N) Σfᵢ|xᵢ − A|
• M.D. is minimum when taken about median

Standard Deviation (σ)

Individual Series

• σ² = (1/n) Σ(xᵢ − x̄)²
• σ = √σ²

Discrete/Continuous Series

• σ² = (1/N) Σfᵢ(xᵢ − x̄)²

Step Deviation Method

• σ² = h² [ (Σfᵢdᵢ² / N) − (Σfᵢdᵢ / N)² ]

Combined Standard Deviation

• σ² = [ n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²) ] / (n₁ + n₂)
• d₁ = x̄₁ − x̄, d₂ = x̄₂ − x̄

Coefficient of Variation (C.V.)

• C.V. = (σ / x̄) × 100
• Lower C.V. ⇒ more consistency

Properties of Variance

• Change of origin does not affect variance
• If observations are multiplied by k, variance becomes k² Var(X)
• If xᵢ = a + huᵢ, then Var(X) = h² Var(U)