Measures of Central Tendency
Arithmetic Mean (AM)
Ungrouped: ¯x = (Σxᵢ)/n Frequency Dist.: ¯x = (Σfᵢxᵢ)/(Σfᵢ) = (Σfᵢxᵢ)/N
Ungrouped: ¯x = (Σxᵢ)/n Frequency Dist.: ¯x = (Σfᵢxᵢ)/(Σfᵢ) = (Σfᵢxᵢ)/N
Geometric Mean (GM)
G = (x₁ x₂ … xₙ)1/n
G = (x₁ x₂ … xₙ)1/n
Harmonic Mean (HM)
Ungrouped: H = n / Σ(1/xᵢ) Frequency Dist.: H = N / Σ(fᵢ/xᵢ)
Ungrouped: H = n / Σ(1/xᵢ) Frequency Dist.: H = N / Σ(fᵢ/xᵢ)
Mode (Grouped Data)
Mode = l + [(f − f₁)/((2f − f₁ − f₂))] × h l = lower limit of modal class, h = class width
Mode = l + [(f − f₁)/((2f − f₁ − f₂))] × h l = lower limit of modal class, h = class width
Median
Ungrouped: n odd → ((n+1)/2)th term
n even → average of (n/2)th & ((n/2)+1)th terms
Grouped → Use cumulative frequency table
Ungrouped: n odd → ((n+1)/2)th term
n even → average of (n/2)th & ((n/2)+1)th terms
Grouped → Use cumulative frequency table
Empirical Relation (Most Important for JEE)
Mode = 3 Median − 2 Mean
Measures of Dispersion
Mean Deviation about A
M.D. = (1/N) Σ fᵢ |xᵢ − A| (A can be mean, median or mode)
M.D. = (1/N) Σ fᵢ |xᵢ − A| (A can be mean, median or mode)
Standard Deviation (σ) & Variance
σ = √[ (1/N) Σ fᵢ (xᵢ − ¯x)² ]
Variance = σ²
σ = √[ (1/N) Σ fᵢ (xᵢ − ¯x)² ]
Variance = σ²
Coefficient of Variation (CV)
CV = (σ / ¯x) × 100
Higher CV ⇒ more variable (less consistent)
Basic Probability Concepts
Random Experiment → Sample Space S → Event (subset of S)
| Verbal Description | Set Notation | |
|---|---|---|
| Not A | Aᶜ | |
| A or B (at least one) | A ∪ B | |
| A and B | A ∩ B | |
| A but not B | A ∩ Bᶜ | |
| Exactly one of A, B | (A ∩ Bᶜ) ∪ (Aᶜ ∩ B) |
Classical Definition (Equally Likely Outcomes)
P(A) = n(A) / n(S)
P(A) = n(A) / n(S)
Axioms of Probability
0 ≤ P(A) ≤ 1
P(S) = 1
If A ∩ B = φ ⇒ P(A ∪ B) = P(A) + P(B)