CA Foundation June 2024 - Statistics Formula Marathon Lecture

Introduction

• Session Title: Formula Marathon of Statistics
• CA Foundation Paper 3: Quantitative Techniques (Statistics)
• Goal: Review all formulas from the Statistics portion
• Notes: Available in the provided PDF link in the description
• Phases:
  • Phase 1: Chapters 1-6 (61 Marks) revision completed
  • Phase 2: Formula Marathon of Mathematics completed
  • Current Session: Formula Marathon of Statistics
• Session Format:
  • Review concepts interactively
  • Notes and formulas to be filled in during the session
  • Approximately 97 formulas to revise

Key Points Covered

Introduction and Basics

• Class Boundaries:
  • Exclusive Classification: No effort needed, UCB = UCL, LCB = LCL
  • Inclusive Classification: Adjustments required (UCB: +0.5, LCB: -0.5)
• Midpoint (Class Mark):
  • Simplified as (LCB + UCB) / 2
• Class Length (Width):
  • Formula: UCB - LCB
• Frequency Density:
  • Formula: Class Frequency / Class Length
• Relative Frequency and Percentage Frequency:
  • Relative Frequency: Class Frequency / Total Frequency
  • Percentage Frequency: (Class Frequency / Total Frequency)

Central Tendency Formulas

• Arithmetic Mean (Discrete Data):
  • Formula: Σx / n
• Arithmetic Mean (Grouped Data):
  • Formula: Σ(fx) / n
• Combined Mean:
  • Formula: (n1x̄1 + n2x̄2) / (n1 + n2)
• Arithmetic Mean (Assumed Mean or Step-Deviation Method):
  • Formula: A + (Σ(fd) / n) × C
• Median (Discrete Data):
  • Sort data in ascending order, apply
    • Odd n: Middle term
    • Even n: Average of two middle terms
• Median (Grouped Data):
  • Formula: L1 + ((n/2) - F) / f) × C
• Partition Values General Formula (Quartiles, Deciles, Percentiles):
  • Formula: (n+1) × p/100 (for Percentiles)
• Mode (Discrete Data):
  • Most frequent observation
• Mode (Grouped Frequency Distribution):
  • Formula: L1 + ((f0 - f-1) / (2f0 - f-1 - f1) × C)
• Relationship Between Mean, Median, and Mode:
  • Symmetrical Data: Mean = Median = Mode
• Geometric Mean (Discrete Positive Observations):
  • Formula: (x1 × x2 × ... × xn)^(1/n)
• Geometric Mean (Frequency Distribution):
  • Formula: (x1^f1 × x2^f2 × ... × xn^fn)^(1/N)
• Harmonic Mean (Discrete Data):
  • Formula: n / Σ(1/x)
• Harmonic Mean (Frequency Distribution):
  • Formula: N / Σ(f / x)
• Combined Harmonic Mean:
  • Formula: (n1 + n2) / ((n1 / H1) + (n2 / H2))
• Relation Between AM, GM, and HM:
  • If observations are identical: AM = GM = HM
  • If observations are distinct: AM > GM > HM

Dispersion Formulas

• Range:
  • Formula (Discrete): L - S
  • Formula (Grouped): UCB (Last CI) - LCB (First CI)
• Coefficient of Range:
  • Formula: (L - S) / (L + S) × 100
• Mean Deviation:
  • About Mean: Σ|x - x̄| / n
  • About Median/Mode: Σ|x - Median| / n or Σ|x - Mode| / n
• Standard Deviation:
  • Formula (Discrete): √[Σ(x - x̄)^2 / n]
  • Shortcut Formula: √[Σ(x^2)/n - (x̄)^2]
• Coefficient of Variation:
  • Formula: (σ / x̄) × 100
• Variance:
  • Same as Standard Deviation without the square root
• Relationship Between Measures of Dispersion (Approx):
  • AM / GM / HM
• Change of Origin and Scale in Dispersion:
  • No effect on origin, but effect on scale

Probability Formulas

• Basic Probability:
  • P(A) = (Number of Favorable Outcomes) / (Total Outcomes)
  • P(A') = 1 - P(A)
• Compound Probability (Intersection):
  • P(A ∩ B) = P(A) × P(B | A)
  • For Independent Events: P(A ∩ B) = P(A) × P(B)
• Conditional Probability:
  • P(B | A) = P(A ∩ B) / P(A)
• Bayes' Theorem:
  • P(A | B) = [P(B | A) × P(A)] / P(B)
• Expected Value:
  • E(x) = Σ[p(x) × x]
• Variance of Probability Distribution:
  • Var(x) = E(x^2) - [E(x)]^2
• Binomial Probability Function:
  • P(X=x) = nCx p^x (1-p)^(n-x)
• Poisson Distribution:
  • P(X=x) = (e^-λ λ^x) / x!
• Normal Distribution:
  • Probability Density Function and Z-Score Calculation
  • μ = 0 and σ = 1 for standard normal distribution

Correlation and Regression Formulas

• Karl Pearson's Correlation Coefficient:
  • r = Cov(X, Y) / (σX σY)
• Spearman's Rank Correlation:
  • Formula: 1 - [(6 ΣD^2) / (n (n^2 - 1))]
• Regression Coefficients:
  • bxy = Cov(X, Y) / Var(X)
  • byx = Cov(X, Y) / Var(Y)
• Combined Regression Coefficient:
  • r^2 = bxy × byx
• Regression Line Intersection:
  • Mean of X and Y
• Coefficient of Determination:
  • r^2
• Coefficient of Non-Determination:
  • 1 - r^2

Index Numbers Formulas

• Price, Quantity, and Value Relatives:
  • Price Relative: P₁ / P₀
  • Quantity Relative: Q₁ / Q₀
  • Value Relative: V₁ / V₀
• Laspeyres Index:
  • Σ(P₁ Q₀) / Σ(P₀ Q₀) × 100
• Paasche's Index:
  • Σ(P₁ Q₁) / Σ(P₀ Q₁) × 100
• Fisher's Ideal Index:
  • √(Laspeyres' Index × Paasche's Index)
• Marshall-Edgeworth Index:
  • Σ(P₁(Q₀ + Q₁)) / Σ(P₀(Q₀ + Q₁)) × 100

Session Conclusion

• Upcoming Sessions:
  • 25th May: One-shot revision of Sequence and Series (Chapter 6)
• Playlist: Contains all live and recorded session links

iem>Note</em>: View the detailed sessions in the given playlist for better conceptual understanding.