UPSC EPFO APFC • DRDO SAO • RBI Grade B • NABARD Grade A
⚡ Quick Answer & AI Exam Snapshot (TL;DR)
Correlation measures the degree and direction of symmetric linear association between two variables (−1 ≤ r ≤ +1), whereas Regression models the functional cause-and-effect relationship to predict a dependent variable from an independent variable (with r² = bᵧₓ × bₓᵧ). Index Numbers measure changes in price/quantity over time (Fisher’s Ideal Index is the Geometric Mean of Laspeyres and Paasche and satisfies both Time Reversal and Factor Reversal tests). Sampling is divided into Probability Sampling (Random, Stratified, Cluster, Systematic) and Non-Probability Sampling (Quota, Judgment, Convenience, Snowball).
1. Correlation (Karl Pearson’s ‘r’ & Spearman’s Rank Correlation ‘R’)
- Karl Pearson’s Product Moment Correlation Coefficient (r):
r = Cov(X, Y) / (σₓ · σᵧ) = ∑(xᵢ − x̄)(yᵢ − ȳ) / [ N · σₓ · σᵧ ]- Range: Always lies strictly between −1 and +1 inclusive (
−1 ≤ r ≤ +1). - Unit-Free & Symmetric:
rₓᵧ = rᵧₓ. - Independent of BOTH Origin and Scale: Adding/subtracting constants or multiplying by positive constants leaves
rcompletely unchanged (if one variable is multiplied by a negative constant and the other by a positive constant, only the sign ofrflips). - Probable Error of Correlation:
P.E. = 0.6745 × [ (1 − r²) / √N ]. Ifr > 6 × P.E., correlation is statistically significant!
- Range: Always lies strictly between −1 and +1 inclusive (
- Spearman’s Rank Correlation Coefficient (R): Best suited for qualitative attributes (e.g., beauty contest judges, honesty, leadership rankings):
R = 1 − [ 6 ∑D² / n(n² − 1) ](whereD = R₁ − R₂is the difference between ranks, and∑Dis always zero).
2. Regression Analysis: The Two Lines of Regression & Properties of bᵧₓ and bₓᵧ
Memorize the five algebraic properties linking Regression Coefficients (bᵧₓ = r · [σᵧ / σₓ] and bₓᵧ = r · [σₓ / σᵧ]) and Correlation (r):
1. Geometric Mean Property:
2. Sign Rule:
3. Magnitude Rule: If one regression coefficient is greater than 1 (e.g.,
4. Point of Intersection: The two regression lines (Y on X and X on Y) ALWAYS intersect at the Point of Means
5. Angle Between Two Regression Lines: If
r = ± √(bᵧₓ × bₓᵧ) ⇒ r² = bᵧₓ × bₓᵧ ≤ 12. Sign Rule:
bᵧₓ, bₓᵧ, and r MUST ALL have the EXACT SAME ALGEBRAIC SIGN (all positive or all negative).3. Magnitude Rule: If one regression coefficient is greater than 1 (e.g.,
bᵧₓ = 1.6), the other MUST be less than 1 (e.g., bₓᵧ ≤ 0.625) so that their product bᵧₓ × bₓᵧ ≤ 1.4. Point of Intersection: The two regression lines (Y on X and X on Y) ALWAYS intersect at the Point of Means
(X̄, Ȳ).5. Angle Between Two Regression Lines: If
r = 0 (uncorrelated), the two regression lines are perpendicular (90°) to each other. If r = ±1 (perfect correlation), the two regression lines coincide (0° angle).
3. Index Numbers: Laspeyres, Paasche, Fisher & Adequacy Tests
| Index Number Type | Formula & Weighting System | Bias & Adequacy Test Performance |
|---|---|---|
| 1. Laspeyres Price Index (P₀₁ᴸ) | [ ∑(p₁ q₀) / ∑(p₀ q₀) ] × 100Uses Base Year Quantities (q₀) as weights. |
Has an UPWARD BIAS (overstates inflation because it ignores substitution away from goods that became costlier). Fails Time & Factor Reversal tests. |
| 2. Paasche Price Index (P₀₁ᴾ) | [ ∑(p₁ q₁) / ∑(p₀ q₁) ] × 100Uses Current Year Quantities (q₁) as weights. |
Has a DOWNWARD BIAS. Fails Time & Factor Reversal tests. |
| 3. Dorbish-Bowley Index | (L + P) / 2Arithmetic Mean of Laspeyres and Paasche. |
Fails both Time Reversal and Factor Reversal tests. |
| 4. Fisher’s Ideal Index (P₀₁ᶠ) | √(L × P)Geometric Mean of Laspeyres and Paasche. |
Called the “Ideal Index Number” because it is free from bias and satisfies BOTH the Time Reversal Test (P₀₁ × P₁₀ = 1) AND the Factor Reversal Test (P₀₁ × Q₀₁ = V₀₁)! (Fails Circular Test). |
| 5. Marshall-Edgeworth Index | Uses Arithmetic Mean of Base & Current Quantities (q₀ + q₁)/2 as weights. |
Satisfies the Time Reversal Test (but fails Factor Reversal Test). |
4. Sampling Techniques: Probability vs Non-Probability Sampling & Standard Error
This table is tested across UPSC EPFO, DRDO SAO (Marketing Research), RBI Grade B, and NABARD:
| Probability (Random) Sampling (Every unit has a known, non-zero chance of selection) |
Non-Probability (Non-Random / Purposive) Sampling (Selection depends on researcher’s judgment/convenience) |
|---|---|
| 1. Simple Random Sampling (SRS): Lottery method or Tippett’s/Fisher’s Random Number Tables (With or Without Replacement). | 1. Convenience (Chunk / Haphazard) Sampling: Selecting whoever is easiest to reach (e.g., mall intercept). |
| 2. Stratified Random Sampling: Population is heterogeneous; divided into internally homogeneous, externally heterogeneous strata, and random samples are drawn from each stratum. | 2. Judgment (Purposive / Deliberate) Sampling: Researcher handpicks “typical” experts/units based on domain judgment. |
| 3. Cluster (Area) Sampling: Population is divided into naturally occurring geographic clusters (each cluster is internally heterogeneous like a mini-population), and entire selected clusters are surveyed. | 3. Quota Sampling: Non-random equivalent of Stratified Sampling—interviewers are given fixed demographic quotas to fill at their own discretion (very common in market research). |
4. Systematic (Quasi-Random) Sampling: First unit selected at random, and thereafter every k-th unit (k = N / n) is selected at fixed intervals. |
4. Snowball (Chain-Referral) Sampling: Initial respondents refer additional respondents—used for hidden, rare, or hard-to-reach populations. |
Standard Error of Sample Mean:
Exam Rule: Standard Error is inversely proportional to the square root of sample size (√n). To reduce the Standard Error to half (1/2), the sample size
SE(X̄) = σ / √nExam Rule: Standard Error is inversely proportional to the square root of sample size (√n). To reduce the Standard Error to half (1/2), the sample size
n must be increased 4 times (2² = 4)!
🎯 High-Yield Exam One-Liners (Last-Minute Revision Box)
- Karl Pearson’s Correlation Coefficient (r) always lies between −1 and +1; it is independent of both change of origin and change of scale.
- Correlation Coefficient (r) is the Geometric Mean of the two Regression Coefficients: r = ±√(bᵧₓ × bₓᵧ).
- Both regression coefficients (bᵧₓ and bₓᵧ) and the correlation coefficient (r) must ALWAYS carry the same algebraic sign.
- Regression coefficients are independent of Change of Origin (adding/subtracting constants) but ARE affected by Change of Scale.
- The two lines of regression (Y on X and X on Y) always intersect at the point of Arithmetic Means (X̄, Ȳ).
- When r = ±1, the two regression lines coincide; when r = 0, the two regression lines are perpendicular (at 90°) to each other.
- Coefficient of Determination (r²) measures the proportion of total variation in the dependent variable explained by the independent variable.
- Correlation is considered statistically significant if r > 6 × Probable Error (PE), where PE = 0.6745 × (1 − r²) / √N.
- Spearman’s Rank Correlation formula is R = 1 − [6∑D² / n(n² − 1)].
- Geometric Mean is the best average for constructing Index Numbers.
- Laspeyres Index uses Base Year Quantities (q₀) as weights and has an Upward Bias.
- Paasche Index uses Current Year Quantities (q₁) as weights and has a Downward Bias.
- Fisher’s Ideal Index is the Geometric Mean of Laspeyres and Paasche: F = √(L × P).
- Irving Fisher’s Ideal Index satisfies BOTH the Time Reversal Test (P₀₁ × P₁₀ = 1) and the Factor Reversal Test (P₀₁ × Q₀₁ = V₀₁).
- The Circular Test (P₀₁ × P₁₂ × P₂₀ = 1) is NOT satisfied even by Fisher’s Ideal Index; it is satisfied by Simple GM of Price Relatives and Kelly’s Fixed-Weight Index.
- In Stratified Sampling, each stratum is internally homogeneous and externally heterogeneous.
- In Cluster Sampling, each cluster is internally heterogeneous and externally homogeneous.
- Quota Sampling is a Non-Probability sampling method (frequently asked as ‘Which of the following is NOT a random sampling method?’).
- Snowball Sampling is a non-probability chain-referral method used for rare or hidden populations.
- Standard Error of Sample Mean is σ / √n; to halve the Standard Error, sample size (n) must be quadrupled (increased 4 times).
📝 UPSC & Regulatory Level Practice MCQs (With Detailed Solutions)
Q1. In a bivariate regression analysis, the regression coefficient of Y on X (bᵧₓ) is −0.8 and the regression coefficient of X on Y (bₓᵧ) is −0.2. What is the Karl Pearson’s Correlation Coefficient (r) between X and Y?
Correct Answer: (B) — Using r = ±√(bᵧₓ × bₓᵧ) = ±√((−0.8) × (−0.2)) = ±√0.16 = ±0.4. Since both regression coefficients are negative, r must also carry a negative sign, so r = −0.4.
Q2. Two regression lines are given by the equations 3X + 2Y = 26 and 6X + Y = 31. Find the Arithmetic Mean of X (X̄) and the Arithmetic Mean of Y (Ȳ):
Correct Answer: (A) — The two regression lines always intersect at (X̄, Ȳ). Solving simultaneously: multiply the second equation by 2 to get 12X + 2Y = 62. Subtract (3X + 2Y = 26): 9X = 36 => X̄ = 4. Substitute X = 4 into 6(4) + Y = 31 => Ȳ = 7.
Q3. If Laspeyres’ Price Index is 144 and Paasche’s Price Index is 100, what are the values of Dorbish-Bowley’s Price Index and Fisher’s Ideal Price Index respectively?
Correct Answer: (B) — Dorbish-Bowley is the Arithmetic Mean: (144 + 100) / 2 = 122. Fisher’s Ideal Index is the Geometric Mean: √(144 × 100) = 12 × 10 = 120.
Q4. Which of the following sampling methods is a ‘Non-Probability (Non-Random) Sampling’ technique?
Correct Answer: (C) — Stratified, Systematic, and Cluster Sampling are Probability (Random) sampling methods, whereas Quota, Convenience, Judgment/Purposive, and Snowball Sampling are Non-Probability methods.
Q5. If the sample size (n) in a random survey is increased from 100 to 900 (9 times), what happens to the Standard Error of the Sample Mean (σ / √n)?
Correct Answer: (B) — Since SE = σ / √n, multiplying n by 9 divides the Standard Error by √9 = 3 (reducing SE to one-third of its original value).
❓ Frequently Asked Questions (Exam FAQs)
Q: Why is Fisher’s Index called the ‘Ideal Index Number’?
Fisher’s Index is the Geometric Mean of Laspeyres and Paasche indices (balancing upward and downward biases) and is the only standard formula that satisfies both the Time Reversal Test (P₀₁ × P₁₀ = 1) and the Factor Reversal Test (P₀₁ × Q₀₁ = V₀₁).
Q: What is the difference between Stratified Sampling and Cluster Sampling?
In Stratified Sampling, the population is divided into internally homogeneous strata and random samples are drawn from EVERY stratum. In Cluster Sampling, the population is divided into naturally occurring, internally heterogeneous clusters, and a random sample of whole clusters is selected.
Q: Can both regression coefficients be greater than 1 at the same time?
No. Because the product of the two regression coefficients equals r² (bᵧₓ × bₓᵧ = r²) and r² can never exceed 1, at least one regression coefficient must be less than or equal to 1.
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