Part 2 of Elementary Statistics for UPSC EPFO APFC and RBI Grade B covers Sampling Techniques, Correlation & Regression, and Index Numbers. Every aspirant must memorize: (1) Probability Sampling (Simple Random, Stratified, Systematic, Cluster, Multi-Stage) vs Non-Probability Sampling (Quota, Convenience, Purposive/Judgmental, Snowball); (2) Standard Error of Sample Mean [SE = sigma / sqrt(n) — increasing sample size 4-fold halves the Standard Error!]; (3) Karl Pearson’s Correlation Coefficient (-1 <= r <= +1; independent of BOTH origin and scale!) & Regression Coefficients b_yx and b_xy (independent of origin, NOT independent of scale; r = +/- sqrt(b_yx * b_xy); the two regression lines ALWAYS intersect at the point of Means [X-bar, Y-bar]!); and (4) Index Numbers: Laspeyres (Base-year quantity weights q_0, upward bias), Paasche (Current-year quantity weights q_1, downward bias), and Fisher's Ideal Index [F = sqrt(L x P) — satisfies BOTH Time Reversal and Factor Reversal Tests!].
1. Probability (Random) vs Non-Probability (Non-Random) Sampling & Standard Error Formula
In UPSC EPFO APFC and RBI Grade B, identifying whether a sampling technique is Probability vs Non-Probability and knowing which sampling method to use for heterogeneous vs homogeneous populations is tested frequently:
| Category | Sampling Method | Core Mechanism & When to Use (Exam Triggers!) |
|---|---|---|
| 1. Probability (Random) Sampling Every unit has a known, non-zero probability of selection; allows estimation of Sampling Error! |
• 1. Simple Random Sampling (SRS — With or Without Replacement): Lottery method or Random Number Tables (Tippett’s/Fisher & Yates tables). Best when the population is small and homogeneous. • 2. Stratified Random Sampling: Population is divided into mutually exclusive, non-overlapping sub-groups called ‘Strata’, and a random sample is drawn from EVERY stratum! Golden Rule for Strata: Each stratum must be internally HOMOGENEOUS (within-stratum) and externally HETEROGENEOUS (between-strata)! • 3. Systematic (Quasi-Random) Sampling: First unit selected at random between $1$ and $k$, and every $k^{\text{th}}$ unit ($k = N/n$ = Sampling Interval) is selected thereafter. • 4. Cluster (Area) Sampling: Population is divided into natural geographical clusters; a random sample of clusters is chosen and ALL units within the selected clusters are surveyed. Golden Rule for Clusters (Opposite of Strata!): Each cluster should be internally HETEROGENEOUS (a mini-representation of the whole population) and externally HOMOGENEOUS between clusters! |
Standard Error of Sample Mean ($\text{SE}_{\bar{x}}$): $$\mathbf{\text{SE}_{\bar{x}} = \frac{\sigma}{\sqrt{n}}}$$ Crucial Numerical Rule: Standard Error is inversely proportional to the SQUARE ROOT of sample size ($\sqrt{n}$)! To reduce the Standard Error to one-half ($1/2$), sample size $n$ must be increased $4\text{ times}$; to reduce SE to one-third ($1/3$), $n$ must be increased $9\text{ times}$! |
| 2. Non-Probability (Non-Random / Judgment) Sampling Selection relies on researcher discretion; Sampling Error CANNOT be mathematically calculated! |
• 1. Quota Sampling: Non-probability counterpart of Stratified Sampling — interviewers are given fixed demographic quotas to fill at their convenience. • 2. Purposive (Judgmental / Deliberate) Sampling: Researcher handpicks specific typical units based on expert judgment. • 3. Convenience (Chunk / Haphazard) Sampling: Selecting whoever is easiest to reach. • 4. Snowball (Network / Chain-Referral) Sampling: Initial respondents refer/recruit additional respondents from their acquaintance network — best for hidden, rare, or hard-to-reach populations (e.g., rare disease patients, undocumented migrants, drug users)! |
Non-Sampling Errors vs Sampling Errors: When sample size $n$ increases (approaching a full Census), Sampling Error decreases, whereas Non-Sampling Error (response bias, data entry errors, non-response) INCREASES! Thus, a Complete Census has ZERO Sampling Error, but HIGHER Non-Sampling Error! |
2. Correlation ($r$) & Linear Regression ($b_{yx}, b_{xy}$): 7 Algebraic Properties & Formulas
$$r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} = \frac{\sum (x_i – \bar{X})(y_i – \bar{Y})}{\sqrt{\sum (x_i – \bar{X})^2 \cdot \sum (y_i – \bar{Y})^2}} \quad \Big| \quad -1 \le r \le +1$$
$$\text{Spearman’s Rank Correlation:} \quad \rho = 1 – \frac{6 \sum D_i^2}{n(n^2 – 1)} \quad (\text{where } D_i = R_{x_i} – R_{y_i})$$
• Coefficient of Determination ($R^2 = r^2$): Explains the percentage of total variation in the dependent variable ($Y$) explained by the independent variable ($X$). (If $r = -0.8$, then $R^2 = (-0.8)^2 = 0.64 = \mathbf{64\%}$!).
2. The Two Regression Lines ($Y$ on $X$ and $X$ on $Y$) & Regression Coefficients ($b_{yx}, b_{xy}$):
• Regression Line of $Y$ on $X$: $Y – \bar{Y} = b_{yx}(X – \bar{X})$, where slope $\mathbf{b_{yx} = r \cdot \frac{\sigma_Y}{\sigma_X} = \frac{\text{Cov}(X,Y)}{\sigma_X^2}}$.
• Regression Line of $X$ on $Y$: $X – \bar{X} = b_{xy}(Y – \bar{Y})$, where slope $\mathbf{b_{xy} = r \cdot \frac{\sigma_X}{\sigma_Y} = \frac{\text{Cov}(X,Y)}{\sigma_Y^2}}$.
The 6 Unbreakable Exam Rules of Correlation & Regression Coefficients:
1. Geometric Mean Property: Correlation coefficient $r$ is the Geometric Mean of the two regression coefficients:
$$\mathbf{r = \pm \sqrt{b_{yx} \times b_{xy}} \quad (\text{or } r^2 = b_{yx} \times b_{xy} \le 1)}$$
2. Same Algebraic Sign Rule: $b_{yx}$, $b_{xy}$, and $r$ MUST ALL HAVE THE EXACT SAME ALGEBRAIC SIGN (either all three positive or all three negative)!
3. Magnitude Rule: If one regression coefficient is greater than $1$ (e.g., $b_{yx} = 1.6$), the other regression coefficient MUST be strictly less than $1$ (e.g., $b_{xy} \le 0.625$) so that their product $b_{yx} \cdot b_{xy} \le 1$!
4. Point of Intersection Rule: The two regression lines ($Y$ on $X$ and $X$ on $Y$) ALWAYS intersect at the Point of Means $(\bar{X}, \bar{Y})$!
5. Angle Between the Two Regression Lines:
— When $r = 0$ (uncorrelated), the two regression lines are PERPENDICULAR ($90^\circ$) to each other!
— When $r = \pm 1$ (perfect correlation), the two regression lines COINCIDE into a single line ($0^\circ$ angle)!
6. Origin vs Scale Invariance:
— Correlation Coefficient ($r$): Independent of BOTH Change of Origin AND Change of Scale (only sign flips if scales have opposite signs).
— Regression Coefficients ($b_{yx}, b_{xy}$): Independent of Change of Origin, Set NOT independent of Change of Scale!
3. Index Numbers: Laspeyres, Paasche, Bowley, Marshall-Edgeworth & Fisher’s Ideal Index
Memorize the formulas and Irving Fisher’s Adequacy Tests for Index Numbers — a regular question in UPSC EPFO APFC, RBI Grade B, and SEBI Grade A:
| Index Number Formula | Mathematical Formula & Weights Used | Bias & Adequacy Tests Satisfied |
|---|---|---|
| 1. Laspeyres Price Index ($P_{01}^L$) (Etienne Laspeyres, 1871) |
$$P_{01}^L = \frac{\sum (p_1 \cdot q_0)}{\sum (p_0 \cdot q_0)} \times 100$$ Uses BASE-YEAR Quantities ($q_0$) as weights! |
• Has an UPWARD BIAS during inflation (overstates cost of living because it ignores consumers substituting cheaper goods!). • Used in India’s WPI and CPI series. • Fails both Time Reversal and Factor Reversal tests. |
| 2. Paasche Price Index ($P_{01}^P$) (Hermann Paasche, 1874) |
$$P_{01}^P = \frac{\sum (p_1 \cdot q_1)}{\sum (p_0 \cdot q_1)} \times 100$$ Uses CURRENT-YEAR Quantities ($q_1$) as weights! |
• Has a DOWNWARD BIAS. • Fails both Time Reversal and Factor Reversal tests. |
| 3. Dorbish–Bowley Price Index ($P_{01}^{DB}$) | $$P_{01}^{DB} = \frac{L + P}{2}$$ Arithmetic Mean of Laspeyres and Paasche indices! |
Fails both Time Reversal and Factor Reversal tests. |
| 4. Marshall–Edgeworth Index ($P_{01}^{ME}$) | $$P_{01}^{ME} = \frac{\sum p_1(q_0 + q_1)}{\sum p_0(q_0 + q_1)} \times 100$$ Uses average of base and current quantities. |
Satisfies Time Reversal Test, but FAILS Factor Reversal Test! |
| 5. Fisher’s Ideal Price Index ($P_{01}^F$) (Irving Fisher, 1922) |
$$\mathbf{P_{01}^F = \sqrt{L \times P} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100}$$ Geometric Mean of Laspeyres and Paasche indices! |
Called the ‘IDEAL INDEX’ because: 1. It is free from upward/downward bias; 2. It satisfies BOTH the Time Reversal Test ($P_{01} \times P_{10} = 1$) AND the Factor Reversal Test ($P_{01} \times Q_{01} = V_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0}$)! (Note: Fisher’s Index does NOT satisfy the Circular Test; Kelly’s Fixed-Weight Index and Simple GM of price relatives satisfy the Circular Test!). |
🎯 High-Yield Exam One-Liners (Last-Minute Revision Box)
- Probability Sampling methods include Simple Random, Stratified, Systematic, Cluster, and Multi-Stage Sampling; Non-Probability methods include Quota, Convenience, Purposive (Judgmental), and Snowball Sampling.
- Stratified Random Sampling is best for a heterogeneous population: strata should be internally HOMOGENEOUS (within-stratum) and externally HETEROGENEOUS (between-strata).
- In Cluster Sampling, clusters should be internally HETEROGENEOUS (mini-replicas of the population) and externally HOMOGENEOUS between clusters.
- Snowball (Chain-Referral) Sampling is a non-probability sampling technique used to survey rare, hidden, or hard-to-reach populations.
- The Standard Error of the Sample Mean is SE = sigma / sqrt(n); to reduce the Standard Error by half (50%), the sample size (n) must be quadrupled (increased 4 times).
- As sample size increases toward a full Census, Sampling Error decreases to zero, whereas Non-Sampling Error increases.
- Karl Pearson’s Correlation Coefficient (r) always lies between -1 and +1, and is completely independent of BOTH change of origin and change of scale.
- Regression coefficients (b_yx and b_xy) are independent of change of origin, but are NOT independent of change of scale.
- The Correlation Coefficient is the Geometric Mean of the two regression coefficients: r = +/- sqrt(b_yx * b_xy), and r, b_yx, and b_xy must all carry the exact same sign.
- If one regression coefficient is greater than 1, the other regression coefficient MUST be strictly less than 1 so that their product (r^2) never exceeds 1.
- The two regression lines (Y on X and X on Y) ALWAYS intersect at the point of Arithmetic Means (X-bar, Y-bar).
- When two variables are uncorrelated (r = 0), their two regression lines are perpendicular (at 90 degrees) to each other; when r = +/-1, the two lines coincide into a single line.
- Laspeyres Price Index uses Base-Year Quantities (q_0) as weights and exhibits an upward bias, whereas Paasche Price Index uses Current-Year Quantities (q_1) as weights and exhibits a downward bias.
- Dorbish–Bowley Index is the Arithmetic Mean of Laspeyres and Paasche indices [(L + P) / 2], whereas Fisher’s Ideal Index is the Geometric Mean of Laspeyres and Paasche indices [sqrt(L x P)].
- Irving Fisher’s Ideal Index is the ONLY standard index formula that satisfies BOTH the Time Reversal Test (P_01 x P_10 = 1) and the Factor Reversal Test (P_01 x Q_01 = V_01).
- Marshall–Edgeworth Index satisfies the Time Reversal Test, but fails the Factor Reversal Test.
- Fisher’s Ideal Index fails the Circular Test (P_01 x P_12 x P_20 = 1), which is satisfied by the Simple Geometric Mean of Price Relatives and Kelly’s Fixed-Weight Aggregate Index.
📝 UPSC & Regulatory Level Practice MCQs (With Detailed Solutions)
❓ Frequently Asked Questions (Exam FAQs)
Q: Can the two regression coefficients in a linear regression be b_yx = 1.5 and b_xy = 0.8?
No, that pair is mathematically impossible! Because r^2 = b_yx * b_xy must always be less than or equal to 1, here 1.5 x 0.8 = 1.20 > 1, which violates the Cauchy-Schwarz inequality (-1 <= r <= +1).
Q: What is the difference betweenStratified Sampling and Quota Sampling?
Both divide the heterogeneous population into homogeneous sub-groups (strata). However, in Stratified Random Sampling, units within each stratum are selected using random (probability) sampling, whereas in Quota Sampling, interviewers select units within each quota non-randomly based on convenience or personal judgment.
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