What are the two lines of regression in bivariate analysis?

The two lines are: (1) Regression of Y on X: Y - Ȳ = byx(X - X̄), used to estimate Y given X; and (2) Regression of X on Y: X - X̄ = bxy(Y - Ȳ), used to estimate X given Y.

What is the relation between correlation coefficient r and regression coefficients?

The correlation coefficient r is the geometric mean of the two regression coefficients: r = ±√(byx · bxy). The sign of r must match the signs of byx and bxy.

At what point do the two regression lines intersect?

The two regression lines always intersect at the mean point (X̄, Ȳ). Therefore, solving the two regression equations simultaneously gives the means of X and Y.

Home SSC CGL Syllabus Statistics Linear Regression & Correlation

Linear Regression & Correlation: Theory & 15 Solved Questions

Specialized Study Guide for SSC CGL Tier-II Paper-II (Junior Statistical Officer - JSO & Statistical Investigator Grade-II). Covers regression lines, coefficients, angle between lines, and 15 exam questions.

Bivariate Linear Regression Overviewद्विचरीय रैखिक प्रतिगमन अवलोकन

While correlation measures the strength and direction of a mutual relationship between two variables, regression expresses the functional relationship between a dependent (criterion) variable and an independent (predictor) variable to enable prediction.

जबकि सहसंबंध दो चरों के बीच पारस्परिक संबंध की मजबूती और दिशा को मापता है, प्रतिगमन भविष्यवाणी को सक्षम करने के लिए एक आश्रित चर और एक स्वतंत्र चर के बीच कार्यात्मक संबंध को व्यक्त करता है।

The Two Lines of Regressionप्रतिगमन की दो रेखाएँ

1. Line of Regression of $Y$ on $X$

Used to estimate the value of $Y$ for a specified value of $X$:

$$Y - \bar{Y} = b_{yx}(X - \bar{X})$$

where $b_{yx}$ is the regression coefficient of $Y$ on $X$:

$$b_{yx} = r \frac{\sigma_y}{\sigma_x} = \frac{\text{Cov}(X, Y)}{\sigma_x^2} = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sum (X - \bar{X})^2} = \frac{N\sum XY - (\sum X)(\sum Y)}{N\sum X^2 - (\sum X)^2}$$

2. Line of Regression of $X$ on $Y$

Used to estimate the value of $X$ for a specified value of $Y$:

$$X - \bar{X} = b_{xy}(Y - \bar{Y})$$

where $b_{xy}$ is the regression coefficient of $X$ on $Y$:

$$b_{xy} = r \frac{\sigma_x}{\sigma_y} = \frac{\text{Cov}(X, Y)}{\sigma_y^2} = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sum (Y - \bar{Y})^2} = \frac{N\sum XY - (\sum X)(\sum Y)}{N\sum Y^2 - (\sum Y)^2}$$

Fundamental Properties of Regression Coefficients (SSC JSO Must-Know)प्रतिगमन गुणांकों के मौलिक गुण

1. Same Signs: $b_{yx}$, $b_{xy}$, and $r$ must all share the identical algebraic sign. If both coefficients are negative, $r$ is strictly negative.
2. Geometric Mean Relationship: The correlation coefficient is the geometric mean of the regression coefficients: $$r = \pm\sqrt{b_{yx} \cdot b_{xy}}$$
3. Arithmetic Mean Dominance: The arithmetic mean of the regression coefficients is numerically greater than or equal to the absolute correlation coefficient: $$\frac{b_{yx} + b_{xy}}{2} \ge |r|$$
4. Reciprocal Bound: Since $r^2 \le 1$, the product $b_{yx} \cdot b_{xy} \le 1$. If one regression coefficient exceeds 1 in magnitude, the other must be less than 1. Both cannot simultaneously exceed 1.
5. Point of Intersection: The two regression lines always intersect at the mean coordinates $(\bar{X}, \bar{Y})$.
6. Origin & Scale Invariance: Regression coefficients are independent of change of origin, but dependent on change of scale. If $U = \frac{X - a}{h}$ and $V = \frac{Y - b}{k}$, then: $$b_{yx} = \frac{k}{h} b_{vu} \quad \text{and} \quad b_{xy} = \frac{h}{k} b_{uv}$$

Angle Between Two Regression Linesदो प्रतिगमन रेखाओं के बीच का कोण

If $\theta$ is the acute angle between the two lines of regression, then:

$$\tan \theta = \left|\frac{1 - r^2}{r}\right| \cdot \left(\frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}\right)$$

Case I ($r = \pm 1$): $\tan \theta = 0 \implies \theta = 0^\circ$. The two lines coincide. Perfect linear correlation.
Case II ($r = 0$): $\tan \theta \to \infty \implies \theta = 90^\circ$. The two lines are perpendicular to each other ($Y = \bar{Y}$ and $X = \bar{X}$). Zero correlation.

Exam Shortcuts for SSC Statistical Investigatorएसएससी सांख्यिकीय अन्वेषक के लिए शॉर्टकट

Shortcut 1: Instantly Find $\bar{X}$ and $\bar{Y}$

Whenever an SSC question provides two regression equations like $2X + 3Y = 18$ and $4X + Y = 16$ and asks for the mean of $X$ and $Y$, do not calculate slopes. Simply treat them as simultaneous linear equations and solve for $(X, Y)$ using substitution or elimination. The solution is identically $(\bar{X}, \bar{Y})$.

Shortcut 2: How to Tell Which Line is $Y$ on $X$ and Which is $X$ on $Y$

Assume the first equation is $Y$ on $X$ (write in form $Y = m_1 X + c_1 \implies b_{yx} = m_1$) and the second is $X$ on $Y$ (write in form $X = m_2 Y + c_2 \implies b_{xy} = m_2$).
Calculate the product $m_1 \cdot m_2$.

If $m_1 \cdot m_2 \le 1$, your assumption was correct!
If $m_1 \cdot m_2 > 1$, your assumption was reversed. Swap the two equations.

Shortcut 3: Negative Sign Trap in $r = \pm\sqrt{b_{yx} \cdot b_{xy}}$

If $b_{yx} = -0.8$ and $b_{xy} = -0.2$, then: $$r = -\sqrt{(-0.8)(-0.2)} = -\sqrt{0.16} = -0.4$$ SSC frequently sets positive $+0.4$ as Option (A) to trap unwary students. Always look at the signs of $b_{yx}$ and $b_{xy}$ first!

Key Concept Revision & Frequently Asked Questionsमहत्वपूर्ण नोट्स एवं अक्सर पूछे जाने वाले प्रश्न

Why are there two regression lines instead of just one?

Each regression line minimizes the sum of squared errors in a different direction. The line of $Y$ on $X$ minimizes the vertical sum of squared residuals ($\sum (y_i - \hat{y}_i)^2$), treating $X$ as error-free. The line of $X$ on $Y$ minimizes horizontal residuals ($\sum (x_i - \hat{x}_i)^2$), treating $Y$ as error-free. They only coincide when $|r| = 1$, where there are zero residuals in both directions.

What is the Standard Error of Estimate?

The Standard Error of Estimate measures the dispersion of observed values around the regression line: $$S_{yx} = \sigma_y \sqrt{1 - r^2} \quad \text{and} \quad S_{xy} = \sigma_x \sqrt{1 - r^2}$$ When $r = \pm 1$, $S_{yx} = 0$, meaning every point lies directly on the regression line. When $r = 0$, $S_{yx} = \sigma_y$, meaning the regression line offers no predictive improvement over the simple mean $\bar{Y}$.

What is the Coefficient of Determination?

The square of the correlation coefficient, $r^2$, is known as the Coefficient of Determination. It represents the proportion of total variation in the dependent variable that is explained by the independent variable. The remainder $(1 - r^2)$ is the Coefficient of Non-Determination (unexplained variance).

SSC Statistical Investigator Practice Questions (Q1 – Q8)अभ्यास प्रश्न

Q1. If the two lines of regression are $x + 2y - 5 = 0$ and $2x + 3y - 8 = 0$, what are the means $\bar{X}$ and $\bar{Y}$?

Q1. यदि दो प्रतिगमन रेखाएं $x + 2y - 5 = 0$ और $2x + 3y - 8 = 0$ हैं, तो माध्य $\bar{X}$ और $\bar{Y}$ क्या हैं?

(A) $\bar{X} = 1, \bar{Y} = 2$(A) $\bar{X} = 1, \bar{Y} = 2$

(B) $\bar{X} = 2, \bar{Y} = 1$(B) $\bar{X} = 2, \bar{Y} = 1$

(D) $\bar{X} = 1, \bar{Y} = 3$(D) $\bar{X} = 1, \bar{Y} = 3$

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution: Since regression lines intersect at $(\bar{X}, \bar{Y})$, solve:
$x + 2y = 5 \implies x = 5 - 2y$
Substitute into second equation: $2(5 - 2y) + 3y = 8 \implies 10 - 4y + 3y = 8 \implies -y = -2 \implies y = 2$.
Then $x = 5 - 2(2) = 1$. Thus $\bar{X} = 1, \bar{Y} = 2$.

Topper's Shortcut: Plug options into both equations. Option (A): $1 + 2(2) = 5$ (Valid) and $2(1) + 3(2) = 8$ (Valid). Instant 5-second answer!

Q2. Given the regression coefficients $b_{yx} = -0.64$ and $b_{xy} = -0.36$. What is the correlation coefficient $r$?

Q2. यदि प्रतिगमन गुणांक $b_{yx} = -0.64$ और $b_{xy} = -0.36$ हैं, तो सहसंबंध गुणांक $r$ क्या होगा?

(A) 0.48(A) 0.48

(B) -0.48(B) -0.48

(D) 0.50(D) 0.50

View Solution हल देखें

Correct Option: (B)सही विकल्प: (B)

Step-by-Step Solution: $r = \pm\sqrt{b_{yx} \cdot b_{xy}} = \pm\sqrt{(-0.64)(-0.36)} = \pm\sqrt{0.2304} = \pm 0.48$. Since both $b_{yx}$ and $b_{xy}$ are negative, $r$ must also be negative: $r = -0.48$.

Topper's Shortcut: $\sqrt{0.64 \times 0.36} = 0.8 \times 0.6 = 0.48$. With negative signs, answer is $-0.48$.

Q3. If $b_{yx} = 1.6$, which of the following values can $b_{xy}$ NOT take?

Q3. यदि $b_{yx} = 1.6$ है, तो $b_{xy}$ निम्नलिखित में से कौन सा मान नहीं ले सकता है?

(A) 0.5(A) 0.5

(B) 0.6(B) 0.6

(D) 0.4(D) 0.4

View Solution हल देखें

Correct Option: (C)सही विकल्प: (C)

Step-by-Step Solution: Since $r^2 \le 1$, we must have $b_{yx} \cdot b_{xy} \le 1 \implies 1.6 \cdot b_{xy} \le 1 \implies b_{xy} \le \frac{1}{1.6} = 0.625$. Therefore, $b_{xy}$ cannot be $0.7$, because $1.6 \times 0.7 = 1.12 > 1$.

Q4. If $\sigma_x = 5$, $\sigma_y = 10$, and correlation $r = 0.8$, find the regression coefficient $b_{xy}$.

Q4. यदि $\sigma_x = 5$, $\sigma_y = 10$, और $r = 0.8$ है, तो प्रतिगमन गुणांक $b_{xy}$ ज्ञात कीजिए।

(A) 1.6(A) 1.6

(B) 0.4(B) 0.4

(D) 0.2(D) 0.2

View Solution हल देखें

Correct Option: (B)सही विकल्प: (B)

Step-by-Step Solution: $b_{xy} = r \frac{\sigma_x}{\sigma_y} = 0.8 \times \frac{5}{10} = 0.8 \times 0.5 = 0.4$. (Note: $b_{yx} = 0.8 \times \frac{10}{5} = 1.6$).

Q5. If the two lines of regression are perpendicular to each other, the correlation coefficient $r$ is:

Q5. यदि प्रतिगमन की दो रेखाएं एक दूसरे के लंबवत हैं, तो सहसंबंध गुणांक $r$ है:

(A) 1(A) 1

(B) -1(B) -1

(D) 0.5(D) 0.5

View Solution हल देखें

Correct Option: (C)सही विकल्प: (C)

Step-by-Step Solution: From $\tan \theta = \left|\frac{1 - r^2}{r}\right| \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$, when lines are perpendicular, $\theta = 90^\circ \implies \tan 90^\circ = \infty \implies r = 0$.

Q6. Let $U = 2X - 5$ and $V = -3Y + 4$. If the regression coefficient of $Y$ on $X$ is $b_{yx} = 0.6$, what is the regression coefficient of $V$ on $U$ ($b_{vu}$)?

Q6. मान लें $U = 2X - 5$ और $V = -3Y + 4$। यदि $b_{yx} = 0.6$ है, तो $b_{vu}$ क्या होगा?

(A) -0.9(A) -0.9

(B) 0.9(B) 0.9

(D) 0.4(D) 0.4

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution: Here $U = \frac{X - 2.5}{1/2} \implies h = 1/2$. And $V = \frac{Y - 4/3}{-1/3} \implies k = -1/3$.
We know $b_{vu} = \frac{\text{scale of } V}{\text{scale of } U} \times b_{yx} = \frac{-3}{2} \times 0.6 = -1.5 \times 0.6 = -0.9$.

Q7. In a bivariate distribution, if $b_{yx} = 0.8$ and $b_{xy} = 0.2$, what is the arithmetic mean of the regression coefficients?

Q7. द्विचरीय वितरण में, यदि $b_{yx} = 0.8$ और $b_{xy} = 0.2$ हैं, तो प्रतिगमन गुणांकों का समांतर माध्य क्या है?

(A) 0.50(A) 0.50

(B) 0.40(B) 0.40

(D) 0.60(D) 0.60

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution: Arithmetic Mean $= \frac{b_{yx} + b_{xy}}{2} = \frac{0.8 + 0.2}{2} = \frac{1.0}{2} = 0.50$. Note that $r = \sqrt{0.8 \times 0.2} = 0.40$, verifying $\text{AM} \ge r$ ($0.50 \ge 0.40$).

Q8. The coefficient of determination between two variables $X$ and $Y$ is $0.64$. What proportion of variation in $Y$ is left unexplained by $X$?

Q8. दो चरों $X$ और $Y$ के बीच निर्धारण गुणांक $0.64$ है। $Y$ में भिन्नता का कितना अनुपात $X$ द्वारा अस्पष्टीकृत रह गया है?

(A) 0.36(A) 0.36

(B) 0.64(B) 0.64

(D) 0.20(D) 0.20

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution: Unexplained variation $= 1 - r^2 = 1 - 0.64 = 0.36$ (or $36\%$).

Solved Previous Year Questions (SSC JSO / Statistical Investigator Paper-II)विगत वर्षों के हल प्रश्न

PYQ 1 (SSC JSO 2023): The two regression equations are given by $3X + 2Y = 26$ and $6X + Y = 31$. Find the value of correlation coefficient $r$.

PYQ 1 (SSC JSO 2023): दो प्रतिगमन समीकरण $3X + 2Y = 26$ और $6X + Y = 31$ दिए गए हैं। सहसंबंध गुणांक $r$ का मान ज्ञात कीजिए।

(A) -0.5(A) -0.5

(B) 0.5(B) 0.5

(D) -0.75(D) -0.75

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution:
Let $3X + 2Y = 26 \implies 2Y = -3X + 26 \implies Y = -1.5 X + 13 \implies b_{yx} = -1.5$.
Let $6X + Y = 31 \implies 6X = -Y + 31 \implies X = -\frac{1}{6}Y + \frac{31}{6} \implies b_{xy} = -0.1667$.
Product $b_{yx} \cdot b_{xy} = (-1.5) \times (-1/6) = 1.5 / 6 = 0.25 \le 1$.
Valid assumption! Thus $r = -\sqrt{0.25} = -0.5$.

Topper's Shortcut: Both slopes are negative $\implies r$ must be negative! Eliminate Option (B) instantly.

PYQ 2 (SSC Statistical Investigator): If the two lines of regression coincide, the correlation coefficient between the variables is:

PYQ 2 (SSC सांख्यिकीय अन्वेषक): यदि प्रतिगमन की दो रेखाएं संपाती हैं, तो चरों के बीच सहसंबंध गुणांक है:

(A) 0(A) 0

(B) $\pm 1$(B) $\pm 1$

(D) $\infty$(D) $\infty$

View Solution हल देखें

Correct Option: (B)सही विकल्प: (B)

Step-by-Step Solution: The lines coincide when the angle $\theta = 0^\circ \implies \tan \theta = 0 \implies 1 - r^2 = 0 \implies r = \pm 1$.

PYQ 3 (SSC JSO 2022): For two variables $X$ and $Y$, the regression equation of $Y$ on $X$ is $Y = 0.57X + 1.2$ and of $X$ on $Y$ is $X = 1.12Y - 0.8$. If $\text{Var}(Y) = 16$, what is the standard deviation of $X$ ($\sigma_x$)?

PYQ 3 (SSC JSO 2022): यदि $Y$ on $X$ की रेखा $Y = 0.57X + 1.2$ और $X$ on $Y$ की रेखा $X = 1.12Y - 0.8$ है, और $\text{Var}(Y) = 16$ है, तो $X$ का मानक विचलन ($\sigma_x$) क्या है?

(A) 5.6(A) 5.6

(B) 4.0(B) 4.0

(D) 3.2(D) 3.2

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution:
We have $b_{yx} = 0.57$, $b_{xy} = 1.12$, and $\sigma_y = \sqrt{16} = 4$.
We know $\frac{b_{xy}}{b_{yx}} = \frac{r (\sigma_x / \sigma_y)}{r (\sigma_y / \sigma_x)} = \frac{\sigma_x^2}{\sigma_y^2}$.
Thus $\frac{\sigma_x^2}{16} = \frac{1.12}{0.57} \approx 1.9649 \implies \sigma_x^2 = 16 \times 1.9649 = 31.438 \implies \sigma_x \approx 5.6$.

PYQ 4 (SSC JSO Paper-II): If $\bar{X} = 40$, $\bar{Y} = 50$, $b_{yx} = 0.6$, what is the estimated value of $Y$ when $X = 60$?

PYQ 4 (SSC JSO Paper-II): यदि $\bar{X} = 40$, $\bar{Y} = 50$, $b_{yx} = 0.6$ है, तो जब $X = 60$ हो तो $Y$ का अनुमानित मान क्या है?

(A) 62(A) 62

(B) 58(B) 58

(D) 72(D) 72

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution: Using $Y - \bar{Y} = b_{yx}(X - \bar{X}) \implies Y - 50 = 0.6(60 - 40) = 0.6(20) = 12 \implies Y = 50 + 12 = 62$.

PYQ 5 (SSC Statistical Investigator): Which of the following statements about regression coefficients is FALSE?

PYQ 5 (SSC सांख्यिकीय अन्वेषक): प्रतिगमन गुणांक के बारे में निम्नलिखित में से कौन सा कथन असत्य है?

(A) Both regression coefficients can exceed unity simultaneously.(A) दोनों प्रतिगमन गुणांक एक साथ इकाई (1) से अधिक हो सकते हैं।

(B) Regression coefficients are independent of change of origin.(B) प्रतिगमन गुणांक मूल बिंदु के परिवर्तन से स्वतंत्र होते हैं।

(D) The arithmetic mean of regression coefficients is greater than or equal to $r$.(D) प्रतिगमन गुणांकों का समांतर माध्य $r$ से अधिक या उसके बराबर होता है।

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution: Statement (A) is FALSE because $b_{yx} \cdot b_{xy} = r^2 \le 1$. If one coefficient is greater than 1, the other must be less than 1.

PYQ 6 (SSC CGL Paper-II): If the acute angle between the two regression lines is $\theta$, and $r = 0$, what is the value of $\theta$?

PYQ 6 (SSC CGL Paper-II): यदि दो प्रतिगमन रेखाओं के बीच का न्यून कोण $\theta$ है, और $r = 0$ है, तो $\theta$ का मान क्या है?

(A) $0^\circ$(A) $0^\circ$

(B) $45^\circ$(B) $45^\circ$

(D) $60^\circ$(D) $60^\circ$

View Solution हल देखें

Correct Option: (C)सही विकल्प: (C)

Step-by-Step Solution: When $r = 0$, $\tan \theta \to \infty$, which corresponds to $\theta = 90^\circ$ (the two lines are orthogonal / perpendicular).

PYQ 7 (SSC JSO): In a study with 10 pairs of observations, $\sum X = 60$, $\sum Y = 40$, $\sum X^2 = 400$, and $\sum XY = 220$. What is the regression coefficient $b_{yx}$?

PYQ 7 (SSC JSO): 10 जोड़ी अवलोकनों के एक अध्ययन में, $\sum X = 60$, $\sum Y = 40$, $\sum X^2 = 400$, और $\sum XY = 220$। प्रतिगमन गुणांक $b_{yx}$ क्या है?

(A) -0.5(A) -0.5

(B) -0.2(B) -0.2

(D) 0.2(D) 0.2

View Solution हल देखें

Correct Option: (A)सही विकल्प: (A)

Step-by-Step Solution:
$b_{yx} = \frac{N \sum XY - (\sum X)(\sum Y)}{N \sum X^2 - (\sum X)^2} = \frac{10(220) - (60)(40)}{10(400) - (60)^2} = \frac{2200 - 2400}{4000 - 3600} = \frac{-200}{400} = -0.5$.