Correlation and Regression SSC CGL Tier 2 Paper 2

In the SSC CGL Tier 2 for SSC JSO paper, Correlation and Regression is an important topic under Statistics. These concepts help to understand the relationship between two or more variables, how one variable changes when another changes.

1. What is Correlation?

Definition

Correlation measures the degree and direction of relationship between two variables. If two variables change together either in the same or opposite direction they are said to be correlated.

Type of Correlation	Direction	Example
Positive Correlation	Both variables move in the same direction.	Height and Weight – as height increases, weight increases.
Negative Correlation	Variables move in opposite directions.	Price and Demand – as price increases, demand decreases.
Zero Correlation	No relationship between variables.	Shoe size and intelligence.

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2. Methods to Study Correlation

(a) Scatter Diagram (Graphical Method)

• It is a simple visual method to show correlation between two variables.
• Values of one variable are plotted on the X-axis and the other on the Y-axis.

Pattern	Type of Correlation	Diagram Description
Points lie close to an upward-sloping straight line	Positive Correlation	Both variables increase together
Points lie close to a downward-sloping straight line	Negative Correlation	One increases, the other decreases
Points scattered randomly	No Correlation	No visible relationship

(b) Karl Pearson’s Coefficient of Correlation (r)

It gives a quantitative measure of correlation between two variables. r=Σ(x−xˉ)(y−yˉ)Σ(x−xˉ)2Σ(y−yˉ)2r = \frac{\Sigma (x – \bar{x})(y – \bar{y})}{\sqrt{\Sigma (x – \bar{x})^2 \Sigma (y – \bar{y})^2}}r=Σ(x−xˉ)2Σ(y−yˉ​)2​Σ(x−xˉ)(y−yˉ​)​

Where:

• x,yx, yx,y = Variables
• xˉ,yˉ\bar{x}, \bar{y}xˉ,yˉ​ = Mean of x and y
• rrr ranges between −1 and +1

Value of r	Interpretation
+1	Perfect positive correlation
−1	Perfect negative correlation
0	No correlation

The closer |r| is to 1, the stronger the relationship.

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(c) Spearman’s Rank Correlation (rₛ)

Used when data are in ranks or qualitative form (like preference, performance, etc.). rs=1−6Σd2n(n2−1)r_s = 1 – \frac{6 \Sigma d^2}{n(n^2 – 1)}rs​=1−n(n2−1)6Σd2​

Where:

• ddd = Difference between ranks of each pair
• nnn = Number of observations

rₛ Value	Meaning
+1	Perfect positive rank correlation
−1	Perfect negative rank correlation
0	No rank correlation

Example:
If 5 students’ marks in Maths and English are ranked and Σd2=10\Sigma d^2 = 10Σd2=10, rs=1−6(10)5(52−1)=1−60120=0.5r_s = 1 – \frac{6(10)}{5(5^2 – 1)} = 1 – \frac{60}{120} = 0.5rs​=1−5(52−1)6(10)​=1−12060​=0.5

→ Moderate positive correlation.

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3. What is Regression?

Definition

Regression shows the functional relationship between two variables, it helps predict the value of one variable based on another.

Concept	Explanation
Dependent Variable (Y)	The variable to be predicted
Independent Variable (X)	The variable used for prediction

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4. Regression Lines

There are two regression lines:

1. Regression Line of Y on X: Y−Yˉ=byx(X−Xˉ)Y – \bar{Y} = b_{yx}(X – \bar{X})Y−Yˉ=byx​(X−Xˉ)
2. Regression Line of X on Y: X−Xˉ=bxy(Y−Yˉ)X – \bar{X} = b_{xy}(Y – \bar{Y})X−Xˉ=bxy​(Y−Yˉ)

Where,

• byxb_{yx}byx​ and bxyb_{xy}bxy​ are regression coefficients

Formulas:

byx=r×σyσx,bxy=r×σxσyb_{yx} = r \times \frac{\sigma_y}{\sigma_x}, \quad b_{xy} = r \times \frac{\sigma_x}{\sigma_y}byx​=r×σx​σy​​,bxy​=r×σy​σx​​

Property	Description
Both lines intersect at (𝑋̄, 𝑌̄).
byx×bxy=r2b_{yx} \times b_{xy} = r^2byx​×bxy​=r2
If r = 0 → lines are perpendicular.
If r = ±1 → both lines coincide.

Use: Regression helps in forecasting for example, predicting sales based on advertisement spend.

5. Multiple Correlation

Definition

When we study the relationship between one dependent variable and two or more independent variables, it is called multiple correlation.

Example:
Predicting a student’s performance (Y) based on study hours (X₁) and attendance (X₂).

Multiple Correlation Coefficient (R):

R=ryx12+ryx22−2ryx1ryx2rx1x21−rx1x22R = \sqrt{r_{yx1}^2 + r_{yx2}^2 – 2r_{yx1}r_{yx2}r_{x1x2} \over 1 – r_{x1x2}^2}R=1−rx1x22​ryx12​+ryx22​−2ryx1​ryx2​rx1x2​​​

Where:

• ryx1,ryx2r_{yx1}, r_{yx2}ryx1​,ryx2​ = Correlation of Y with X₁ and X₂
• rx1x2r_{x1x2}rx1x2​ = Correlation between X₁ and X₂

Range: 0 ≤ R ≤ 1

• R close to 1 → strong relationship
• R close to 0 → weak relationship

6. Key Differences Between Correlation and Regression

Basis	Correlation	Regression
Meaning	Measures degree of relationship between variables	Expresses the relationship mathematically
Purpose	To find strength & direction	To predict one variable from another
Number of Lines	One (no distinction)	Two (Y on X, X on Y)
Interchangeability	No dependent or independent variable	One variable is dependent on another
Value Range	−1 to +1	Any real value

Key Takeaways

Below are the key takeaways:

• Correlation → Measures relationship strength.
• Regression → Provides predictive equations.
• Karl Pearson’s coefficient (r) and Spearman’s rank (rₛ) are most common in exams.
• Regression lines always pass through means (𝑋̄, 𝑌̄).
• Multiple correlation deals with more than two variables.
• Formula shortcuts and properties are frequently asked in Paper II (Statistics).

FAQs on Correlation and Regression

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