Correlation and Regression Notes for SSC Statistics

Correlation and regression study the relation between two variables, such as the height and weight of students, or the price and demand of a product. This chapter is a regular part of the SSC CGL Statistics paper and gives direct formula-based questions. These correlation and regression notes for SSC explain the coefficient of correlation, the rank correlation, the regression lines and their properties in simple language with solved examples.

What Is Correlation?

• Correlation shows how strongly two variables move together.
• Positive correlation: both variables increase or decrease together, such as income and spending.
• Negative correlation: when one increases, the other decreases, such as price and demand.
• Zero correlation: there is no linear relation between the variables.

Karl Pearson Coefficient of Correlation

Example 1. If y = 2x + 1 for all the data points, what is the correlation between x and y?
The points lie exactly on a straight line with a positive slope, so r = +1.

Spearman Rank Correlation

When data is given as ranks, use Spearman's formula: r = 1 − 6Σd² / [n(n² − 1)], where d is the difference between the two ranks of the same item.

Example 2. For 5 students, Σd² = 4. Find the rank correlation.
r = 1 − (6 × 4) / (5 × 24) = 1 − 24 / 120 = 0.8.

Regression

Regression is used to estimate the value of one variable from the other. There are two regression lines.

Properties of Regression

• Both regression lines pass through the point (mean of x, mean of y).
• r² = b_yx × b_xy, so r is the geometric mean of the two regression coefficients.
• Both regression coefficients and r have the same sign.
• If one regression coefficient is greater than 1, the other must be less than 1.
• The arithmetic mean of the two regression coefficients is greater than or equal to r.
• If r = ±1, the two regression lines coincide; if r = 0, they are perpendicular to each other.

Example 3. If r = 0.8, σx = 5 and σy = 10, find b_yx and b_xy.
b_yx = 0.8 × 10 / 5 = 1.6, and b_xy = 0.8 × 5 / 10 = 0.4. Check: 1.6 × 0.4 = 0.64 = r².

Difference Between Correlation and Regression

Quick Tips

• Remember that r always lies between −1 and +1; an answer outside this range is wrong.
• Use r² = b_yx × b_xy to check your answer quickly.
• The sign of r is the sign of the regression coefficients.
• Learn the Spearman formula with the constant 6 and the denominator n(n² − 1).

Quick Revision Points

• r lies from −1 to +1.
• Spearman: r = 1 − 6Σd² / [n(n² − 1)].
• b_yx = r σy / σx and b_xy = r σx / σy.
• Regression lines intersect at the means of x and y.
• r² = b_yx × b_xy.

Practice Questions With Answers

1. What is the range of the correlation coefficient?
   Answer: −1 to +1
2. If r = 0.6, σx = 4 and σy = 8, find b_yx.
   Answer: 1.2
3. If b_yx = 0.9 and b_xy = 0.4, find r.
   Answer: 0.6 (the square root of 0.36)
4. At which point do the two regression lines meet?
   Answer: At (mean of x, mean of y)
5. For 10 items, Σd² = 33. Find the rank correlation.
   Answer: 0.8 (1 − 198/990)
6. What is the correlation between x and y if y = −3x + 5?
   Answer: −1
7. What does r = 0 mean?
   Answer: No linear correlation
8. If one regression coefficient is 1.5, can the other be 1.2?
   Answer: No, because one of them must be less than 1
9. Does the correlation change if each value of x is multiplied by 2?
   Answer: No
10. What is the angle between the regression lines when r = 0?
    Answer: 90 degrees

How to Practise This Chapter

Learn the formulas, then solve one example of each type daily: Pearson, Spearman and regression coefficients. Solve timed sets from our practice papers and the previous year papers page. Read the previous chapter, Mean, Median and Mode, and see all topics on the Statistics notes page. You can also read the Quantitative Aptitude notes for maths practice.

Frequently Asked Questions

What is correlation?

Correlation is a measure of the strength and direction of the linear relation between two variables.

What is the range of the correlation coefficient?

The correlation coefficient lies between minus 1 and plus 1.

What is positive and negative correlation?

In positive correlation, both variables move in the same direction. In negative correlation, they move in opposite directions.

What is the formula of Spearman rank correlation?

It is r equals 1 minus 6 times the sum of d squared divided by n times (n squared minus 1).

What is regression?

Regression is a method to estimate the value of one variable from the value of another variable using a line of best fit.

How many regression lines are there?

There are two lines: the regression of y on x and the regression of x on y.

What is the relation between r and the regression coefficients?

The square of r equals the product of the two regression coefficients, and all three have the same sign.

Where do the two regression lines intersect?

They intersect at the point whose coordinates are the means of x and y.

What is the difference between correlation and regression?

Correlation shows the strength of the relation, while regression is used to estimate one variable from the other.

Does correlation show cause and effect?

No. A high correlation only shows that the variables move together, not that one causes the other.

What happens if r is plus or minus 1?

The two regression lines coincide, and all the points lie on a straight line.

How many questions come from this chapter in SSC exams?

Usually two to four questions come from correlation and regression, but the number changes from year to year.