CA Foundation · Quantitative Aptitude

Correlation and Regression for CA Foundation Quantitative Aptitude

Correlation measures the strength and direction of a linear relationship between two variables, with r between -1 and +1. Regression gives an equation to estimate one variable from the other. To solve questions, identify the method, apply the formula, then check the sign and range of your answer.

What this chapter covers

This chapter is about two variables measured together, such as advertising spend and sales. Correlation tells you whether they move together and how strongly. Regression goes one step further. It gives a line that lets you estimate one variable from the other.

The chapter has two halves. The first half covers ways to measure correlation: scatter diagram, Karl Pearson's coefficient, Spearman's rank correlation and the concurrent deviation method. The second half covers the two regression lines, their equations and regression coefficients. The two halves are linked. The correlation coefficient r is the geometric mean of the two regression coefficients, with their common sign.

It connects to the rest of the paper through Statistics. You will reuse means, standard deviations and variance from earlier chapters. Questions are mostly numerical, so calculation speed and accuracy matter in this objective paper.

Correlation and Regression is a compact chapter with a small set of formulas and many predictable question patterns. Most MCQs are direct: find r, find a rank correlation, find a regression coefficient, or identify a property. Once you know the formulas and a few relations, you can answer quickly and avoid negative marking of 0.25 per wrong answer. The effort is modest compared with the return, and the maths also supports other Statistics topics.

Correlation and Regression: topics in the order to study them

1Meaning and Types of CorrelationStart with the vocabulary: positive, negative, zero, linear and non-linear correlation. Everything later depends on it.
2Scatter Diagram MethodIt gives a visual feel for direction and strength before you use formulas.
3Karl Pearson's Coefficient of CorrelationThis is the core formula of the chapter and the most tested. Regression builds on it.
4Spearman's Rank CorrelationIt is a variation for ranked data. Learn it after Pearson so you can compare the two.
5Concurrent Deviation MethodIt is the shortest method and easy to learn once you know the idea of direction of change.
6Regression Lines and EquationsMove to regression only after r is clear, since the lines use means and the same sums.
7Regression Coefficients and Their PropertiesThis ties both halves together: r equals the signed geometric mean of the two regression coefficients.

How to prepare Correlation and Regression

Aim to be fluent with the formulas first, then fast with calculations. Practise in short timed sets, as in the real objective paper.

Read each topic once and write its formula and the conditions for using it on one page.
Draw scatter diagrams by hand for small data sets until you can read direction and strength at a glance.
Solve Pearson's r problems using the formula with deviations from assumed means, and check that your answer lies between -1 and +1.
Practise Spearman's and concurrent deviation questions, including repeated ranks, until the steps become automatic.
For regression, always decide first which variable is dependent. Then write the correct line (y on x or x on y) before you substitute.
Learn the properties: r² = bxy × byx, both coefficients share the sign of r, and the lines pass through the means.
Do timed MCQ sets. If a question needs long arithmetic, test the options or skip it and return later.

Common mistakes in Correlation and Regression

Using the wrong regression line, such as y on x when the question asks for x on y.
Fix: Write the dependent variable first, then pick the matching coefficient: byx for y on x, bxy for x on y.
Giving r with the wrong sign when computing it from regression coefficients.
Fix: Take the sign from either regression coefficient, since both share it. If both are negative, r is negative.
Forgetting the tie correction in Spearman's rank correlation.
Fix: Scan the data for repeats first. Add (m³ - m) ÷ 12 to Σd² for each tied group.
Treating correlation as proof of cause and effect.
Fix: Remember that correlation shows association only. Choose options that avoid claiming causation.
Mixing up n in the concurrent deviation method, using the number of observations instead of the number of pairs of deviations.
Fix: Count the deviation pairs directly. For N observations, n = N - 1.
Doing lengthy calculations on every question and running out of time.
Fix: Use change of origin and scale, check the range of r, eliminate impossible options, and skip very long questions until the end.

Last-day revision: Correlation and Regression

Pearson's r always lies between -1 and +1; r has no unit.
r = Cov(x, y) ÷ (σx × σy), where Cov(x, y) = Σ(x - x̄)(y - ȳ) ÷ n.
r is independent of change of origin and scale in magnitude. If x and y are scaled by factors of opposite sign, r changes sign; if both factors have the same sign, r is unchanged.
Spearman's R = 1 - 6Σd² ÷ [n(n² - 1)], where d is the difference in ranks.
For repeated ranks, add a correction of (m³ - m) ÷ 12 to Σd² for each group of m tied items.
Concurrent deviation: r = ±√(±(2c - n) ÷ n), where c is the number of concurrent deviations and n is the number of pairs of deviations. The sign outside the root is the sign of (2c - n), and (2c - n) ÷ n must be positive for the root to be taken.
Regression of y on x: y - ȳ = byx (x - x̄), with byx = r × σy ÷ σx.
Regression of x on y: x - x̄ = bxy (y - ȳ), with bxy = r × σx ÷ σy.
r² = bxy × byx, so r = ±√(bxy × byx), with the sign of the coefficients.
Both regression lines pass through the point (x̄, ȳ); the means can be found by solving the two line equations together.
Regression coefficients are unaffected by change of origin but are affected by change of scale.
The two regression coefficients have the same sign. Both cannot be greater than 1; if one is greater than 1, the other must be less than 1 (since bxy × byx = r² ≤ 1).

Correlation and Regression practice questions

Correlation and Regression: frequently asked questions

What is the difference between correlation and regression?

Correlation measures how strongly two variables move together, using a single number r. Regression gives an equation to estimate one variable from the other. Correlation treats both variables alike, while regression separates them into dependent and independent.

Which formulas should I memorise for Correlation and Regression?

Learn Pearson's r, Spearman's R with the tie correction, the concurrent deviation formula, and the two regression coefficients byx and bxy. Also learn r² = bxy × byx. These cover most MCQs in this chapter.

Can the correlation coefficient be greater than 1?

No. Pearson's r always lies between -1 and +1. If your calculation gives a value outside this range, you have made an error and should recheck your sums.

How do I find the means from two regression equations?

Both regression lines pass through the point (x̄, ȳ). Solve the two equations together as simultaneous equations. The values you get for x and y are the means.

How much time should I give this chapter?

Because the formulas are few, most students can cover it in a few focused sessions plus timed practice. Spend extra time on regression properties and on getting calculations fast and accurate.