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CMA Foundation · Fundamentals of Business Mathematics and Statistics

Correlation and Regression for CMA Foundation Paper 3

Correlation measures the direction and strength of a linear relationship between two variables, shown by r from -1 to +1. Regression gives an equation to estimate one variable from the other. To solve MCQs, learn the formulas for r, rank correlation and regression coefficients, and use the link r² = bxy × byx.

What this chapter covers

This chapter studies how two variables move together. Correlation tells you whether they rise and fall together, move in opposite directions, or show no linear pattern. It also tells you how strong the link is. Regression goes one step further. It gives you a line so you can estimate one variable when you know the other.

The chapter has two halves. The first half covers correlation: its meaning and types, Karl Pearson's coefficient, Spearman's rank correlation, probable error and the coefficient of determination. The second half covers regression: the two regression lines and the properties of regression coefficients. The two halves are tied together by one key link: r is the geometric mean of the two regression coefficients, with their common sign.

The chapter connects to the rest of Paper 3. You need averages, standard deviation and variance from Statistics, because r and b use them. You also need basic algebra from Business Mathematics to solve the two regression equations for means. Questions here are mostly numerical, so speed with simple calculations counts.

Correlation and regression give you formula-driven MCQs with a single clear answer, which is the easiest kind to score once the formulas are in your head. There is no negative marking, so every question is worth attempting, and many questions can be solved in under a minute using a property instead of a full calculation. The chapter also feeds later study, because the idea of relating two variables is used in economics and in cost estimation. Since each paper needs at least 40% to pass, a chapter where you can reliably collect marks helps protect your Paper 3 score.

Correlation and Regression: topics in the order to study them

  1. 1Meaning and Types of CorrelationStart here to learn the vocabulary: positive, negative, zero, linear and non-linear, simple and multiple. Every later formula depends on these ideas.
  2. 2Karl Pearson's Coefficient of CorrelationThis is the main formula of the chapter. It is also the base for probable error and for the regression link, so learn it before anything else.
  3. 3Spearman's Rank CorrelationIt works on ranks instead of values, so it is easy once you know Pearson's r. Learn it next, including the tied-rank correction.
  4. 4Probable Error and Coefficient of DeterminationBoth are small add-ons that use r. Do them after you are comfortable computing r.
  5. 5Regression Lines and EquationsNow move to regression. You need to know which variable is dependent in each line and how the two lines pass through the means.
  6. 6Regression Coefficients and Their PropertiesStudy this last because it joins both halves. The properties link b values with r and let you solve many MCQs without full calculation.

How to prepare Correlation and Regression

Aim to know every formula by heart and to practise numbers until the arithmetic is quick. Most marks come from a few repeated question patterns.

  1. Read the meaning and types of correlation once and write down the sign and range of r. Be able to say what r = 0, +1 and -1 each mean.
  2. Learn Pearson's formula in its short forms, including the one using deviations from the mean. Solve three or four small data sets by hand with clean numbers.
  3. Practise Spearman's rank correlation with and without tied ranks. Write the tie correction on a card and revise it.
  4. Learn probable error and the coefficient of determination as two-line formulas. Practise questions that ask for r² as a percentage.
  5. For regression, practise writing both lines, identifying which is Y on X and which is X on Y, and finding the means from two given equations.
  6. Memorise the properties: r² = bxy × byx, both b values share the sign of r, and the arithmetic mean of the two b values is greater than or equal to the absolute value of r, i.e. (bxy + byx) ÷ 2 ≥ |r|. Use them to eliminate options.
  7. Finish with timed sets of 20 MCQs in 20 minutes, then review every error and note whether it was a formula slip or an arithmetic slip.

Common mistakes in Correlation and Regression

  • Taking the wrong regression line for a question.

    Fix: Write the form first. Y on X has Y alone on the left, and X on Y has X alone on the left. Then use the matching b value.

  • Giving r a positive sign when both b values are negative.

    Fix: Take the sign of r from the b values. If both are negative, r is negative.

  • Forgetting the tie correction in Spearman's rank correlation.

    Fix: Always scan the data for repeated values before ranking. Give tied items the average rank and add the correction term for each group.

  • Using the wrong formula for probable error.

    Fix: Remember it as 0.6745 times (1 - r²) over root n, and check that n is the number of pairs.

  • Treating a high correlation as proof of cause.

    Fix: For theory MCQs, pick the option that says correlation shows association only.

  • Making arithmetic slips with deviations and squares.

    Fix: Check that the sum of deviations from the mean is zero before moving on. Rework any column that fails this check.

Last-day revision: Correlation and Regression

  • r always lies between -1 and +1, and it has no unit.
  • r = +1 is perfect positive, r = -1 is perfect negative, and r = 0 means no linear relationship.
  • Pearson's r = Cov(X, Y) ÷ (σx × σy).
  • Correlation does not prove cause and effect.
  • Spearman's R = 1 - 6Σd² ÷ [n(n² - 1)], where d is the difference in ranks.
  • For tied ranks, add a correction of (m³ - m) ÷ 12 to Σd² for each tied group of m items.
  • Probable error = 0.6745 × (1 - r²) ÷ √n.
  • Coefficient of determination = r², and it shows the share of variation explained.
  • Regression of Y on X: Y - Ȳ = byx (X - X̄), where byx = r × σy ÷ σx.
  • Regression of X on Y: X - X̄ = bxy (Y - Ȳ), where bxy = r × σx ÷ σy.
  • r = ±√(bxy × byx), and the sign is the same as that of both b values.
  • Both regression lines pass through the point (X̄, Ȳ).

Correlation and Regression practice questions

Correlation and Regression in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Correlation and Regression: frequently asked questions

Which formulas matter most in Correlation and Regression?

Pearson's r, Spearman's rank correlation, the two regression coefficients and r² = bxy × byx matter most. Learn probable error and the coefficient of determination as short extras. Together these cover most of the numerical MCQs.

How do I find r when two regression equations are given?

First decide which equation is Y on X and which is X on Y, then read off byx and bxy. Multiply them and take the square root. Give r the common sign of the two b values, and check that the product does not exceed 1.

Can the regression coefficients both be greater than 1?

No. Their product equals r², which cannot exceed 1, so at most one of them can be greater than 1. If you get two values above 1, you have probably picked the wrong equation or made an error.

Is this chapter easy to score in a one-hour objective paper?

It can be, because most questions are direct formula applications with a single correct answer. With no negative marking you should attempt every question. Practise with simple numbers so the calculations stay fast.