Management Accounting · Analytical techniques in budgeting and forecasting

Correlation Coefficient and Coefficient of Determination Explained

Updated 11 October 2026 · Fact-checked

The correlation coefficient r measures the strength and direction of a linear relationship between two variables, from -1 to +1. The coefficient of determination r² is r squared. It shows the proportion of variation in y explained by x. To solve questions, find r, square it, then interpret it.

Understand Correlation Coefficient and Coefficient of Determination

Correlation asks one question: when x changes, does y tend to change in a straight-line way? For example, does advertising spend move with sales? The answer is a single number called the correlation coefficient, written r.

r always lies between -1 and +1. A value near +1 means strong positive correlation: as x rises, y rises. A value near -1 means strong negative correlation: as x rises, y falls. A value near 0 means there is little or no linear relationship. The sign gives direction. The size, ignoring the sign, gives strength.

The coefficient of determination, r², is r multiplied by itself. It is always between 0 and 1. It tells you the proportion of the variation in y that is explained by variation in x. If r = 0.9, then r² = 0.81. So 81% of the variation in y is explained by x. The other 19% is due to other factors or random variation.

Because r² is squared, it loses the sign. If r = -0.9, r² is still 0.81. So r² shows the strength of the linear fit (how much variation is explained), not the direction, and so gives an indication of forecast reliability.

Correlation and regression are different. Correlation measures how strong the linear link is. Regression gives the equation of the line, y = a + bx, which you use to forecast. A high r² gives more confidence in a regression forecast. Also, correlation does not prove cause. Two variables can move together by chance or because of a third factor.

Key formulas to remember

Range of r: -1 ≤ r ≤ +1; +1 is perfect positive, -1 is perfect negative, 0 means no linear correlation.
Coefficient of determination: r² = r × r; Always between 0 and 1. Shown as a proportion or a percentage of variation in y explained by x.
Correlation coefficient from r²: r = ±√r²; Take the sign from the slope b of the regression line. Positive b gives positive r.
Unexplained variation: 1 - r²; The proportion of variation in y not explained by x.
Correlation coefficient formula: r = [nΣxy - ΣxΣy] ÷ √{[nΣx² - (Σx)²][nΣy² - (Σy)²]}; You can apply it by hand or use the calculator's linear regression mode. Know the structure, but most questions give you r.

How to solve Correlation Coefficient and Coefficient of Determination questions

Use this method for any question on r or r².

1Identify what is given: r, r², or raw data. Note which variable is x and which is y.
2If you have raw data and must calculate r, use the formula carefully, or the calculator's linear regression mode.
3Read the sign of r to state direction: positive means both move together, negative means they move opposite ways.
4Read the size of r to state strength: close to 1 or -1 is strong, close to 0 is weak.
5To find r², square r. Convert to a percentage if the question asks for the proportion explained.
6State what r² means: the proportion of variation in y explained by x. The remainder, 1 - r², is unexplained.
7If asked about forecasting, say a high r² gives more reliable regression forecasts, and that correlation does not prove cause.

Quickest way: Square it, then say what it means

When to use it: Use this for multiple choice and number entry questions where r is given and you need r² or an interpretation.

Check the sign first. It decides direction, and you can rule out options that get it wrong.
Square r on your calculator. Do not square the percentage.
Convert to a percentage by multiplying by 100 if asked.
Match the meaning: r² is the share of variation in y explained by x.
For 'unexplained', use 1 - r².

Common mistakes in Correlation Coefficient and Coefficient of Determination

Treating r² as the correlation coefficient itself, for example saying r = 0.64 when r² = 0.64.
The two figures look similar and questions often give only one of them.
Fix: Read the label. If given r², take the square root to find r, and choose the sign from the slope.
Saying r = -0.9 is a weak correlation because it is negative.
Students link negative with bad or small.
Fix: Strength depends on size, ignoring the sign. -0.9 is strong negative.
Saying r² of 0.81 means 81% of y is caused by x.
Explained variation is confused with proof of cause.
Fix: Say 81% of the variation in y is explained by variation in x. Correlation does not prove cause.
Forgetting that r² is always positive and giving a negative r².
Students square the number but keep the sign in mind.
Fix: A squared number is never negative. Restore the sign only when finding r from r², using the slope.
Confusing correlation with regression.
Both use the same x and y data.
Fix: Correlation measures strength of the link. Regression gives the line y = a + bx for forecasting.
Thinking r close to 0 means no relationship at all.
The word 'no correlation' is over-read.
Fix: A low r means no linear relationship. A curved relationship may still exist.

Worked examples

Example 1

A company finds that the correlation coefficient between advertising spend and sales is r = 0.8. What proportion of the variation in sales is explained by advertising spend, and what proportion is not?

Show the solution

r² = 0.8 × 0.8 = 0.64.
Explained variation = 0.64, or 64%.
Unexplained variation = 1 - 0.64 = 0.36, or 36%.

Answer: 64% of the variation in sales is explained by advertising spend. 36% is explained by other factors or random variation.

Example 2

A regression of cost per unit (y) on output level (x) has a coefficient of determination of 0.49 and a regression slope b that is negative. What is the correlation coefficient, and how should it be described?

Show the solution

r = ±√0.49 = ±0.7.
The slope is negative, so r is negative: r = -0.7.
Size 0.7 is moderately strong, ignoring the sign.
The sign shows that as output rises, cost per unit tends to fall.

Answer: r = -0.7. It is a moderately strong negative linear correlation. 49% of the variation in cost per unit is explained by output level.

Exam tips

In multiple choice, check the sign of r before any calculation. It often removes two options.
If a question gives r² and asks for r, always look for the slope or the description of direction to choose the sign.
Use wording such as 'explained by' for r². Avoid saying 'caused by'.
In multiple response questions, select the stated number of statements only. Reject any that claim correlation proves cause or that r can exceed 1.
For number entry, follow the rounding instruction exactly, and enter r² as a decimal or percentage as the question asks.

Practice questions from Analytical techniques in budgeting and forecasting

Correlation Coefficient and Coefficient of Determination in other exams

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

Correlation Coefficient and Coefficient of Determination: frequently asked questions

What is the difference between correlation and regression?

Correlation gives one number, r, that shows the strength and direction of a linear relationship. Regression gives an equation, y = a + bx, that you use to forecast y from x. Both use the same paired data.

How do I interpret the coefficient of determination?

Take r² as a percentage. It is the share of the variation in y explained by changes in x. For example, r² = 0.75 means 75% is explained and 25% is unexplained.

Can r² be negative?

No. It is a squared number, so it lies between 0 and 1. Only r can be negative.

Does a high correlation prove that x causes y?

No. Two variables can move together by coincidence or because of a third factor. High correlation only shows a strong linear association.