Karl Pearson's Coefficient of Correlation: Method and Interpretation
Updated 17 October 2026 · Fact-checked
Correlation measures the degree and direction of the linear relationship between two variables. Karl Pearson's coefficient r lies between -1 and +1; r = Sum(dx dy) / sqrt(Sum dx^2 x Sum dy^2) by the actual mean method. A positive value means the variables move together, a negative value that they move in opposite directions, and correlation does not prove causation.
A downward band gives r near -1, a shapeless cloud r near 0, an upward band r near +1; our data (r = 0.81) is strongly positive.
Illustration drawn for QuestionBuddy.
Understand Karl Pearson's Coefficient of Correlation
Meaning. Two variables are correlated if a change in one is accompanied by a change in the other. Correlation measures only association, not cause and effect: ice-cream sales and drowning accidents may rise together because both depend on hot weather.
Types. Positive (both move in the same direction, e.g. height and weight), negative (opposite directions, e.g. price and quantity demanded), zero (no relationship). Linear (points close to a straight line) versus non-linear. Simple (two variables) versus multiple. A scatter diagram plots each pair as a point: an upward band suggests positive, a downward band negative, a shapeless cloud none.
Karl Pearson's coefficient of correlation (r). Measures the degree and direction of linear association.
- Actual mean method: r = Σ(dx·dy) ÷ √(Σdx² × Σdy²), where dx = X − mean of X and dy = Y − mean of Y.
- Direct method: r = [nΣXY − ΣXΣY] ÷ √{[nΣX² − (ΣX)²] × [nΣY² − (ΣY)²]}.
- The same value comes from either.
Worked data. X: 10, 12, 14, 16, 18; Y: 20, 25, 22, 30, 28. Means: X̄ = 70 ÷ 5 = 14, Ȳ = 125 ÷ 5 = 25.
| X | Y | dx | dy | dx·dy | dx² | dy² |
| 10 | 20 | −4 | −5 | 20 | 16 | 25 |
| 12 | 25 | −2 | 0 | 0 | 4 | 0 |
| 14 | 22 | 0 | −3 | 0 | 0 | 9 |
| 16 | 30 | 2 | 5 | 10 | 4 | 25 |
| 18 | 28 | 4 | 3 | 12 | 16 | 9 |
Σdx·dy = 42, Σdx² = 40, Σdy² = 68. r = 42 ÷ √(40 × 68) = 42 ÷ √2720 = 42 ÷ 52.15 = 0.81 (strong positive).
Direct check: ΣX = 70, ΣY = 125, ΣXY = 1792, ΣX² = 1020, ΣY² = 3193; numerator = 5 × 1792 − 70 × 125 = 8960 − 8750 = 210; denominator = √{(5100 − 4900)(15965 − 15625)} = √(200 × 340) = √68000 = 260.77; r = 210 ÷ 260.77 = 0.81, the same value.
Properties. (1) r lies between −1 and +1. (2) r = +1 perfect positive, −1 perfect negative, 0 no linear relation. (3) r is independent of change of origin and scale. (4) r is a pure number (no units). (5) rxy = ryx. (6) Squares: r² shows the proportion of variation shared.
Interpretation (rough guide). |r| above 0.7 strong, 0.3 to 0.7 moderate, below 0.3 weak. Sign shows direction.
Limitations. Measures only linear association; sensitive to extreme values; correlation is not causation.
Draw: scatter diagrams for r near +1, −1 and 0.
Karl Pearson's Coefficient of Correlation: key facts
- Actual mean method
- r = Sum(dx dy) / sqrt(Sum dx^2 x Sum dy^2)
- dx = X - mean X; dy = Y - mean Y
- Direct method
- r = (n Sum XY - Sum X Sum Y) / sqrt((n Sum X^2 - (Sum X)^2)(n Sum Y^2 - (Sum Y)^2))
- Same value
- Range
- -1 <= r <= +1
- Pure number
- Invariance
- r unchanged by change of origin and scale
- Properties
How to answer Karl Pearson's Coefficient of Correlation questions
Questions give paired data and ask for r and its interpretation.
- 1Compute means and deviations dx and dy.
- 2Fill columns dx·dy, dx², dy² and sum them.
- 3Substitute into r = Σdxdy / √(Σdx²Σdy²).
- 4Interpret the sign and size.
- 5State that correlation does not imply causation.
Karl Pearson's Coefficient of Correlation in real life
The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.
Study hours and marks
Suppose a teacher notes hours of study and marks for ten students and finds r = 0.99.
What does the value mean and what does it not mean?
Try it first, then show the solutionSolution
- Positive and close to 1: more study hours go with higher marks.
- It shows association, not that hours alone cause the marks.
- Other factors such as ability and sleep also matter.
Answer: Strong positive correlation; not proof of causation.
Memory hook: Moves together is not caused by.
Tricks to remember Karl Pearson's Coefficient of Correlation
Short memory aids. Say each one aloud once, then use it on a question the same day.
dx dy over root
r = sum dxdy / root of (sum dx2 times sum dy2).
Use it when: Formula
Minus one to plus one
r is between -1 and +1.
Use it when: Range
Association not cause
Correlation is not causation.
Use it when: Interpretation
Karl Pearson's Coefficient of Correlation: solved examples
Ordered from board level to HOTS. Try each one before you open the solution.
Example 1: Meaning and types · 3 marks · Short answerBoard level
Define correlation and distinguish positive, negative and zero correlation with an example of each.
Try it first, then show the solutionSolution
- Correlation: association between two variables so that change in one goes with change in the other (1 mark).
- Positive: same direction, e.g. height and weight; negative: opposite, e.g. price and quantity demanded (1 mark).
- Zero: no relationship, e.g. shoe size and marks (1 mark).
Answer: Association; same, opposite, none.
Example 2: Compute r · 5 marks · Long answerHarder board level
Compute the Karl Pearson coefficient for X: 10, 12, 14, 16, 18 and Y: 20, 25, 22, 30, 28 by the actual mean method.
Try it first, then show the solutionSolution
- Means: X = 14 and Y = 25 (1 mark).
- dx: -4, -2, 0, 2, 4; dy: -5, 0, -3, 5, 3 (1 mark).
- Sum dxdy = 20 + 0 + 0 + 10 + 12 = 42; Sum dx2 = 40; Sum dy2 = 68 (1 mark).
- r = 42 / sqrt(40 x 68) = 42 / 52.15 (1 mark).
- r = 0.81, a strong positive correlation (1 mark).
Answer: r = 0.81.
Example 3: Perfect correlation · 3 marks · Short answerHarder board level
Show that for X: 1, 2, 3, 4, 5 and Y = 2X + 3, r = +1, and state what r is if Y = 10 - X.
Try it first, then show the solutionSolution
- Y = 5, 7, 9, 11, 13; dx = -2, -1, 0, 1, 2; dy = -4, -2, 0, 2, 4 (1 mark).
- Sum dxdy = 20; Sum dx2 = 10; Sum dy2 = 40; r = 20 / sqrt(400) = 1 (1 mark).
- For Y = 10 - X, every dy = -dx, so r = -1 (1 mark).
Answer: +1; -1.
Example 4: Change of origin and scale · 3 marks · Short answerHarder board level
A student transforms X and Y of the first sum to (X - 14)/2 and (Y - 25)/5. What happens to r and why?
Try it first, then show the solutionSolution
- r is unchanged: 0.81 (1 mark).
- Subtracting a constant (origin) does not change deviations from the mean (1 mark).
- Dividing by positive constants rescales dx and dy equally in numerator and denominator, so it cancels (1 mark).
Answer: r unchanged.
Example 5: Causation trap · 4 marks · Short answerHOTS / competency
Suppose a town finds r = 0.9 between ice-cream sales and the number of sunburn cases. Evaluate the claim that ice-cream causes sunburn.
Try it first, then show the solutionSolution
- The high r shows strong association (1 mark).
- Both depend on a third factor: hot sunny weather (1 mark).
- Association does not prove cause (1 mark).
- A proper study would control the third factor or test the cause directly (1 mark).
Answer: Common cause; correlation is not causation.
Where students lose marks in Karl Pearson's Coefficient of Correlation
The slips examiners see most often, and what to write instead.
Mistake: r greater than 1
Square root slip.
Wrong working
r = 1.4.
Right working
r must lie within -1 and +1; recheck calculation.
Do this: Check range.
Usually costs: 1 mark.
Mistake: Causation claim
Association misread.
Wrong working
High r proves X causes Y.
Right working
It shows association only.
Do this: No cause.
Usually costs: 1 mark.
Mistake: dx from wrong mean
Mean error.
Wrong working
Using median for dx.
Right working
Use arithmetic means.
Do this: Means.
Usually costs: 1 mark.
Exam tips
- Set out the column table.
- Check r lies within -1 and 1.
- State the type and strength.
- Say that correlation is not causation.
Practice questions on Karl Pearson's Coefficient of Correlation
Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.
Q1 · 1 mark · MCQBoard level
The value of the correlation coefficient can lie between
(a) 0 and 1 (b) -1 and +1 (c) -infinity and +infinity (d) 0 and 100Hint
Bounded.
Answer
(b).
Q2 · 1 mark · Assertion-reasonHarder board level
Assertion (A): r = 0 means no linear relationship.
Reason (R): There may still be a non-linear relationship.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.Hint
Linear only.
Answer
(b) Both A and R are true but R is not the correct explanation of A.
Q3 · 2 marks · Very short answerBoard level
If the covariance-based sums are Sum dxdy = -36, Sum dx2 = 25 and Sum dy2 = 64, find r and interpret.
Hint
Root of 1600.
Answer
r = -36 / sqrt(25 x 64) = -36 / 40 = -0.9, a strong negative correlation.
Q4 · 4 marks · Case-basedHarder board level
Read and answer.
Suppose a scatter diagram shows points falling from top left to bottom right in a narrow band.
(a) Name the type of correlation. (1)
(b) Give the likely sign of r. (1)
(c) Give an economic example. (2)Hint
Falling band.
Answer
(a) Negative correlation. (b) Negative sign, close to -1. (c) Price and quantity demanded of a normal good.
Q5 · 3 marks · Short answerHOTS / competency
Suppose X and Y have r = 0.8. If every X is converted from rupees to paise (multiplied by 100) and Y is unchanged, what is the new r? What if Y is replaced by -Y?
Hint
Scale and sign.
Answer
Multiplying X by 100 is a positive change of scale, so r stays 0.8. Replacing Y by -Y reverses the direction, so r becomes -0.8.
Test your recall
Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.
1.Range of r?Show answer
-1 to +1.
2.r = 0 means?Show answer
No linear relation.
3.dx formula?Show answer
X - mean X.
4.Negative correlation example?Show answer
Price and quantity demanded.
5.Scatter diagram shows?Show answer
Pairs as points.
6.Is r affected by origin?Show answer
No.
Karl Pearson's Coefficient of Correlation: frequently asked questions
What is correlation?
Association between two variables.
What is r?
Karl Pearson's coefficient of correlation.
What does r = -1 mean?
Perfect negative linear relation.
Is correlation causation?
No.
What is a scatter diagram?
A plot of paired values.