Key takeaways
- Correlation shows direction and strength of linear association, not cause.
- A scatter diagram shows the pattern before any calculation.
- Pearson’s r lies between −1 and +1, has no unit and is unchanged by origin and scale.
- Spearman’s rs = 1 − 6Σd² ÷ n(n² − 1) works with ranks.
- Tied ranks need a correction of (m³ − m) ÷ 12 each.
This is Chapter 6 of the Statistics for Economics chapter list for Class 11 Economics. Where Chapter 5 summarised one variable, this chapter relates two.
6.1Types of Relationship and What Correlation Measures
Correlation measures the direction and strength of a linear relationship between two variables. It is positive when they move in the same direction and negative when they move in opposite directions. It measures movement together, not cause: a high correlation does not prove that one variable causes the other.
Hotel bookings rise with a hill station’s tourist season; the price of tomatoes falls as arrivals at the mandi rise. Such pairs show covariation. Some relationships are causal, as with rainfall and crop yield. Others are coincidence, or the work of a third variable: ice-cream sales and swimming accidents both rise in summer because of heat, though neither causes the other.
| Type | Movement | Example |
|---|---|---|
| Positive | Both rise or both fall | Income and consumption |
| Negative | One rises, the other falls | Price and quantity demanded |
| Zero (none) | No linear pattern | Shoe size and rainfall |
| Perfect (r = +1 or −1) | All points on a straight line | Distance and fare at a fixed rate |
Correlation shows association, not causation. Before inferring cause, ask whether a third factor, chance, or the direction of influence explains the link.
Nonsense and spurious correlations
Two variables can show a high correlation with no real link. Both may rise over time, for example the number of mobile subscriptions and the sales of a brand of biscuits, and so move together by coincidence. A third factor may drive both, such as summer heat raising ice-cream sales and swimming-pool visits. Such results are called spurious correlations.
To avoid being misled, ask whether there is a sensible reason for the link, whether a third factor could explain it, and whether the pattern holds in different time periods and places. Theory and common sense should accompany every coefficient.
6.2Scatter Diagram
A scatter diagram plots each pair of values as a point on a graph. Points rising from left to right show positive correlation, falling points show negative correlation, and a shapeless cloud shows none. The closer the points lie to a straight line, the stronger the linear relationship. It gives a visual idea before any calculation.
Always draw the scatter diagram first. Pearson’s r measures only linear association, so if the points follow a curve, r can be near zero even though the variables are tightly related.
Outliers and clusters in a scatter diagram
A scatter diagram may show more than a general slope. A single point far from the others, an outlier, can pull the coefficient up or down sharply. It should be checked for a recording error before being kept or dropped. Points may also form separate clusters, for example urban and rural households, in which case combining them into one coefficient can hide the pattern within each group.
Looking at the diagram first also shows whether the relationship is straight, curved or bunched. These features decide whether Pearson’s coefficient is appropriate, and the diagram is the cheapest check against a misleading number.
6.3Karl Pearson’s Coefficient of Correlation
Karl Pearson’s coefficient r is the covariance of two variables divided by the product of their standard deviations. It lies between −1 and +1, has no unit, and is unaffected by changes of origin and scale. A value near +1 or −1 shows a strong linear relationship, and near 0 a weak one.
r = Σxy ÷ √(Σx² × Σy²), where x = X − X̄ and y = Y − Ȳ
Equivalent: r = [NΣXY − ΣXΣY] ÷ √{{[NΣX² − (ΣX)²] [NΣY² − (ΣY)²]}}
Worked case
For six students, X = weekly study hours (2, 4, 5, 7, 8, 10) and Y = marks (38, 41, 55, 58, 70, 72): ΣX = 36, ΣY = 334, ΣXY = 2201, ΣX² = 258, ΣY² = 19598. Then NΣXY − ΣXΣY = 6 × 2201 − 36 × 334 = 1182, and the denominator is √[(6 × 258 − 1296)(6 × 19598 − 111556)] = 1232.91. So r = 0.959, a very strong positive correlation.
Properties of r
- r is a pure number; units cancel.
- −1 ≤ r ≤ +1. A value outside this range signals an error.
- r = 0 means no linear relation, not necessarily no relation.
- r is unchanged if every X is shifted by a constant or multiplied by a positive constant, so the step deviation method can be used.
- r is the same for X on Y and Y on X.
Step deviation method
With u = (X − A) ÷ h and v = (Y − B) ÷ k, rXY = rUV. For X = 60, 70, 80, 90, 100 and Y = 210, 240, 250, 290, 310, take A = 80, h = 10, B = 250, k = 10: u = −2, −1, 0, 1, 2 and v = −4, −1, 0, 4, 6. Then Σu = 0, Σv = 5, Σuv = 25, Σu² = 10, Σv² = 69, and r = (5 × 25 − 0) ÷ √[(5 × 10)(5 × 69 − 25)] = 125 ÷ √(50 × 320) = 125 ÷ 126.49 = 0.988.
| Value of r | Meaning |
|---|---|
| 0.7 to 1 | Strong positive |
| 0.3 to 0.7 | Moderate positive |
| −0.3 to 0.3 | Weak or none |
| −0.7 to −0.3 | Moderate negative |
| −1 to −0.7 | Strong negative |
Interpreting r in words
A value of r close to +1 means that the points lie close to a rising straight line, so higher values of one variable go with higher values of the other. A value near −1 describes a falling line. A value near 0 means that no straight-line pattern is visible. The square of r tells what proportion of the variation in one variable goes along with variation in the other.
In the study-hours example, r is 0.96, so r squared is about 0.92: roughly 92 per cent of the variation in marks goes along with differences in study hours, and the rest is linked to other factors such as ability, health and the difficulty of the paper.
A checklist for calculating r
Before calculating, draw the scatter diagram. List the pairs and compute the totals of X, Y, XY, X squared and Y squared. Substitute into the formula and compute the numerator and denominator separately. Check that r lies between −1 and +1. Finally, state the result in words, giving its direction and strength and noting that it does not prove cause.
6.4Spearman’s Rank Correlation
Spearman’s rank correlation measures the strength of association between two sets of ranks. It is used when items can be placed in order but not measured, such as honesty or beauty, or when only ranks are available. Its value is 1 minus 6 times the sum of squared rank differences divided by n times n squared minus 1.
rs = 1 − [6Σd²] ÷ [n(n² − 1)]
d = difference between the two ranks of an item. If m items share a rank, add (m³ − m) ÷ 12 to Σd² for each such group.
Two judges rank six singers: judge 1 gives ranks 1 to 6; judge 2 gives 2, 1, 4, 3, 6, 5. The differences are −1, 1, −1, 1, −1, 1, so Σd² = 6 and rs = 1 − 6 × 6 ÷ (6 × 35) = 1 − 36 ÷ 210 = 0.829: the judges largely agree.
With tied ranks, each tied item gets the average of the ranks it covers. If two items tie for 3rd and 4th place, both get 3.5, and a correction of (2³ − 2) ÷ 12 = 0.5 is added to Σd².
When ranks are better than values
Ranks are useful when the values themselves are not reliable or not available. Judges of a competition give scores that differ in strictness, but their rankings can be compared. A few extreme values can strongly affect Pearson’s r, whereas ranks are not affected by how large an outlier is.
Ranks also allow correlation to be measured for qualities that can only be put in order, such as quality of service or preference for brands. The price paid is the loss of information on the size of differences between items.
Worked Examples
Example 1: Pearson’s r
- Given
- Study hours 2, 4, 5, 7, 8, 10; marks 38, 41, 55, 58, 70, 72.
- Formula
- r = [NΣXY − ΣXΣY] ÷ √{[NΣX² − (ΣX)²][NΣY² − (ΣY)²]}
- Steps
- ΣX = 36; ΣY = 334; ΣXY = 2201; ΣX² = 258; ΣY² = 19598. Numerator = 1182; denominator = 1232.91.
- Answer
- r = 0.959: strong positive correlation.
Example 2: Spearman’s coefficient
- Given
- Ranks of six singers by two judges: (1, 2), (2, 1), (3, 4), (4, 3), (5, 6), (6, 5).
- Formula
- rs = 1 − 6Σd² ÷ n(n² − 1)
- Steps
- d² = 1 for each pair, so Σd² = 6. rs = 1 − 36 ÷ 210.
- Answer
- rs = 0.829.
Example 3: Spearman with a tied rank
- Given
- Rank sets (1, 2), (2, 1), (3.5, 4), (3.5, 3), (5, 5); n = 5; two items tie.
- Formula
- Add (m³ − m) ÷ 12 to Σd² for the tie of m = 2
- Steps
- d² = 1, 1, 0.25, 0.25, 0 gives 2.5. Correction = 6 ÷ 12 = 0.5, so Σd² = 3. rs = 1 − 6 × 3 ÷ (5 × 24) = 1 − 0.15.
- Answer
- rs = 0.85.
Common Mistakes
- Reading a high r as proof of cause and effect. Association is not causation.
- Using Pearson’s r for a curved relationship. Check the scatter diagram first.
- Giving r a unit or reporting a value above 1. r is a pure number between −1 and +1.
- Forgetting the tie correction in Spearman’s formula when ranks are repeated.
10 Important Questions for Class 11 Board and School Examinations
Questions are written in original words. Marks are shown beside each question.
Q1The value of Karl Pearson’s coefficient of correlation lies between (a) 0 and 1 (b) −1 and +1 (c) −∞ and +∞ (d) 0 and 100 1 mark · MCQ
The correct option is (b) −1 and +1.
Q2If r = −0.9, the two variables are (a) strongly positively related (b) unrelated (c) strongly negatively related (d) perfectly related 1 mark · MCQ
The correct option is (c) strongly negatively related.
Q3Assertion (A): A high correlation between two variables proves that one causes the other. Reason (R): Correlation measures covariation only. Which is correct? 1 mark · Assertion-reason
A is false and R is true.
Correlation shows that variables move together; it does not prove cause.
Q4What is a scatter diagram? What does it show? 3 marks
A scatter diagram plots pairs of values of two variables as points on a graph to show the pattern of their relationship.
- Points rising from left to right show positive correlation.
- Points falling from left to right show negative correlation.
- A shapeless cloud shows no correlation, and points on a straight line show perfect correlation.
It is drawn before calculating a coefficient, because it shows whether the relationship is straight or curved.
Q5State three properties of the correlation coefficient. 3 marks
The correlation coefficient is a pure number, lies between −1 and +1 and is unaffected by changes of origin and scale.
- r has no unit, since the units of the two variables cancel.
- A value outside the range of −1 to +1 indicates a calculation error.
- Adding a constant to every X, or multiplying by a positive constant, does not change r.
A positive r means the variables move together, and a negative r means they move in opposite directions.
Q6Distinguish between positive and negative correlation with two examples each. 4 marks
In positive correlation the variables move in the same direction, and in negative correlation they move in opposite directions.
- Positive: income and consumption rise together; rainfall and crop yield rise together.
- Negative: the price of a good and the quantity demanded; hours of study and the chance of failing.
- The strength is judged by the size of r, with values near +1 or −1 being strong.
Neither shows cause on its own.
Strong correlation in either direction means that the points lie close to a straight line.
Q7When is Spearman’s rank correlation preferred to Pearson’s coefficient? 4 marks
Spearman’s rank correlation is preferred when items can be ranked but not measured, or when only ranks are available.
- Qualities such as honesty or beauty cannot be measured, but people can be placed in order.
- It suits data with extreme values or small samples, where ranks are more reliable than raw values.
- Pearson’s coefficient needs measured data and a roughly linear relationship.
With tied ranks, each tied item gets the average of the ranks it covers and a correction is added to the sum of squared rank differences.
Q8Calculate Karl Pearson’s r for X = 2, 4, 5, 7, 8, 10 and Y = 38, 41, 55, 58, 70, 72. 6 marks
The coefficient of correlation is 0.959, showing a strong positive correlation between study hours and marks.
- Given: X = 2, 4, 5, 7, 8, 10 and Y = 38, 41, 55, 58, 70, 72, with N = 6.
- Sums: ΣX = 36, ΣY = 334, ΣXY = 2,201, ΣX² = 258, ΣY² = 19,598.
- Numerator: NΣXY − ΣXΣY = 6 × 2,201 − 36 × 334 = 13,206 − 12,024 = 1,182.
- Denominator: √[(6 × 258 − 36²)(6 × 19,598 − 334²)] = √(252 × 6,032) = 1,232.9.
- r = 1,182 ÷ 1,232.9 = 0.959.
Students who study more tend to score higher marks.
Pearson’s r is valid here because the scatter diagram shows a roughly straight-line pattern, and the value is high. It does not by itself prove that study causes the marks, since other factors such as ability also matter.
Q9Explain why a high correlation between two variables may not imply a cause-and-effect relationship. 6 marks
A high correlation may not imply cause and effect because the link may come from a third factor, coincidence or the opposite direction of influence.
- A third factor can drive both variables: heat raises both ice-cream sales and swimming accidents, but neither causes the other.
- Two series that both trend upwards can show a high r by chance.
- The direction of influence may be opposite to what is assumed.
- Cause needs theory, controls and evidence beyond a coefficient.
Correlation measures only how closely two variables move together, so it should be read with care.
For example, the number of doctors sent to villages during an epidemic may be positively correlated with the number of deaths, because doctors go where the epidemic is worst, not because they cause deaths. Careful study of timing, controls and other variables is needed before concluding cause.
Q10Two judges rank five candidates as (1, 3), (2, 1), (3, 2), (4, 5), (5, 4). Find Spearman’s rank correlation. 4 marks · Numerical
The rank correlation is 0.60.
Steps: d = −2, 1, 1, −1, 1 so d² = 4, 1, 1, 1, 1 and Σd² = 8.
Formula: rs = 1 − 6Σd² ÷ n(n² − 1) = 1 − 48 ÷ 120.
Final answer: rs = 0.60.
10 Points for State PSC and UPSC Aspirants
- Karl Pearson (1857–1936) devised the product-moment coefficient; Charles Spearman proposed the rank coefficient in 1904.Why it matters: name-and-concept questions.Source: NCERT; standard statistics textbooks
- The Phillips curve describes a negative association between inflation and unemployment (A. W. Phillips, 1958).Why it matters: a classic negative correlation in macroeconomics.Source: Standard macroeconomics textbooks
- The law of demand is a negative relation between price and quantity demanded.Why it matters: ties correlation to Microeconomics.Source: NCERT Introductory Microeconomics
- Engel’s law: the share of income spent on food falls as income rises, a negative correlation.Why it matters: visible in NSS consumption data.Source: Standard economics textbooks; MoSPI
- The quantity theory of money links money supply and the price level positively, other things equal.Why it matters: used to read inflation and money supply series.Source: Reserve Bank of India
- Human Development Index and per capita income are strongly, but not perfectly, positively related across countries.Why it matters: shows that correlation is not identity.Source: UNDP Human Development Reports
- Spurious correlation arises when two trending series correlate without a link; test with theory and controls.Why it matters: an essay and critique point.Source: Standard statistics textbooks
- Okun’s law reports a negative relation between unemployment and output gap in many economies.Why it matters: macro link for essay answers.Source: Standard macroeconomics textbooks
- The Economic Survey commonly uses scatter plots to relate indicators such as spending and outcomes across states.Why it matters: learn to read such plots.Source: Economic Survey
- Rank correlation suits composite rankings, such as ranking states on an index.Why it matters: think of Niti Aayog’s index-based rankings of states.Source: NITI Aayog
These are revision notes, not official predictions of exam content.
Quick Revision
- Correlation measures direction and strength of linear association.
- Correlation ≠ causation.
- −1 ≤ r ≤ +1; r has no unit.
- r is unchanged by change of origin and scale.
- Spearman: rs = 1 − 6Σd² ÷ n(n² − 1).
- Tie correction: (m³ − m) ÷ 12 per group.
- Draw the scatter first.
Related Glossary Terms and Chapters
Glossary:
CorrelationScatter diagramSpearman rank correlationKarl Pearson coefficient
Related chapters: Measures of Central Tendency, which summarise one variable; Index Numbers, where price indices are compared with money supply.
Sources
- NCERT textbooks: Statistics for Economics, Class 11, National Council of Educational Research and Training. Syllabus reference. Accessed 23 October 2026.
- Human Development Reports, United Nations Development Programme. Accessed 23 October 2026.
- Economic Survey, Ministry of Finance, Government of India. Accessed 23 October 2026.
- Reserve Bank of India. Accessed 23 October 2026.
- NITI Aayog, Government of India. Accessed 23 October 2026.
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