Correlation: ISC Class 11 Economics Notes and Numericals

What Correlation is about

Weightage: Correlation is part of the unit Statistics of the ISC Class XI Economics syllabus. CISCE gives no Class XI unit weightage for Economics: Paper I (Theory) carries 80 marks, and the school sets the paper.Source: CISCE ISC Economics syllabus, examination year 2028, Class XI paper pattern

Key concepts

Meaning and significance of correlation

Correlation is a statistical measure of the degree and direction of the linear relationship between two variables: when one changes, does the other tend to change, and by how much?

The coefficient of correlation (r) lies between −1 and +1.

Significance (uses) in economics:

1. Tests economic relationships: price and quantity demanded (expected negative), income and consumption (positive).
2. Helps prediction: if two variables are highly correlated, one helps estimate the other (the basis of regression).
3. Helps policy: knowing that credit and investment are positively related guides monetary policy.
4. Helps business decisions: advertising and sales, price and sales.

Correlation and causation: correlation shows that variables move together; it does not prove that one causes the other. Two variables can be correlated because a third factor drives both (ice-cream sales and drowning cases both rise in summer), or by pure chance (spurious or nonsense correlation).

Worked example

Suppose a study finds a high positive correlation between the number of umbrellas sold and the number of people with colds in a city. Does buying umbrellas cause colds? [1 mark]

Try it first, then show the solutionSolution

1. Both rise in the rainy season: a third factor (rain or season) affects both.

Answer: No. The correlation is due to a common cause (the rainy season), not causation.

Types and degrees of correlation

Types:

1. Positive (direct): both variables move in the same direction (income and consumption; price and supply).
2. Negative (inverse): they move in opposite directions (price and quantity demanded; interest rate and investment).
3. Linear: the ratio of change is constant; points lie near a straight line.
4. Non-linear (curvilinear): the ratio of change varies; points follow a curve.
5. Simple: two variables; partial and multiple: more than two (beyond this syllabus).

Degrees (a common rule of thumb):

Value of r	Degree
+1 or −1	Perfect correlation
0.75 to 1 (or −0.75 to −1)	High degree
0.50 to 0.75	Moderate degree
0.25 to 0.50	Low degree
Below 0.25	Very low degree
0	No (linear) correlation

An r of 0 means no linear relationship; a non-linear relationship may still exist.

Worked example

Classify: (i) r = −0.92 between price and demand; (ii) r = 0.40 between study hours and marks. [1 mark]

Try it first, then show the solutionSolution

1. (i) Negative and above 0.75 in size: high negative correlation.
2. (ii) Positive between 0.25 and 0.5: low positive correlation.

Answer: (i) High negative; (ii) low positive.

Scatter diagram

A scatter diagram plots each pair of values (X, Y) as a dot. The pattern of dots shows the direction and approximate strength of correlation.

Pattern of dots	Correlation
On a rising straight line	Perfect positive (r = +1)
Closely around a rising line	High positive
Widely around a rising line	Low positive
On a falling straight line	Perfect negative (r = −1)
Closely around a falling line	High negative
No pattern, scattered	No correlation (r ≈ 0)
Along a curve	Non-linear

Merits: simple and visual; shows direction, outliers and non-linearity at a glance; not affected by extreme values the way r is.

Demerits: gives no exact numerical value; judgement of 'closeness' is subjective.

Worked example

Suppose plotting rainfall against wheat yield gives dots that rise at first and then fall at very high rainfall. What does this show? [1 mark]

Try it first, then show the solutionSolution

1. The dots follow a curve, not a straight line.

Answer: A non-linear (curvilinear) relationship: yield rises with rain up to a point, then falls with excess rain.

Karl Pearson's coefficient of correlation

Karl Pearson's coefficient (the product moment coefficient) for two variables with ungrouped data:

Actual mean method: $r=\frac{\sum xy}{\sqrt{\sum x^2\times \sum y^2}}$, where $x=X-\bar{X}$ and $y=Y-\bar{Y}$.

Assumed mean (short-cut) method: $r=\frac{N\sum dxdy-\sum dx\sum dy}{\sqrt{N\sum d{x}^2-(\sum dx{)}^2}\times \sqrt{N\sum d{y}^2-(\sum dy{)}^2}}$

Example: income (X, ₹ thousand) and consumption (Y, ₹ thousand) of 5 households:

X	Y	x	y	x²	y²	xy
20	18	−10	−7	100	49	70
25	21	−5	−4	25	16	20
30	26	0	1	0	1	0
35	28	5	3	25	9	15
40	32	10	7	100	49	70

Means: 30 and 25. Σx² = 250, Σy² = 124, Σxy = 175.

r = 175 ÷ √(250 × 124) = 175 ÷ √31,000 = 175 ÷ 176.07 = +0.99: very high positive correlation, as theory predicts.

Properties: (1) −1 ≤ r ≤ +1; (2) independent of change of origin and scale (subtracting or dividing by constants does not change r, which is why the short-cut methods work); (3) r is the same whether X is correlated with Y or Y with X; (4) r = 0 means no linear relationship.

Merits: gives both direction and degree as a precise number. Demerits: assumes a linear relationship; affected by extreme values; tedious; can be misread as causation.

Worked example

Suppose that for the rate of interest (X) and private investment (Y) over several years, Σxy = −96, Σx² = 40 and Σy² = 360 (deviations from the means). Find r and interpret it. [2 marks]

Try it first, then show the solutionSolution

1. r = −96 ÷ √(40 × 360) = −96 ÷ √14,400 = −96 ÷ 120 = −0.80.
2. High negative correlation (above 0.75 in size): years of higher interest rates go with lower investment, as theory predicts.

Answer: r = −0.80: high negative correlation between the interest rate and investment.

Spearman's rank correlation

When variables are qualitative (beauty, honesty, taste) or only ranks are available, or when there are extreme values, use Spearman's rank correlation coefficient:

$r_s=1-\frac{6\sum D^2}{N^3-N}$

where D = difference between the two ranks of each item and N = number of pairs.

Steps: (1) rank each variable (highest = 1, or lowest = 1, but the same rule for both); (2) find D = R1 − R2 (check that ΣD = 0); (3) square and add; (4) apply the formula.

Example (no ties): two judges rank 10 dancers:

Dancer	A	B	C	D	E	F	G	H	I	J
Judge 1	1	2	3	4	5	6	7	8	9	10
Judge 2	3	1	4	2	6	5	9	7	10	8
D²	4	1	1	4	1	1	4	1	1	4

ΣD² = 22; $r_s$ = 1 − (6 × 22) ÷ (1,000 − 10) = 1 − 132 ÷ 990 = 0.87: high agreement between the judges.

Tied ranks: give each tied item the average of the ranks they would occupy (two items tied for 2nd and 3rd both get 2.5). Then add a correction factor for each tie group:

$r_s=1-\frac{6\left[\sum D^2+\sum \frac{m^3-m}{12}\right]}{N^3-N}$

where m = number of items sharing a rank in a group.

Merits: simple; suits qualitative data; not affected by extreme values. Demerits: uses ranks, so information in the actual values is lost; tedious for large N.

Worked example

Suppose for 7 pairs of ranks ΣD² = 14. Find Spearman's rank correlation. [1 mark]

Try it first, then show the solutionSolution

1. r_s = 1 − (6 × 14) ÷ (343 − 7) = 1 − 84 ÷ 336 = 1 − 0.25 = 0.75.

Answer: 0.75 (high positive).

Choosing between Pearson and Spearman

Basis	Karl Pearson's r	Spearman's r_s
Data	Quantitative values	Ranks or qualitative data
Uses	Actual values	Only the order
Extreme values	Affected	Not much affected
Assumption	Linear relationship	Monotonic relationship (consistently rising or falling)
Calculation	Longer	Simpler

For the same data the two can differ, because ranking discards the size of differences between values.

Interpreting r: always state direction (positive or negative), degree (high, moderate, low) and the economic meaning (for example, 'consumption rises with income'). Add a caution that correlation is not causation where relevant.

Worked example

Which coefficient would you use to measure the agreement between two food critics who rank 8 restaurants? Why? [1 mark]

Try it first, then show the solutionSolution

1. The data are ranks of a qualitative attribute (taste).

Answer: Spearman's rank correlation, because only ranks of a qualitative characteristic are available.

Correlation in real life

Each concept, met in an everyday situation. Read the situation, answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Topics in this chapter

• Spearman's Rank Correlation, Tied Ranks and Choosing the Method
• Correlation: Types, Scatter Diagram and Karl Pearson's Coefficient

Correlation: key facts

Karl Pearson (actual mean)

$r=\frac{\sum xy}{\sqrt{\sum x^2\times \sum y^2}}$

x and y are deviations from the respective means.

Karl Pearson (assumed mean)

$r=\frac{N\sum dxdy-\sum dx\sum dy}{\sqrt{N\sum d{x}^2-(\sum dx{)}^2}\times \sqrt{N\sum d{y}^2-(\sum dy{)}^2}}$

Spearman's rank correlation

$r_s=1-\frac{6\sum D^2}{N^3-N}$

Spearman with tied ranks

$r_s=1-\frac{6\left[\sum D^2+\sum \frac{m^3-m}{12}\right]}{N^3-N}$

One correction term for each group of tied ranks.

Correlation: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Correlation

The question patterns to expect, and the method that works for each.

Tricks to remember Correlation

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Same direction, plus; opposite, minus

  The sign of r tells the direction; the size tells the strength.

  Use it when: Interpreting r.

• Spearman: 1 − 6ΣD² over N cubed minus N

  'One minus six D-squared, over n-cubed minus n.'

  Use it when: Spearman's formula.

• Ties share the average rank

  Two tied for 3rd and 4th → 3.5 each; add (m³ − m)/12 for each tie.

  Use it when: Tied ranks.

• Correlation ≠ causation

  Ice-cream and drowning both rise in summer.

  Use it when: Limits of correlation.

Correlation: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section C style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 2 + 1 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Correlation

The slips examiners see most often, and what to write instead.

• ✗Mistake: Ranking one variable from highest and the other from lowest

  Opposite ranking reverses the sign of r_s.

  ✓Do this: Use the same rule (highest = 1) for both.

  Usually costs: 2 marks.

• ✗Mistake: Forgetting the correction for tied ranks

  Without it, r_s is overstated.

  ✓Do this: Add (m³ − m) ÷ 12 for every tie group in either series.

  Usually costs: 1 mark.

• ✗Mistake: Getting r greater than 1

  A sign of an arithmetic slip; r must lie between −1 and +1.

  ✓Do this: Recheck Σx², Σy² (always positive) and Σxy.

  Usually costs: 1-2 marks.

• ✗Mistake: Interpreting correlation as causation

  Correlation shows only association.

  ✓Do this: Say 'is associated with' and mention possible third factors.

  Usually costs: 1 mark.

Practice questions on Correlation

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   If the points of a scatter diagram lie on a straight line falling from left to right, the correlation is:
   (a) perfect positive
   (b) perfect negative
   (c) zero
   (d) low negative

   Hint

   Exactly on a falling line.

   Answer

   (b) perfect negative.

2. Q2 · 1 mark · Very short answerExam level

   Give one example each of positive and negative correlation from economics.

   Hint

   Income; price.

   Answer

   Positive: income and consumption (or price and supply). Negative: price and quantity demanded (or interest rate and investment).

3. Q3 · 1 mark · Very short answerExam level

   Calculate Spearman's rank correlation when ΣD² = 30 and N = 10.

   Hint

   N³ − N = 990.

   Answer

   r_s = 1 − 180 ÷ 990 = 0.82.

4. Q4 · 4 marks · Short answerHarder exam level

   Calculate Karl Pearson's r for X: 3, 5, 7, 9, 11 and Y: 15, 12, 10, 6, 2.

   Hint

   Means 7 and 9.

   Answer

   x = −4, −2, 0, 2, 4; y = 6, 3, 1, −3, −7; Σx² = 40; Σy² = 36 + 9 + 1 + 9 + 49 = 104; Σxy = −24 − 6 + 0 − 6 − 28 = −64. r = −64 ÷ √(40 × 104) = −64 ÷ 64.50 = −0.99.

5. Q5 · 2 marks · Short answerHarder exam level

   State four properties of Karl Pearson's coefficient of correlation.

   Hint

   Range, origin, scale, symmetry.

   Answer

   It lies between −1 and +1; it is independent of change of origin; it is independent of change of scale; it is symmetric (r between X and Y equals r between Y and X). Also, r = 0 means no linear relationship.

6. Q6 · 2 marks · Short answerHarder exam level

   Suppose Σdx = 0, Σdy = 0, Σdx² = 28, Σdy² = 32 and Σdxdy = 20 for 7 pairs (deviations from assumed means equal to the actual means). Find r.

   Hint

   Assumed = actual, so the simple formula applies.

   Answer

   r = 20 ÷ √(28 × 32) = 20 ÷ √896 = 20 ÷ 29.93 = 0.67: moderate positive correlation.

7. Q7 · 4 marks · Short answerHOTS / competency

   Suppose three judges ranked 6 singers. Judges 1 and 2 have ΣD² = 50, judges 1 and 3 have ΣD² = 14, and judges 2 and 3 have ΣD² = 70. Which pair agrees most and which least?

   Hint

   N³ − N = 210.

   Answer

   r(1,2) = 1 − 300 ÷ 210 = −0.43; r(1,3) = 1 − 84 ÷ 210 = 0.60; r(2,3) = 1 − 420 ÷ 210 = −1.00. Judges 1 and 3 agree most (0.60); judges 2 and 3 have exactly opposite rankings (−1).

8. Q8 · 2 marks · Short answerHarder exam level

   What is spurious correlation? Give an example.

   Hint

   Nonsense or chance.

   Answer

   A correlation that has no meaningful cause-and-effect basis, arising by chance or through a third variable. Example: sales of ice cream and the number of drowning cases both rise in summer; neither causes the other.

9. Q9 · 6 marks · Long answerHOTS / competency

   'A high value of r between two variables proves that one causes the other.' Critically examine, with reference to the uses and limitations of correlation in economics.

   Hint

   Association vs causation; third factor; chance; reverse causation.

   Answer

   The statement is incorrect. Correlation measures how closely two variables move together, which is useful: it tests expected relationships (price and demand), helps prediction and guides policy. But a high r can arise from (1) a third factor affecting both (rain affects both umbrella sales and colds); (2) pure chance in small samples (spurious correlation); (3) reverse causation (does higher income cause more education, or education cause higher income?); (4) a mixture of influences. Also r measures only linear association; a strong curved relationship may give a low r. So economic theory and further analysis are needed before claiming causation.

10. Q10 · 2 marks · Short answerHOTS / competency

    Suppose the rank correlation between two variables for 8 pairs is 0.75. Find ΣD².

    Hint

    Rearrange the formula; N³ − N = 504.

    Answer

    0.75 = 1 − 6ΣD² ÷ 504 → 6ΣD² = 0.25 × 504 = 126 → ΣD² = 21.

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. 1.Range of r?Show answer

   −1 to +1.

2. 2.Pearson's formula (actual means)?Show answer

   Σxy ÷ √(Σx² × Σy²).

3. 3.Spearman's formula?Show answer

   1 − 6ΣD² ÷ (N³ − N).

4. 4.Correction for a tie of m items?Show answer

   (m³ − m) ÷ 12.

5. 5.Effect of change of scale on r?Show answer

   None.

6. 6.Correlation implies causation?Show answer

   No; it shows association only.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Correlation: direction and degree of linear relationship; −1 ≤ r ≤ +1.
• ☐Positive (same direction), negative (opposite), linear or non-linear.
• ☐Scatter diagram: visual, no exact value.
• ☐r = Σxy ÷ √(Σx²Σy²); independent of origin and scale.
• ☐r_s = 1 − 6ΣD² ÷ (N³ − N); ties: average rank + (m³ − m)/12.
• ☐Correlation ≠ causation.

Correlation: frequently asked questions