Class | Subject | Book | Chapter | Session | Questions covered |
|---|---|---|---|---|---|
XI | Economics | Statistics for Economics (NCERT) | 6 – Correlation | 2026-27 | 20 of 20 |
Chapter at a glance
Topic | Key facts |
|---|---|
Correlation | Studies the direction and intensity of the relationship between two variables |
Types | Positive (move together), negative (move in opposite directions), linear and non-linear |
Scatter diagram | Visual tool; shows any kind of relationship but gives no numerical value |
Karl Pearson’s r | (deviations from means), or   |
Properties of r | Unit-free; lies between −1 and +1; unaffected by change of origin and scale; measures only linear relation |
Spearman’s rank correlation | ; used for qualitative data or when data are not precise  |
Caution | Correlation does not imply causation |
Exercise Questions
The unit of correlation coefficient between height in feet and weight in kgs is
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- kg/feet (ii) percentage (iii) non-existent
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The correlation coefficient is a pure number without units.
- non-existent
The range of simple correlation coefficient is
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- 0 to infinity (ii) minus one to plus one (iii) minus infinity to infinity
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- minus one to plus one
If is positive the relation between X and Y is of the type
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- When Y increases X increases (ii) When Y decreases X increases (iii) When Y increases X does not change
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Positive correlation means both variables move in the same direction.
- When Y increases X increases
If the variable X and Y are
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- linearly related (ii) not linearly related (iii) independent
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r = 0 means there is no linear relationship; a non-linear relationship may still exist, so the variables need not be independent.
- not linearly related
Of the following three measures which can measure any type of relationship
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- Karl Pearson’s coefficient of correlation (ii) Spearman’s rank correlation (iii) Scatter diagram
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Both coefficients measure only linear (or monotonic) relationships; the scatter diagram shows any type of relationship, linear or non-linear.
- Scatter diagram
If precisely measured data are available the simple correlation coefficient is
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- more accurate than rank correlation coefficient (ii) less accurate than rank correlation coefficient (iii) as accurate as the rank correlation coefficient
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With precise data, Karl Pearson’s r uses the actual values, whereas rank correlation uses only ranks and loses some information.
- more accurate than rank correlation coefficient
Why is r preferred to covariance as a measure of association?
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- Covariance depends on the units of measurement of the variables (e.g. cm × kg) and on their scale, so its value can be any size. It cannot be compared across data sets and does not tell how strong the relationship is.
- r is covariance divided by the product of the standard deviations; it is a unit-free pure number that lies between −1 and +1.
So r shows both the direction and degree of linear association and can be compared across different pairs of variables.
Because r is unit-free, lies between −1 and +1 and shows the strength of the relation, while covariance depends on units and scale.
Can r lie outside the –1 and 1 range depending on the type of data?
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No. Whatever the data, r always lies between −1 and +1. A value outside this range indicates a mistake in calculation.
No; r always lies between −1 and +1.
Does correlation imply causation?
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No. Correlation only measures the co-variation of two variables; it does not show that one causes the other. A high correlation may arise:
- by chance (spurious correlation, e.g. between the number of storks and births);
- because both are affected by a third variable (e.g. ice-cream sales and drowning both rise in summer);
- because of mutual influence, where the direction of cause is unclear.
Causation must be established by theory and other evidence.
No; correlation shows only co-movement, not cause and effect.
When is rank correlation more precise than simple correlation coefficient?
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Rank correlation is more appropriate (and gives better results) when:
- The variables are qualitative and cannot be measured precisely (beauty, honesty, intelligence), but can be ranked.
- Data are not precisely measured or contain errors.
- Data contain extreme values: ranks are unaffected by them, whereas r is distorted.
- The relationship is non-linear but monotonic.
When variables are qualitative or imprecisely measured, or when extreme values distort the data.
Does zero correlation mean independence?
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No. Zero correlation (r = 0) means only the absence of a linear relationship. The variables may still have a strong non-linear relationship. For example, for X = −3, −2, −1, 1, 2, 3 and Y = X², r = 0 (see Question 19), yet Y depends completely on X.
If two variables are independent, r is zero, but the reverse is not true.
No; r = 0 only rules out a linear relation; a non-linear relation may exist.
Can simple correlation coefficient measure any type of relationship?
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No. Karl Pearson’s coefficient measures only the linear relationship between two variables. It cannot properly measure a non-linear (curvilinear) relationship; for that, a scatter diagram is more useful.
No; it measures only linear relationships.
Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.
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This is a practical activity; each student uses their own data. A worked illustration with prices of three of the vegetables (₹ per kg, illustrative) is given below. The other pairs are worked out in the same way.
Day | Potato (X) | Onion (Y) | Tomato (Z) |
|---|---|---|---|
Mon | 30 | 40 | 60 |
Tue | 32 | 42 | 55 |
Wed | 31 | 41 | 58 |
Thu | 34 | 45 | 50 |
Fri | 35 | 46 | 48 |
Sat | 33 | 44 | 52 |
Sun | 36 | 48 | 45 |
Using :
- r (potato, onion) ≈ +0.99: very high positive correlation; the prices of potato and onion rose together during the week.
- r (potato, tomato) ≈ −1.00 (−0.998): very high negative correlation; tomato prices fell while potato prices rose.
Interpretation: prices of vegetables that are affected by the same supply conditions tend to move together; seasonal arrivals of one vegetable can make its price fall while others rise. With real data, the values will vary.
Compute r for each pair from the week’s prices; values near +1 show prices moving together, near −1 moving oppositely, near 0 unrelated.
Measure the height of your classmates. Ask them the height of their benchmate. Calculate the correlation coefficient of these two variables. Interpret the result.
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This is a practical activity. An illustration with eight students (heights in cm, illustrative):
Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
Height (X) | 160 | 152 | 165 | 170 | 158 | 162 | 155 | 168 |
Benchmate’s height (Y) | 158 | 155 | 163 | 166 | 160 | 159 | 152 | 170 |
Applying Karl Pearson’s formula gives r ≈ 0.90.
Interpretation: a high positive value would suggest that students of similar height sit together (for example, because seats are allotted by height). If seating is random, r will be close to zero, showing no relationship. Correlation here would not mean that one student’s height causes the other’s.
Calculate r from the paired heights; a high positive r means similar-height students sit together, r near 0 means no relation.
List some variables where accurate measurement is difficult.
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Variables that are qualitative or subjective: beauty, intelligence, honesty, happiness, leadership, health status, taste, loyalty, morality, satisfaction, preference for a product, and the quality of a performance. For such variables, rank correlation is used.
Beauty, intelligence, honesty, happiness, leadership, taste, satisfaction and similar qualitative attributes.
Interpret the values of r as 1, –1 and 0.
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- r = +1: perfect positive correlation; the variables move in the same direction in exact proportion; all points lie on a rising straight line.
- r = −1: perfect negative correlation; the variables move in opposite directions in exact proportion; all points lie on a falling straight line.
- r = 0: no linear correlation between the variables (there may be a non-linear relation).
+1: perfect positive; −1: perfect negative; 0: no linear correlation.
Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
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- Rank correlation uses only the ranks, not the actual values, so information about the size of the differences between values is lost.
- Extreme values affect Pearson’s r but not the ranks.
- Pearson’s r measures linear association between values, whereas rank correlation measures the association of order (monotonic relationship).
Hence, generally, the rank correlation coefficient is not equal to Karl Pearson’s coefficient for the same data, though they are usually close when the relationship is linear and there are no outliers.
Because rank correlation uses only ranks (losing information on actual values and ignoring extremes), whereas Pearson’s r uses actual values and measures linear association.
Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y)
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X | 65 | 66 | 57 | 67 | 68 | 69 | 70 | 72 |
|---|---|---|---|---|---|---|---|---|
Y | 67 | 56 | 65 | 68 | 72 | 72 | 69 | 71 |
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Given: N = 8. Take assumed means A = 68 for X and B = 69 for Y; , .
Formula:
X | Y | dx | dy | dx² | dy² | dx·dy |
|---|---|---|---|---|---|---|
65 | 67 | −3 | −2 | 9 | 4 | 6 |
66 | 56 | −2 | −13 | 4 | 169 | 26 |
57 | 65 | −11 | −4 | 121 | 16 | 44 |
67 | 68 | −1 | −1 | 1 | 1 | 1 |
68 | 72 | 0 | 3 | 0 | 9 | 0 |
69 | 72 | 1 | 3 | 1 | 9 | 3 |
70 | 69 | 2 | 0 | 4 | 0 | 0 |
72 | 71 | 4 | 2 | 16 | 4 | 8 |
−10 | −12 | 156 | 212 | 88 |
Numerator
Denominator
Note: the textbook gives r = 0.603. With the data as printed, the correct value is 0.44; we verified this by both the direct and the assumed-mean methods.
Comment: moderate positive correlation; taller fathers tend to have taller sons.
r ≈ 0.44 (moderate positive correlation)
Calculate the correlation coefficient between X and Y and comment on their relationship:
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X | −3 | −2 | −1 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
Y | 9 | 4 | 1 | 1 | 4 | 9 |
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; .
X | Y | XY | X² |
|---|---|---|---|
−3 | 9 | −27 | 9 |
−2 | 4 | −8 | 4 |
−1 | 1 | −1 | 1 |
1 | 1 | 1 | 1 |
2 | 4 | 8 | 4 |
3 | 9 | 27 | 9 |
0 | 28 | 0 | 28 |
Numerator:
Since the numerator is zero, r = 0.
Comment: there is no linear relationship. However, Y = X² exactly, so there is a perfect non-linear relationship. This shows that zero correlation does not mean independence.
r = 0: no linear relation, though Y = X² is a perfect non-linear relation.
Calculate the correlation coefficient between X and Y and comment on their relationship
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X | 1 | 3 | 4 | 5 | 7 | 8 |
|---|---|---|---|---|---|---|
Y | 2 | 6 | 8 | 10 | 14 | 16 |
Answer
N = 6; , , , , .
Comment: perfect positive correlation; Y = 2X in every case, so all points lie on a straight line.
r = 1 (perfect positive correlation; Y = 2X)
Topic checklist
- Meaning and types of correlation
- Scatter diagram
- Karl Pearson’s coefficient and its properties
- Spearman’s rank correlation (including tied ranks)
- Correlation vs causation; r = 0 vs independence