NCERT Solutions for Class 11 Economics Chapter 6: Correlation (NCERT 2026–27)
These Class 11 Economics Chapter 6 solutions cover Correlation from the NCERT textbook Statistics for Economics (continued for the 2026–27 session). The chapter explains how to examine the relationship between two variables — its direction (positive or negative) and its intensity — using three tools: the scatter diagram, Karl Pearson’s coefficient of correlation and Spearman’s rank correlation. Below you get every end-of-chapter Exercise question reproduced verbatim, with theory answers in exam-ready prose and all numericals solved step by step with full working tables and verified values, plus key formulas, extra practice, MCQs, Assertion–Reason and FAQs.
Class: 11Subject: EconomicsBook: Statistics for EconomicsChapter: 6Chapter Name: CorrelationSession: 2026–27
Chapter 6, Correlation, studies the relationship between two variables, such as temperature and ice-cream sales, or the supply of tomatoes and their price. Correlation measures covariation, not causation — it tells us whether two variables move together and how strongly, but it never proves that one causes the other. A relationship is positive when both variables move in the same direction and negative when they move in opposite directions. Three tools are used: the scatter diagram gives a visual picture without a numerical value; Karl Pearson’s coefficient of correlation (r) gives a precise measure of linear relationship and lies between −1 and +1; and Spearman’s rank correlation is used when variables are based on ranks or attributes such as honesty or beauty that cannot be measured exactly, and when the data contain extreme values. Repeated ranks need correction factors. The chapter ends by reminding us that statistical measures are no substitute for common sense in interpreting data.
Key Terms & Important Formulas
Correlation: a statistical tool that studies and measures the direction and intensity of the relationship between two variables. It measures covariation, not causation.
Positive correlation: the variables move in the same direction (e.g. income and consumption). Negative correlation: they move in opposite directions (e.g. price of apples and demand for apples).
Scatter diagram: a graph on which paired values of two variables are plotted as points; the closeness and overall direction of the points show the nature of the relationship without a numerical value.
Karl Pearson’s coefficient (r): the product-moment correlation coefficient; a precise numerical measure of linear relationship between two variables. It has no unit, is unaffected by change of origin and scale, and always lies in −1 ≤ r ≤ 1.
Spearman’s rank correlation (rs): a correlation coefficient computed from ranks instead of values; useful for qualitative attributes, when ranks are given, and when data contain extreme values. It also lies between −1 and +1.
Covariance: the average of the products of the deviations of X and Y from their means; its sign decides the sign of r.
Karl Pearson’s r (actual mean): r = Σ(X − X̄)(Y − Ȳ) ÷ √[Σ(X − X̄)2 × Σ(Y − Ȳ)2]
Karl Pearson’s r (direct/raw-score): r = [NΣXY − (ΣX)(ΣY)] ÷ √[ (NΣX2 − (ΣX)2) × (NΣY2 − (ΣY)2) ]
Spearman’s rank correlation: rs = 1 − [ 6ΣD2 ÷ (n3 − n) ], where D = difference between paired ranks and n = number of pairs.
Spearman’s rs with repeated ranks: rs = 1 − [ 6{ΣD2 + Σ(m3 − m)/12} ÷ (n3 − n) ], where m = number of times each rank is repeated.
NCERT Exercises — Full Solutions
All questions below are reproduced verbatim from the NCERT textbook’s end-of-chapter Exercises. Answers are original; numericals are solved step by step with working tables and verified values.
1. The unit of correlation coefficient between height in feet and weight in kgs is
(i) kg/feet(ii) percentage(iii) non-existent
ANSWER(iii) non-existent. The correlation coefficient (r) is a pure number with no unit, because it is a ratio in which the units of the two variables cancel out. So r between height in feet and weight in kg has no unit at all.
2. The range of simple correlation coefficient is
(i) 0 to infinity(ii) minus one to plus one(iii) minus infinity to infinity
ANSWER(ii) minus one to plus one. The value of r always lies between −1 and +1, i.e. −1 ≤ r ≤ 1. If a calculation gives a value outside this range, it indicates an error.
3. If rxy is positive the relation between X and Y is of the type
(i) When Y increases X increases(ii) When Y decreases X increases(iii) When Y increases X does not change
ANSWER(i) When Y increases X increases. A positive r means the two variables move in the same direction — both rise together and both fall together.
4. If rxy = 0 the variable X and Y are
(i) linearly related(ii) not linearly related(iii) independent
ANSWER(ii) not linearly related. r = 0 means there is no linear relation between X and Y. It does not necessarily mean they are independent, because a non-linear relationship may still exist.
5. Of the following three measures which can measure any type of relationship
(i) Karl Pearson’s coefficient of correlation(ii) Spearman’s rank correlation(iii) Scatter diagram
ANSWER(iii) Scatter diagram. A scatter diagram can show any type of relationship — linear or non-linear — because it simply plots the points; the two coefficients measure only linear (or rank-linear) relationships.
6. If precisely measured data are available the simple correlation coefficient is
(i) more accurate than rank correlation coefficient(ii) less accurate than rank correlation coefficient(iii) as accurate as the rank correlation coefficient
ANSWER(i) more accurate than rank correlation coefficient. When data are precisely measured, Karl Pearson’s coefficient uses all the information in the data, whereas rank correlation uses only the ranks and so loses some information.
7. Why is r preferred to covariance as a measure of association?
ANSWERThe correlation coefficient r is preferred to covariance because covariance depends on the units of measurement of the two variables, so its value can be very large or very small simply because of the units chosen, and it has no fixed limits. The coefficient r, on the other hand, is a pure number with no unit; it is obtained by dividing the covariance by the product of the two standard deviations, so the units cancel out.Because r always lies between −1 and +1, two correlation coefficients can be directly compared to judge which relationship is stronger. Covariance cannot be compared in this way. Hence r is a far more convenient and meaningful measure of the degree of association.
8. Can r lie outside the −1 and 1 range depending on the type of data?
ANSWERNo. The value of the correlation coefficient can never lie outside the range −1 to +1, whatever the type of data. By its very construction (covariance divided by the product of the standard deviations), r is mathematically bound by −1 ≤ r ≤ 1.If, in any exercise, you obtain a value such as 1.2 or −1.5, it does not mean the data are unusual — it means there is a mistake in the calculation, and the working must be checked.
9. Does correlation imply causation?
ANSWERNo, correlation does not imply causation. Correlation only measures covariation — the fact that two variables tend to change together — not that one causes the other.A high correlation may arise for several reasons: it may be a genuine cause-and-effect relation, a pure coincidence (e.g. the arrival of migratory birds and the local birth rate), or the effect of a third variable on both (e.g. rising temperature causes both brisk ice-cream sales and more deaths by drowning, so these two appear correlated without one causing the other). Therefore correlation must never be interpreted as cause and effect.
10. When is rank correlation more precise than simple correlation coefficient?
ANSWERRank correlation is more useful (and effectively more precise) than the simple correlation coefficient in the following situations:(i) when the variables cannot be measured numerically but can only be ranked — such as honesty, beauty or intelligence; (ii) when the data contain extreme values (outliers), because rank correlation is not affected by them while Karl Pearson’s r is distorted; and (iii) when the relationship is non-linear but its direction is clear. In such cases ranks give a more reliable picture of association.
11. Does zero correlation mean independence?
ANSWERNo. Zero correlation (r = 0) means only that there is no linear relationship between the two variables. It does not necessarily mean that the variables are independent.Two variables can have r = 0 and yet be related in a non-linear way. For example, for X = −3, −2, −1, 1, 2, 3 and Y = 9, 4, 1, 1, 4, 9 (where Y = X2), the correlation coefficient is 0, but Y clearly depends on X. So zero correlation rules out a linear relation, not every relation.
12. Can simple correlation coefficient measure any type of relationship?
ANSWERNo. The simple (Karl Pearson’s) correlation coefficient can measure only a linear relationship between two variables — one that can be represented by a straight line.If the true relationship between the variables is non-linear (curved), Karl Pearson’s coefficient is misleading and should not be used. To examine such relationships, a scatter diagram should be drawn first. Hence the simple correlation coefficient cannot measure every type of relationship.
13. Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.
ANSWERThis is a data-collection activity, so your figures will vary. Record the daily price (Rs/kg) of any two vegetables over seven days, then apply Karl Pearson’s formula r = [NΣXY − (ΣX)(ΣY)] ÷ √[(NΣX2 − (ΣX)2)(NΣY2 − (ΣY)2)].Model interpretation: if the prices of two vegetables rise and fall together over the week (for example because of common weather or transport costs), r will be positive; if one rises while the other falls, r will be negative; and if there is no clear pattern, r will be close to zero. State which pair you found and whether the relationship was strong (r near ±1) or weak (r near 0). (Use your own collected data.)
14. Measure the height of your classmates. Ask them the height of their benchmate. Calculate the correlation coefficient of these two variables. Interpret the result.
ANSWERThis too is an activity with your own data. Take X = a student’s height and Y = that student’s benchmate’s height for, say, ten pairs, and compute r using Karl Pearson’s formula.Model interpretation: heights of randomly seated benchmates are usually not related, so r will normally be close to zero, showing no linear relationship. If, however, students were seated by height, you might find a positive r. Report your value and interpret its sign and strength accordingly. (Use your own measured data.)
15. List some variables where accurate measurement is difficult.
ANSWERVariables that are qualitative attributes are difficult to measure accurately and can usually only be ranked. Examples include: intelligence, honesty, beauty, physical appearance, bravery, kindness, leadership ability, friendliness, efficiency of workers, employee morale and consumer satisfaction. For such variables, Spearman’s rank correlation is used instead of Karl Pearson’s coefficient.
16. Interpret the values of r as 1, −1 and 0.
ANSWERr = 1: perfect positive correlation. The two variables move in exactly the same direction in a fixed proportion, and on a scatter diagram all the points lie on a straight upward-sloping line.r = −1: perfect negative correlation. The two variables move in exactly opposite directions in a fixed proportion, and all points lie on a straight downward-sloping line.r = 0: no linear correlation. There is no linear relationship between the variables; the points are scattered with no upward or downward straight-line trend (though a non-linear relation may still exist).
17. Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
ANSWERThe rank correlation coefficient differs from the Pearsonian coefficient because the two are computed from different information. Karl Pearson’s coefficient uses the actual measured values of the variables and therefore all the information in the data, whereas Spearman’s coefficient uses only the ranks of the items.Because ranking discards the exact magnitudes (and the gaps between consecutive values are usually not equal), rank correlation generally gives a less accurate value, and in general rs ≤ r. The two would give identical results only if the first differences of the ordered values were constant. Also, rank correlation is unaffected by extreme values, while Pearson’s r is, so the two diverge whenever outliers are present.
18. Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y)
X 65 66 57 67 68 69 70 72Y 67 56 65 68 72 72 69 71
ANSWERWe use Karl Pearson’s direct (raw-score) formula with N = 8. First build the working table.
X
Y
X2
Y2
XY
65
67
4225
4489
4355
66
56
4356
3136
3696
57
65
3249
4225
3705
67
68
4489
4624
4556
68
72
4624
5184
4896
69
72
4761
5184
4968
70
69
4900
4761
4830
72
71
5184
5041
5112
ΣX = 534
ΣY = 540
ΣX2 = 35788
ΣY2 = 36644
ΣXY = 36118
Numerator: NΣXY − (ΣX)(ΣY) = 8 × 36118 − 534 × 540 = 288944 − 288360 = 584.First bracket: NΣX2 − (ΣX)2 = 8 × 35788 − (534)2 = 286304 − 285156 = 1148.Second bracket: NΣY2 − (ΣY)2 = 8 × 36644 − (540)2 = 293152 − 291600 = 1552.Denominator: √(1148 × 1552) = √1781696 ≈ 1334.80.r = 584 ÷ 1334.80 ≈ 0.44. The heights of fathers and sons are positively correlated, but the relationship is only moderate (the value is well short of +1).Note: some older NCERT printings show the answer as r = 0.603. Re-working the given data carefully by both the direct method and the deviation method (X̄ = 66.75, Ȳ = 67.5) gives the verified value r ≈ 0.44, so the printed 0.603 is a textbook misprint for this data set.
19. Calculate the correlation coefficient between X and Y and comment on their relationship:
X −3 −2 −1 1 2 3Y 9 4 1 1 4 9
ANSWERHere N = 6. Notice ΣX = 0, which simplifies the direct formula. Build the table.
X
Y
X2
Y2
XY
−3
9
9
81
−27
−2
4
4
16
−8
−1
1
1
1
−1
1
1
1
1
1
2
4
4
16
8
3
9
9
81
27
ΣX = 0
ΣY = 28
ΣX2 = 28
ΣY2 = 196
ΣXY = 0
Numerator: NΣXY − (ΣX)(ΣY) = 6 × 0 − 0 × 28 = 0.Since the numerator is 0, r = 0.Comment: the correlation coefficient is zero, so there is no linear relationship between X and Y. However, the data are not independent — clearly Y = X2, a perfect non-linear (parabolic) relation. This shows that r = 0 rules out only a linear relation, not every relation.
20. Calculate the correlation coefficient between X and Y and comment on their relationship
X 1 3 4 5 7 8Y 2 6 8 10 14 16
ANSWERHere N = 6. Build the working table.
X
Y
X2
Y2
XY
1
2
1
4
2
3
6
9
36
18
4
8
16
64
32
5
10
25
100
50
7
14
49
196
98
8
16
64
256
128
ΣX = 28
ΣY = 56
ΣX2 = 164
ΣY2 = 656
ΣXY = 328
Numerator: NΣXY − (ΣX)(ΣY) = 6 × 328 − 28 × 56 = 1968 − 1568 = 400.First bracket: NΣX2 − (ΣX)2 = 6 × 164 − (28)2 = 984 − 784 = 200.Second bracket: NΣY2 − (ΣY)2 = 6 × 656 − (56)2 = 3936 − 3136 = 800.Denominator: √(200 × 800) = √160000 = 400.r = 400 ÷ 400 = 1.Comment: r = +1 shows perfect positive correlation. X and Y move in exactly the same direction in a fixed proportion (here Y = 2X); on a scatter diagram all points lie on a single upward-sloping straight line.
Extra Practice Questions
Short Answer Type Questions
Q1. Define correlation. Does it measure causation?
ANSWERCorrelation is a statistical tool that studies and measures the direction and intensity of the relationship between two variables. It measures only covariation — how variables move together — and never proves that one variable causes the other. Hence correlation does not measure causation.
Q2. Distinguish between positive and negative correlation with one example each.
ANSWERIn positive correlation both variables move in the same direction — e.g. as income rises, consumption also rises. In negative correlation the variables move in opposite directions — e.g. as the price of apples rises, the demand for apples falls.
Q3. What is a scatter diagram and what is its main advantage?
ANSWERA scatter diagram plots the paired values of two variables as points on a graph. Its main advantage is that it gives a quick visual idea of the nature and direction of the relationship — including non-linear ones — without any calculation, though it does not give a numerical value.
Q4. State any two properties of the correlation coefficient.
ANSWER(i) r is a pure number with no unit and is independent of the units of measurement. (ii) The value of r always lies between −1 and +1, and it is unaffected by a change of origin and scale.
Q5. When is Spearman’s rank correlation preferred over Karl Pearson’s coefficient?
ANSWERSpearman’s rank correlation is preferred when the variables are qualitative attributes that can only be ranked (such as honesty or beauty), when the data contain extreme values (outliers) that would distort Pearson’s r, and when only ranks are available rather than exact measurements.
Long Answer Type Questions
Q1. Explain the three main techniques of measuring correlation.
ANSWERThree tools are used to study correlation. (1) The scatter diagram plots paired values as points; the closeness and overall slope of the points reveal whether the correlation is positive, negative, perfect or absent, and it can show non-linear relations too — but it gives no numerical value. (2) Karl Pearson’s coefficient of correlation (r) gives a precise numerical measure of the linear relationship between two variables; it is computed from the covariance divided by the product of the standard deviations, is a pure number, and lies between −1 and +1. It should be used only when the relation is linear. (3) Spearman’s rank correlation (rs) measures the linear association between the ranks of items; it is used for attributes that cannot be measured exactly, when the data contain extreme values, or when only ranks are given. Together these tools describe both the direction and the intensity of a relationship.
Q2. The ranks of five students given by two judges are A: 1,2,3,4,5 and B: 2,4,1,5,3. Calculate the rank correlation coefficient.
ANSWERWe use Spearman’s formula rs = 1 − [6ΣD2 ÷ (n3 − n)] with n = 5.
Rank A
Rank B
D = A − B
D2
1
2
−1
1
2
4
−2
4
3
1
2
4
4
5
−1
1
5
3
2
4
ΣD2
14
n3 − n = 125 − 5 = 120. So rs = 1 − (6 × 14)/120 = 1 − 84/120 = 1 − 0.70 = 0.30. There is a weak positive agreement between the two judges’ rankings.
Q3. Explain why correlation does not imply causation, using suitable examples.
ANSWERCorrelation measures only that two variables vary together; it cannot tell us why they vary together, so it does not prove cause and effect. A high correlation can arise in three ways. First, it may be a genuine cause-and-effect relation — e.g. years of schooling and yield per acre. Second, it may be pure coincidence — e.g. the arrival of migratory birds in a sanctuary and the birth rate of the locality are correlated but have nothing to do with each other. Third, a hidden third variable may influence both — e.g. brisk sales of ice-cream and deaths by drowning are positively correlated, but neither causes the other; rising temperature drives up both ice-cream sales and the number of people swimming. There can also be a case where the cause-and-effect runs the opposite way to what we assume. For all these reasons, statisticians warn that correlation is no substitute for common sense, and it must never be interpreted as causation without further investigation.
MCQs & Assertion–Reason
1. Correlation measures:
(a) causation (b) covariation (c) the mean (d) dispersion
2. The correlation coefficient r is:
(a) measured in the units of X (b) measured in the units of Y (c) a pure number with no unit (d) always positive
3. The value of r lies between:
(a) 0 and 1 (b) −1 and +1 (c) −∞ and +∞ (d) 0 and ∞
4. Which technique can show a non-linear relationship?
For each Assertion–Reason question, choose: (A) Both true and the Reason correctly explains the Assertion; (B) Both true but the Reason is not the correct explanation; (C) Assertion true, Reason false; (D) Assertion false, Reason true.
A-R 1. Assertion: The correlation coefficient has no unit.
Reason: r is a ratio in which the units of the two variables cancel out, making it a pure number.
A-R 2. Assertion: A high correlation between two variables proves that one causes the other.
Reason: Correlation measures covariation, not causation.
A-R 3. Assertion: Zero correlation means the variables are always independent.
Reason: A non-linear relation can exist between two variables even when r = 0.
A-R 4. Assertion: Spearman’s rank correlation is useful when data contain extreme values.
Reason: Rank correlation is not affected by extreme values, unlike Karl Pearson’s coefficient.
A-R 5. Assertion: If a calculation gives r = 1.4, the data are highly correlated.
Reason: The value of r can never lie outside the range −1 to +1.
Answer key: 1-(A), 2-(D), 3-(D), 4-(A), 5-(D).
Exam Tips & Common Mistakes
How to score full marks in this chapter
Always write the formula first, then the working table, then the substitution, then the final value with one line of interpretation — markers award method marks even if arithmetic slips. Remember the three key facts: r has no unit, lies between −1 and +1, and measures covariation not causation. For numericals, prefer the direct (raw-score) formula when means are not whole numbers, and the deviation formula when means are whole numbers. In Spearman’s method, when ranks repeat, do not forget the correction factor Σ(m3 − m)/12. Quote textbook examples — ice-cream and drowning, migratory birds and birth rate — to explain “correlation is not causation”.
Common mistakes to avoid
Writing a unit (like kg/feet) for r — it is a pure number with no unit.
Concluding cause and effect from a high correlation — correlation measures only covariation.
Saying r = 0 means the variables are independent — it only rules out a linear relation.
Using Karl Pearson’s coefficient for a clearly non-linear relationship.
Forgetting the correction factor for repeated ranks in Spearman’s formula.
Mistaking squares: writing (ΣX)2 instead of ΣX2, or vice versa, in the direct formula.
Frequently Asked Questions
What is Chapter 6 of Class 11 Economics (Statistics for Economics) about?
Chapter 6, Correlation, explains how to measure the relationship between two variables — its direction and intensity — using the scatter diagram, Karl Pearson’s coefficient of correlation and Spearman’s rank correlation. It stresses that correlation measures covariation, not causation.
What is the difference between Karl Pearson’s and Spearman’s correlation?
Karl Pearson’s coefficient uses the actual measured values and gives a precise measure of linear relationship, while Spearman’s rank correlation uses only the ranks of items. Spearman’s method is preferred for qualitative attributes and when the data contain extreme values, but it is generally less accurate than Pearson’s method.
Why does correlation not imply causation?
A correlation only shows that two variables move together. The link may be a coincidence, or both variables may be influenced by a hidden third factor (for example, rising temperature raises both ice-cream sales and deaths by drowning). So a high correlation never proves that one variable causes the other.
