Chapter 6: Correlation

Q11Written answer

Does zero correlation mean independence?

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ANSWER No. 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.

Q12Written answer

Can simple correlation coefficient measure any type of relationship?

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ANSWER No. 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.

Q13Written answer

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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ANSWER This 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.)

Q14Written answer

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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ANSWER This 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.)

Q15Written answer

List some variables where accurate measurement is difficult.

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ANSWER Variables 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.

Q16Written answer

Interpret the values of r as 1, −1 and 0.

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ANSWER r = 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).

Q17Written answer

Why does rank correlation coefficient differ from Pearsonian correlation coefficient?

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ANSWER The 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 r_s ≤ 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.

Q18Written answer

Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y) X 65 66 57 67 68 69 70 72 Y 67 56 65 68 72 72 69 71

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ANSWER We 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.

Q19Written answer

Calculate the correlation coefficient between X and Y and comment on their relationship: X −3 −2 −1 1 2 3 Y 9 4 1 1 4 9

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ANSWER Here 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.

Q20Written answer

Calculate the correlation coefficient between X and Y and comment on their relationship X 1 3 4 5 7 8 Y 2 6 8 10 14 16

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ANSWER Here 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.

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