Chapter 6: Correlation

Q1Written answer

The unit of correlation coefficient between height in feet and weight in kgs is (i) kg/feet (ii) percentage (iii) non-existent

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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.

Q2Written answer

The range of simple correlation coefficient is (i) 0 to infinity (ii) minus one to plus one (iii) minus infinity to infinity

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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.

Q3Written answer

If r_xy 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

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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.

Q4Written answer

If r_xy = 0 the variable X and Y are (i) linearly related (ii) not linearly related (iii) independent

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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.

Q5Written answer

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

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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.

Q6Written answer

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

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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.

Q7Written answer

Why is r preferred to covariance as a measure of association?

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ANSWER The 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.

Q8Written answer

Can r lie outside the −1 and 1 range depending on the type of data?

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ANSWER No. 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.

Q9Written answer

Does correlation imply causation?

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ANSWER No, 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.

Q10Written answer

When is rank correlation more precise than simple correlation coefficient?

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ANSWER Rank 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.

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