Correlation | ISC Class 11 Economics Notes

This note covers the meaning and significance of correlation, the types and degrees of correlation, the scatter diagram, Karl Pearson's coefficient of correlation (including the step deviation method) and Spearman's rank correlation coefficient, worked through numerical examples, along with the limits of what correlation can tell us.

What is correlation and why does it matter?

In earlier statistical work a single variable is summarised using measures such as the mean or standard deviation. Correlation analysis goes a step further: it examines the relationship between two variables together.

As the summer heat rises, hill stations get more visitors and ice-cream sales become brisk, so temperature is related to the number of visitors and to ice-cream sales.

Similarly, when the supply of tomatoes increases in a local market, price drops, sometimes from ₹40 per kg to ₹4 per kg or even less, so supply is related to price.

Definition: Correlation. Correlation studies and measures the direction and intensity of the relationship among variables. It measures covariation, not causation, and should never be interpreted as implying a cause-and-effect relationship.

Correlation analysis deals with questions such as whether there is any relationship between two variables, whether a change in the value of one is associated with a change in the other, whether both variables move in the same direction, and how strong the relationship is.

Some relationships, such as that between low rainfall and low agricultural productivity, may be given a cause-and-effect interpretation.

Others, such as the relation between the arrival of migratory birds in a sanctuary and the birth rate in the locality, are simple coincidence, and relationships such as that between the size of a person's shoes and the money in their pocket are difficult to explain even if they exist.

Sometimes a third variable's impact on two variables gives rise to a relation between the two.

A brisk sale of ice-creams may be related to a higher number of deaths by drowning, but the victims are not drowned by eating ice-creams: rising temperature leads to brisk ice-cream sales and, at the same time, more people going to swimming pools to beat the heat, which raises the number of drowning deaths.

Temperature is behind the high correlation between ice-cream sales and drowning deaths, which is why correlation must never be read as proof of causation.

What types of correlation are there?

For simplicity, correlation is usually assumed to be linear, meaning the relative movement of the two variables can be represented by a straight line on graph paper. Correlation is commonly classified into positive and negative correlation.

Positive and negative correlation

Positive correlation occurs when the variables move together in the same direction: when income rises, consumption also rises, and when income falls, consumption also falls. Sale of ice-cream and temperature move in the same direction.

Negative correlation occurs when the variables move in opposite directions: when the price of apples falls its demand increases, and when the price rises its demand decreases.

Spending more time in studying is associated with a decline in the chances of failing, while spending less time in studying is associated with an increase in the chances of scoring low marks or grades; these are instances of negative correlation, since the variables move in opposite directions.

Degree of correlation

The degree of correlation ranges from perfect correlation, where all points lie exactly on a line and correlation is said to be in unity, through strong or weak correlation depending on how closely the points cluster around a line, down to no correlation where there is no upward-rising or downward-sloping line around which the points are scattered.

A careful observation of how closely the plotted points cluster around a line, and of the direction in which that line runs, gives an idea of both the nature and the intensity of the relationship between the two variables being studied, whether they are economic variables such as income and consumption or two subjects' marks.

What is a scatter diagram?

A scatter diagram is a useful technique for visually examining the form of relationship between two variables, without calculating any numerical value. The values of the two variables are plotted as points on graph paper.

The degree of closeness of the scatter points and their overall direction enable the relationship to be examined.

Definition: Scatter diagram. A scatter diagram visually presents the nature of the association between two variables without giving any specific numerical value; the values of the two variables are plotted as points on a graph.

If all the points lie on a line, the correlation is perfect. If the scatter points are widely dispersed around the line, the correlation is low. The correlation is said to be linear if the scatter points lie near a line or on a line.

The horizontal axis shows variable X and the vertical axis shows variable Y. The points are scattered around an upward-rising line: when X rises, Y also rises.

The horizontal axis shows variable X and the vertical axis shows variable Y. The points are scattered around a downward-sloping line: when X rises, Y falls, and vice versa.

With no correlation, points are scattered with no upward or downward line visible. With perfect positive or perfect negative correlation, the points themselves lie exactly on an upward-rising or downward-sloping line rather than merely scattered around one.

A relationship is said to be linear if it can be represented by a straight line.

When the true relation between two variables is non-linear, for example a positive or negative curved relation, a scatter diagram will show the points clustering around a curve rather than a straight line, and in this case it is advisable not to try to use Karl Pearson's coefficient of correlation, since this would be misleading.

How is Karl Pearson's coefficient of correlation calculated?

Karl Pearson's coefficient of correlation, also known as the product moment correlation coefficient or simple correlation coefficient, gives a precise numerical value for the degree of linear relationship between two variables X and Y.

It should be used only when there is a linear relation between the variables; where the relation is non-linear, calculating it can be misleading, so the scatter diagram should be examined first.

Definition: Karl Pearson's coefficient of correlation. A numerical measure of the linear relationship between two variables X and Y, denoted r, obtained from the covariance of X and Y divided by the product of their standard deviations.

Mean, variance and covariance

Let X₁, X₂, ..., Xɴ be N values of X and Y₁, Y₂, ..., Yɴ the corresponding values of Y. The arithmetic mean of X is X̄ = ΣX/N and of Y is Ȳ = ΣY/N.

The variance of X is σₓ² = Σ(X − X̄)²/N and of Y is σᵥ² = Σ(Y − Ȳ)²/N; the standard deviations σₓ and σᵥ are the positive square roots of the variances.

Writing x = X − X̄ and y = Y − Ȳ for the deviations of each value from its mean, the covariance of X and Y is defined as Cov(X,Y) = Σxy/N.

The sign of the covariance between X and Y determines the sign of the correlation coefficient. Standard deviations are always positive, so if the covariance is zero, the correlation coefficient is also zero.

The formula for r

Karl Pearson's r = Σxy / (Nσₓσᵥ), which can equivalently be written as r = Σxy / √(Σx²Σy²), using the deviations from the actual means, or using raw totals as r = (NΣXY − ΣXΣY) / √[...] where the denominator contains the product of √[NΣX² − (ΣX)²] and √[NΣY² − (ΣY)²].

Worked example 1. Years of schooling of farmers (X) and annual yield per acre in ₹'000 (Y) for seven farmers: X = 0, 2, 4, 6, 8, 10, 12 and Y = 4, 4, 6, 10, 10, 8, 7.

The table gives ΣX = 42, X̄ = 6, Σ(X − X̄)² = 112, ΣY = 49, Ȳ = 7, Σ(Y − Ȳ)² = 38, Σ(X − X̄)(Y − Ȳ) = 42.

Answer: Using r = Σ(X − X̄)(Y − Ȳ) / √[Σ(X − X̄)² × Σ(Y − Ȳ)²], r = 42 / √(112 × 38) = 0.644.

Years of education of farmers and annual yield per acre are positively correlated, and the value of r is large, implying that more years farmers invest in education, the higher the yield per acre is likely to be.

The data behind Worked example 1 are set out below exactly as tabulated.

What do the properties of r tell us?

Karl Pearson's r has several useful properties for interpretation.

The eight properties

1. r has no unit; it is a pure number, so units of measurement are not part of r. For instance, r between height in feet and weight in kilograms could be, say, 0.7.
2. A negative value of r indicates an inverse relation: a change in one variable is associated with a change in the other in the opposite direction, as when a rise in the price of a commodity is associated with a fall in its demand, or a rise in the rate of interest is associated with a fall in the demand for funds because funds have become costlier.
3. A positive value of r indicates the two variables move in the same direction, as when a rise in the price of coffee, a substitute of tea, is associated with a rise in the demand for tea, or when improvement in irrigation facilities is associated with higher yield.
4. The value of r always lies between minus one and plus one, −1 ≤ r ≤ 1; a calculated value outside this range indicates an error in calculation.
5. The magnitude of r is unaffected by a change of origin or a change of scale, which is the property used in the step deviation method.
6. If r = 0, the two variables are uncorrelated and there is no linear relation between them, though other types of relation may exist.
7. If r = 1 or r = −1, the correlation is perfect and there is an exact linear relation.
8. A high value of r, close to +1 or −1, indicates a strong linear relationship, while a low value close to zero indicates a weak linear relation, though a non-linear relation may still exist.

Reading the size of r

Interpretation also depends on magnitude: a correlation coefficient between marks in English and Statistics of 0.1 means the marks are positively correlated but the relationship is weak, so a student scoring high marks in English may score relatively low marks in Statistics; had r been 0.9, high marks in English would invariably go with high marks in Statistics.

Similarly, between arrival of vegetables in a local mandi and their price, an r of −0.9 means large supply is accompanied by much lower price, whereas an r of −0.1 means large supply is accompanied by only a slightly lower price; if r were zero, there would be no fall in price even after large supplies, which is possible if the extra supply is absorbed by a good transport network carrying it to other markets.

How is the step deviation method used?

When the values of the variables are large, the burden of calculation can be reduced using the property that r is independent of a change of origin and scale. This is the step deviation method.

The variables X and Y are transformed as U = (X − A)/h and V = (Y − B)/k, where A and B are assumed means and h and k are common factors with the same sign. Then rᵤᵥ = r_XY.

Worked example 2. Price index (X): 120, 150, 190, 220, 230. Money supply in ₹ crores (Y): 1800, 2000, 2500, 2700, 3000.

Using assumed mean A = 100, h = 10, B = 1700, k = 100, the transformed values U = (X − 100)/10 give 2, 5, 9, 12, 13 and V = (Y − 1700)/100 give 1, 3, 8, 10, 13.

From these, ΣU = 41, ΣV = 35, ΣU² = 423, ΣV² = 343, ΣUV = 378, with N = 5.

Answer: r = [NΣUV − ΣUΣV] / √{[NΣU² − (ΣU)²][NΣV² − (ΣV)²]} = [5(378) − (41)(35)] / √{[5(423) − 41²][5(343) − 35²]} = 0.98. The strong positive correlation between price index and money supply is an important premise of monetary policy: when money supply grows, the price index also rises.

What is Spearman's rank correlation coefficient?

Spearman's rank correlation was developed by the British psychologist C. E. Spearman. It measures the linear association between ranks assigned to individual items according to their attributes, rather than their actual values, where attributes are variables that cannot be numerically measured, such as intelligence, physical appearance or honesty.

Definition: Spearman's rank correlation coefficient. A measure of linear association between the ranks assigned to units on two variables, used when variables cannot be measured precisely; it has the same range and interpretation as the simple correlation coefficient.

When rank correlation is used

Spearman's method is used in several situations: when heights and weights cannot be measured directly but students can still be ranked by weight and height; when dealing with attributes such as fairness, honesty or beauty that can at most be ranked rather than measured; where the relation is non-linear but its direction is clear; and because, unlike Karl Pearson's coefficient, Spearman's coefficient is not affected by extreme values, so it is useful where the data contain extreme values.

However, it is generally not as accurate as the ordinary method because not all the information in the data is used,.

The rank correlation formula

Spearman's rs = 1 − 6ΣD²/(n³ − n), where n is the number of observations and D is the deviation of the ranks assigned to one variable from those assigned to the other. Like Karl Pearson's r, it lies between 1 and −1.

Worked example 3. Five competitors are ranked by three judges in a beauty contest. Judge A's ranks: 1, 2, 3, 4, 5. Judge B's ranks: 2, 4, 1, 5, 3. Judge C's ranks: 1, 3, 5, 2, 4.

Answer: Between A and B, the deviations D are −1, −2, 2, −1, 2, giving D² values 1, 4, 4, 1, 4, so ΣD² = 14. rs = 1 − 6(14)/(5³ − 5) = 1 − 84/120 = 1 − 0.7 = 0.3.

Between A and C, ΣD² = 10, giving rs = 0.5. Between B and C, the deviations are 1, 1, −4, 3, −1, so ΣD² = 28 and rs = 1 − 168/120 = −0.4. The perceptions of judges A and C are the closest; judges B and C have very different tastes.

How is rank correlation calculated when ranks have to be worked out, and when they repeat?

Rank correlation is calculated under three situations: when the ranks are already given, as in Worked example 3; when the ranks are not given and have to be worked out from the raw data; and when ranks are repeated.

Working out ranks from raw marks

Worked example 4. Percentage marks of five students in Statistics (X) and Economics (Y): A 85/60, B 60/48, C 55/49, D 65/50, E 75/55.

Answer: Ranking the marks in each subject gives Statistics ranks A=1, E=2, D=3, B=4, C=5 and Economics ranks A=1, E=2, D=3, C=4, B=5. The rank deviations are 0, −1, 1, 0 and 0 (students A to E), so ΣD² = 2 and rs = 1 − 6(2)/(5³ − 5) = 1 − 12/120 = 0.9: a strong positive rank correlation between performance in the two subjects.

Correcting for tied ranks

When items are tied, they are given a common rank, which is the mean of the ranks those items would have occupied had they been slightly different from each other; the next item is then assigned the rank following the rank already used.

rs (repeated ranks) = 1 − 6[ΣD² + correction]/(n³−n), where the correction for each group of tied ranks is (m³ − m)/12 and m is the number of items tied at that rank; the correction terms for all tied groups are added together before the formula is applied.

Worked example 5. Twelve paired values of X and Y are ranked; Y has the value 50 at the 9th, 10th and 11th positions, so all three are given the common rank 10, and two other Y values are tied at the 5th and 6th positions, so both take the common rank 5.5. The sum of squared rank deviations works out to ΣD² = 198, with n = 12.

Answer: The correction factor for the tied ranks is (3³ − 3)/12 + (2³ − 2)/12 = 24/12 + 6/12 = 2.5. Substituting, rs = 1 − 6(198 + 2.5)/(12³ − 12) = 1 − 1203/1716 = 1 − 0.70 = 0.30.

There is positive rank correlation between X and Y, both moving in the same direction, though the relationship cannot be described as strong.

What are the limits of correlation analysis?

Correlation measures covariation, not causation. The presence of correlation between two variables X and Y simply means that when the value of one variable changes in one direction, the value of the other is found to change either in the same direction (positive) or the opposite direction (negative), but in a definite way; it does not prove that one causes the other.

A cautionary example: doctors and deaths

During an epidemic, the government sends doctors to affected villages, and the correlation between the number of deaths and the number of doctors sent is found to be positive.

Normally, healthcare provided by doctors would be expected to reduce deaths, showing negative correlation. This positive correlation can arise because the data relate to a specific period, many reported deaths may be terminal cases where doctors could do little, the benefit of doctors' presence may become visible only after some time, or some reported deaths may not even be due to the epidemic, for instance if a tsunami also hits the area and raises the death toll.

Statistical methods are no substitute for understanding the data properly before correlation is calculated and interpreted.

What each technique can and cannot do

The scatter diagram gives a visual presentation of the relationship and is not confined to linear relations.

Karl Pearson's coefficient of correlation and Spearman's rank correlation measure linear relationship among variables; when variables cannot be measured precisely, rank correlation can be used.

Correlation gives an idea of the direction and intensity of change in a variable when the correlated variable changes, but it never by itself proves cause and effect.

Going beyond the syllabus

Correlation is often a first step before regression analysis, which goes further by estimating the actual equation of the line relating one variable to another and allows prediction of one variable's value from the other, a technique studied at a more advanced level.

It is also worth remembering that correlation is calculated between only two variables at a time: where many variables move together, such as income, savings and investment in an economy, each pair has to be examined separately, since a high correlation between one pair does not by itself describe the relationship of either variable with a third.

The data used for the step deviation worked example in this note, on price index and money supply, illustrate why correlation is of interest to policy: a strong positive correlation between the two supports the view, used in forming monetary policy, that a faster growth of money supply tends to go together with a faster rise in the general price level, though, as emphasised throughout, this association alone cannot be read as one variable causing the other.

How does correlation apply to Indian macroeconomic data?

The Economic Survey (2004-05) reported the annual growth of national income at current prices alongside the Gross Domestic Saving as a percentage of GDP for ten successive years, from 1992-93 to 2001-02. This paired Indian data can be used to calculate r between the two series using any of the formulae set out above.

This Indian data set is a good illustration of how correlation is used on macroeconomic variables: the national income growth rate and the Gross Domestic Saving ratio can be treated as the two variables X and Y, and the covariance-based formula, or the step deviation method where the values are large, can be applied to this ten-year series exactly as it was applied to the price index and money supply data in Worked example 2.

A student working through such a table would first find the means of the two columns, then the deviations of each year's figures from those means, and finally substitute the deviations into Karl Pearson's formula. Because both columns here show only modest year-to-year variation, the resulting r would need to be interpreted with the same caution used throughout this note: even a strong numerical correlation between India's growth rate and its saving ratio over this period would describe covariation between the two series and would not, by itself, establish that a higher saving ratio caused faster growth, or the reverse.

Glossary

• Correlation — the study and measurement of the direction and intensity of the relationship between two variables.
• Positive correlation — a relationship in which both variables move in the same direction.
• Negative correlation — a relationship in which the variables move in opposite directions.
• Linear relationship — a relationship between two variables that can be represented by a straight line.
• Scatter diagram — a graph on which the paired values of two variables are plotted as points to show the nature of their association.
• Covariance — the average of the product of the deviations of two variables from their respective means.
• Standard deviation — the positive square root of the variance of a variable, used in Karl Pearson's formula.
• Karl Pearson's coefficient of correlation — a numerical measure, denoted r, of the degree of linear relationship between two variables.
• Step deviation method — a simplified way of calculating r using assumed means and common factors to transform large values.
• Assumed mean — a convenient value chosen in place of the actual mean to simplify calculation, used in the step deviation method.
• Attributes — qualities such as honesty or beauty that cannot be measured numerically but can be ranked.
• Spearman's rank correlation — a coefficient measuring the linear association between the ranks assigned to items on two variables.
• Common rank — the average of the ranks that tied items would have received had they not been equal.
• Correction factor — the adjustment, (m³ − m)/12, added when calculating rank correlation with repeated ranks.
• Perfect correlation — correlation in which r equals +1 or −1 and all points lie exactly on a straight line.

Common errors and misconceptions

• Misconception: A high correlation between two variables proves one causes the other. Correct: Correlation measures covariation only; a third variable, such as temperature behind ice-cream sales and drowning deaths, may explain both.
• Misconception: r can take any value if the data are unusual enough. Correct: r always lies between −1 and +1; a calculated value outside this range means there is an error in the working.
• Misconception: Karl Pearson's coefficient can be used for any relationship. Correct: It should be used only where the relationship is linear; for a non-linear relationship it can be misleading, and the scatter diagram should be checked first.
• Misconception: r = 0 means the variables are completely independent. Correct: r = 0 only means there is no linear relation between them; a non-linear relation may still exist.
• Misconception: Spearman's method is always less useful than Karl Pearson's. Correct: Spearman's coefficient is preferred when attributes cannot be measured precisely, when data contain extreme values, or when the relation is non-linear but monotonic.
• Misconception: Tied ranks can simply be ignored in Spearman's formula. Correct: Tied items are given a common rank (the average of the ranks they would have taken), and a correction factor of (m³ − m)/12 for each tied group must be added.
• Misconception: The unit of r depends on the units of X and Y. Correct: r is a pure number with no unit, regardless of the units in which X and Y are measured.

Exam-style questions with model answers

Q1. State the range within which the simple correlation coefficient always lies. [1 mark]

1. The simple correlation coefficient r always lies between −1 and +1, written as −1 ≤ r ≤ 1.

Q2. Distinguish between positive correlation and negative correlation, giving one example of each from economic variables. [2 marks]

1. Positive correlation exists when the two variables move in the same direction, for example when income rises, consumption also rises.
2. Negative correlation exists when the two variables move in opposite directions, for example when the price of apples falls, its demand increases.

Q3. Explain why a scatter diagram should be drawn before calculating Karl Pearson's coefficient of correlation. [3 marks]

1. A scatter diagram visually shows the nature of the relationship between two variables without any calculation.
2. Karl Pearson's coefficient should be used only when the relationship between the variables is linear.
3. If the scatter diagram shows a non-linear relationship, calculating Karl Pearson's coefficient can be misleading, so the diagram helps decide whether the formula is appropriate before using it.

Q4. Given X = 1, 3, 4, 5, 7, 8 and Y = 2, 6, 8, 10, 14, 16, state what type of correlation exists between X and Y and why. [3 marks]

1. Here Y is exactly double X for every pair of values given (for example when X = 1, Y = 2; when X = 8, Y = 16).
2. Since both variables always rise together in exact proportion, all the points plotted on a scatter diagram would lie exactly on an upward-rising straight line.
3. This means the correlation coefficient r would equal 1, which is perfect positive correlation.

Q5. Explain why correlation does not imply causation, using an example. [4 marks]

1. Correlation only measures covariation, that is, whether two variables tend to change together, not whether one causes the other.
2. A high correlation can arise because a third variable affects both variables being studied.
3. For example, a brisk sale of ice-creams is correlated with a higher number of deaths by drowning, but eating ice-cream does not cause drowning.
4. Rising temperature is the real cause behind both: it raises ice-cream sales and also increases the number of people swimming, which raises drowning deaths, so the correlation between ice-cream sales and drowning is coincidental rather than causal.

Q6. Calculate Karl Pearson's coefficient of correlation between X and Y from the following data, and interpret the result: X = 0, 2, 4, 6, 8 with mean 4, Y = 4, 4, 6, 10, 10 with mean 6·8, Σ(X−X̄)² = 40, Σ(Y−Ȳ)² = 36·8, Σ(X−X̄)(Y−Ȳ) = 36·0. [4 marks]

1. Apply the formula r = Σ(X−X̄)(Y−Ȳ) / √[Σ(X−X̄)² × Σ(Y−Ȳ)²].
2. Substitute the given values: r = 36·0 / √(40 × 36·8) = 36·0 / √1472.
3. √1472 is approximately 38·37, so r ≈ 36·0/38·37 ≈ 0·94.
4. Since r is close to +1, there is a strong positive correlation between X and Y: as X rises, Y also tends to rise strongly.

Q7. Five students' ranks in Statistics and Economics are given below. Calculate Spearman's rank correlation coefficient and interpret it: Statistics ranks 1, 4, 5, 3, 2 and Economics ranks 1, 5, 4, 3, 2 (in the same student order). [5 marks]

1. Find the deviation D between the two ranks for each student: 1−1=0, 4−5=−1, 5−4=1, 3−3=0, 2−2=0.
2. Square each deviation: 0, 1, 1, 0, 0, so ΣD² = 2.
3. Apply the formula rs = 1 − 6ΣD²/(n³ − n), with n = 5, so n³ − n = 125 − 5 = 120.
4. rs = 1 − 6(2)/120 = 1 − 12/120 = 1 − 0·1 = 0·9.
5. A value of 0·9 is close to +1, so there is a strong positive rank correlation: students who rank well in Statistics also tend to rank well in Economics.

Q8. Explain, with reasons, two situations in which Spearman's rank correlation coefficient is preferred to Karl Pearson's coefficient of correlation. [5 marks]

1. Spearman's coefficient is preferred when at least one of the variables cannot be measured numerically but can only be ranked, such as honesty or beauty, because Karl Pearson's formula requires actual numerical values.
2. It is preferred when the variables cannot be measured precisely at all, for example height and weight in a place where no measuring instruments are available, so that only ranking is possible.
3. It is preferred when the data contain extreme values, because Spearman's coefficient is not affected by extreme values, whereas Karl Pearson's coefficient can be distorted by them.
4. It is preferred where the relationship between the variables is non-linear but its direction is clear, since rank correlation can still be calculated meaningfully in such a case, while Karl Pearson's coefficient assumes linearity.
5. However, when precisely measured data are available and the relationship is genuinely linear, Karl Pearson's coefficient is more accurate than rank correlation, because rank correlation does not use all the information in the data.

Key takeaways

• Correlation analysis studies the relationship between two variables, measuring covariation rather than causation.
• Correlation is classified as positive, when variables move in the same direction, or negative, when they move in opposite directions.
• A scatter diagram gives a visual, non-numerical picture of the relationship and shows whether it is linear and how strong it appears to be.
• Karl Pearson's coefficient r gives a precise numerical measure of linear relationship and always lies between −1 and +1.
• The step deviation method uses assumed means and common factors to simplify calculation of r without changing its value.
• Spearman's rank correlation coefficient measures linear association between ranks and is preferred when attributes cannot be measured numerically, data contain extreme values, or precise measurement is impossible.
• Repeated ranks are given a common rank, and a correction factor of (m³ − m)/12 for each tied group is added to Spearman's formula.
• High correlation between two variables can arise from a third variable affecting both, so correlation must never be read as proof of cause and effect.

Test yourself

What does a correlation coefficient of r = 0 mean?

It means there is no linear relation between the two variables, though some other type of relation, such as a non-linear one, may still exist between them.

Why should Karl Pearson's coefficient of correlation not be used when the scatter diagram shows a non-linear relation?

Because Karl Pearson's coefficient measures only linear relationships; applying it to a non-linear relation can give a misleading numerical value.

What is the unit of the correlation coefficient between height in feet and weight in kilograms?

None: r is a pure number with no unit, regardless of the units in which the two variables are measured.

Why is Spearman's rank correlation coefficient generally less accurate than Karl Pearson's coefficient?

Because ranking replaces individual values with their order, so some information contained in the original data is not used in the calculation.

What correction is applied to Spearman's formula when ranks are repeated?

A correction factor of (m³ − m)/12 is added for each group of tied items, where m is the number of items sharing that rank.

Give one example where a positive correlation does not mean one variable causes the other.

The positive correlation between the number of doctors sent to villages during an epidemic and the number of deaths does not mean doctors cause deaths; it reflects the severity of the cases and timing of reporting.

What range of values can Spearman's rank correlation coefficient take?

Like Karl Pearson's coefficient, it lies between −1 and +1, with the same interpretation of direction and strength.