Example 5. The values of X and Y are given as follows. Work out the ranks (giving common ranks to repeated items) and calculate the Spearman's rank correlation coefficient.
Imagine you're tracking two things over time — say, the number of ice creams sold at your school canteen and the outside temperature. On hot days, both go up; on cool days, both drop. They seem to move together. That's the basic idea of correlation: a measure of how two variables move in relation to each other.
But not all co-movement is equal. Sometimes one variable goes up while the other goes down — like the price of a good and the quantity demanded (law of demand). Sometimes they seem to have no connection at all — like the number of students in your class and the price of tea in China. Pearson correlation gives us a single number that captures the strength and direction of this linear relationship.
The Precise Meaning
Pearson correlation coefficient, usually denoted by r, measures the linear relationship between two variables X and Y. It answers: If I know how far X is from its average, how far (and in which direction) is Y from its average, on average?
The formula is:
r=∑(Xi−Xˉ)2⋅∑(Yi−Yˉ)2∑(Xi−Xˉ)(Yi−Yˉ)
Where:
Xi, Yi are individual observations
Xˉ, Yˉ are the means (averages) of X and Y
∑ means "sum over all observations"
The numerator is the covariance — it tells you whether deviations from the mean tend to be in the same direction (positive product) or opposite directions (negative product). The denominator is the product of the standard deviations of X and Y, which scales the result so that r always lies between −1 and +1.
Important
r is unitless and always between −1 and +1:
r=+1: perfect positive linear relationship (all points lie on an upward-sloping line)
r=−1: perfect negative linear relationship (all points lie on a downward-sloping line)
r=0: no linear relationship (but there could still be a non-linear one!)
Why It Matters in Economics
Economics is full of pairs of variables that we suspect move together. Pearson correlation gives us a first, clean check on whether that suspicion holds water.
Example 1: Consumption and Income. Keynes said consumption depends on income. If you plot household consumption against household income for a sample of families, you'd expect a positive r — higher income families tend to consume more. A value close to +0.8 or +0.9 would be strong evidence for that relationship.
Example 2: Price and Quantity Demanded. The law of demand says price and quantity demanded move in opposite directions. A negative r between price and quantity (holding other factors constant) would confirm this. But here's the catch — in real market data, price and quantity are determined simultaneously by supply and demand, so a simple correlation might not show the expected negative sign. That's why economists use more advanced tools (like regression) to isolate the relationship.
Example 3: Investment and Interest Rates. You'd expect a negative correlation — when interest rates are high, borrowing is expensive, so investment falls. But the relationship might be weak (r close to 0) because investment also depends on expectations, technology, and government policy.
Watch out
Correlation does NOT imply causation. Just because ice cream sales and drowning incidents are positively correlated (both peak in summer) does NOT mean ice cream causes drowning. The common cause is hot weather, which makes people both buy ice cream and go swimming. In economics, this is a constant trap — GDP and money supply are correlated, but which causes which? The answer requires theory, not just correlation.
Visualising It
Draw a scatter plot with X on the horizontal axis and Y on the vertical axis. If the points cluster around a straight line sloping upward, r is positive and strong. If they cluster around a line sloping downward, r is negative and strong. If they form a shapeless cloud, r is near zero.
But here's the nuance: a perfect circle of points has r=0 even though X and Y are clearly related (non-linearly). Pearson correlation only captures linear relationships. Two variables could be perfectly related by a U-shaped curve and still have r=0. …
Because the Y-values include repeated (tied) observations, those items are given common average ranks, and Spearman's formula needs an additional correction term for each group of ties. …
X-values are all distinct; Y has ties (90 twice, 75 twice, 50 thrice), so those items get common (average) ranks and a tie-correction 12m3−m is added for each group. Here ∑d2=198 and the total correction =3, giving rs≈+0.297.
Concept first
When items repeat, each is given the average of the ranks it would occupy, and Spearman's formula carries a correction term for every tie group of size m: