Correlation Coefficient Range: From Intuition to Precision
Imagine you're tracking two things: the price of tea and the quantity of tea sold. When tea gets cheaper, people buy more. When it gets pricier, they buy less. These two variables move in opposite directions — one goes up, the other goes down. That's a relationship.
Now think about the price of tea and the temperature outside. On a hot day, more people buy iced tea. On a cold day, fewer do. Here, both price and temperature might move together (both high in summer) or in opposite ways depending on the season. The relationship is weaker.
The correlation coefficient is a single number that tells you how strongly two variables are related — and whether they move in the same direction or opposite directions. Its range is the first thing you need to understand.
The Range: -1 to +1
The correlation coefficient, usually denoted by r, can only take values between −1 and +1, inclusive.
−1≤r≤+1
That's it. No correlation coefficient ever goes outside this range. If you ever calculate one and get r=1.5 or r=−2, you've made a mistake.
What Each Part of the Range Means
r=+1 — Perfect positive correlation. The two variables move exactly together. If one goes up by 10%, the other goes up by exactly the same proportion. In economics, this is rare in real data but appears in identities. For example, if you know a family's total expenditure and you know their spending on food plus their spending on everything else, those two totals are perfectly correlated — they're the same thing measured differently.
r=−1 — Perfect negative correlation. The two variables move exactly opposite. When one goes up, the other goes down by the same proportion. Again, rare in real data but possible in constructed examples.
r=0 — No linear correlation. The variables show no tendency to move together or opposite. Knowing one tells you nothing about the other. For instance, the price of tea in India and the number of students in a school in Brazil — there's no reason they'd be related.
Values between 0 and +1 — Positive correlation, but not perfect. The closer to +1, the stronger the relationship. r=0.8 means a strong positive relationship; r=0.2 means a weak one.
Values between -1 and 0 — Negative correlation. The closer to -1, the stronger the inverse relationship.
The correlation coefficient measures linear relationships only. Two variables could be perfectly related in a curved way (like y=x2) and still have r=0. Always check a scatter diagram before trusting r alone.
Why the Range Matters in Economics
In economics, you're constantly asking: does X cause Y? Correlation doesn't prove causation, but it's the first clue. If you're studying demand, you expect price and quantity demanded to have a negative correlation — that's the law of demand. If you calculate r=−0.9 between price and quantity demanded for a good, you have strong evidence the law holds for that good.
If you're studying consumption and income, you expect a positive correlation — higher income, higher consumption. A correlation of r=0.95 between these two is common in macro data. …