Correlation Coefficient

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Correlation Coefficient: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Correlation coefficient, written r, is a number from −1 to 1 that tells how strongly and in which direction two variables follow a straight-line pattern.

Algebra 1StatisticsS-ID.C.8

What is Correlation Coefficient?

A scatter plot shows a pattern, and the correlation coefficient \(r\) boils it down to one number. The sign gives the direction: positive when the dots rise, negative when they fall. The size gives the strength: the closer \(|r|\) is to 1, the tighter the dots hug a straight line.

The value \(r = 1\) or \(r = -1\) means every point lies exactly on a line, and \(r\) near 0 means no linear pattern. It measures only straight-line association, so a perfect curve can still produce a value near zero, and a strong value never proves that one variable causes the other.

How to spot it

  • A calculator or software prints r after a linear regression for paired data.
  • A question asks how strong or how weak a relationship between two measurements is.
  • You must match a scatter plot to a value such as 0.9, −0.8 or 0.1.
  • A report states that two things are correlated and you must interpret the number.

The method

  1. Look at the sign of the correlation coefficient first: positive for an upward trend, negative for a downward trend.
  2. Judge the size of \(|r|\): near 1 is strong, around 0.5 is moderate, near 0 is weak or none.
  3. Remember that the value only measures linear patterns, so check the scatter plot for curves or outliers.
  4. State the conclusion in context, without claiming that one variable causes the other.
r near 1r near 0r near −1strong positiveno linear patternstrong negativer always lies between −1 and 1
Rising dots give r near 1, a shapeless cloud gives r near 0, falling dots give r near −1.

Correlation Coefficient: solved example

Problem

A city compares each day’s high temperature with the number of ice pops sold at its pool and finds \(r = 0.91\). A second study of hours of TV and quiz average finds \(r = -0.08\). Interpret both.

Solution

The first value is positive and close to 1: a strong positive linear association, with hotter days going along with more sales. It does not prove that heat alone causes the sales.

The second is negative but almost 0: practically no linear association between TV hours and quiz average.

Try it

Question 1

Which relationship is stronger, \(r = -0.95\) or \(r = 0.60\)?

Show the answer

The first. Strength depends on \(|r|\), and 0.95 is larger than 0.60.

Question 2

A student reports \(r = 1.4\). What is wrong?

Show the answer

The value cannot exceed 1 in size, so there was an error in the calculation.

Correlation Coefficient vs. Correlation vs. Causation

The value r only describes how closely two variables move together in a straight pattern. Causation is a claim about why. Even \(r = 0.99\) can come from a hidden third factor, such as hot weather raising both ice cream sales and swimming trips.

Correlation vs. Causation

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