The correlation coefficient, r, is a number from −1 to 1 that measures how closely the points of a scatter plot follow a straight line. The sign gives the direction (positive slopes up, negative slopes down), and the size gives the strength: near ± 1 is strong, near 0 is weak or none.
In Spanish
coeficiente de correlación
Also called
r, Pearson correlation coefficient, correlation coefficient r, Pearson's r
Grade level
Algebra 1
Algebra 1StatisticsCCSS HSS-ID.C.8SATACTAP
Correlation Coefficient in a picture
The sign of r shows the direction of the trend; the **closer to 1 or −1**, the tighter the points hug a line.
Formal definition and notation
For paired data(xi, yi), i = 1, …, n, the Pearson correlation coefficient r is the sum of the products of the deviations (xi − x̄)(yi − ȳ), divided by the square root of the product of the two sums of squared deviations (formula below). It always satisfies −1 ≤ r ≤ 1, equals ± 1 exactly when all points lie on a non-horizontal line, has no units, and measures only linear association.
r = 0 means no linear pattern. A perfect U-shaped curve can still have r = 0.
r has no units, and changing units (inches to centimeters) or swapping x and y does not change it.
One outlier can change r a lot, and a strong r never proves causation.
Correlation coefficient formula
Correlation coefficient formula
r = Σ (x − x̄)(y − ȳ)√Σ (x − x̄)2 · Σ (y − ȳ)2
x − x̄
how far each x-value is from the mean of x
y − ȳ
how far each y-value is from the mean of y
Σ
add over all the data pairs
In practice a calculator or spreadsheet computes r (CORREL in a spreadsheet, LinReg on a graphing calculator). The formula shows why: points above both means or below both means push r up.
Correlation Coefficient examples, step by step
Example 1: Which is strongest?
Warm-up
Three data sets have r = 0.35, r = −0.91 and r = 0.78. Which shows the strongest linear relationship?
Strength depends on the size of r, ignoring the sign.
Sizes: 0.35, 0.91 and 0.78.
The largest is 0.91, from r = −0.91.
Answer: r = -0.91 is strongest (a strong negative relationship).
Example 2: Compute r by hand
Core
Find r for the points (1, 3), (2, 5), (3, 4), (4, 8), (5, 10).
Means: x̄ = 3 and ȳ = 6.
Deviations: x − 3 gives −2, −1, 0, 1, 2; y − 6 gives −3, −1, −2, 2, 4.
Sum of products: 6 + 1 + 0 + 2 + 8 = 17. Sums of squares: 10 for x and 34 for y.
Slope tells how much y changes per unit of x and has units; r tells how well a line fits, with no units, and is always between −1 and 1.
Correlation Coefficientvs.Coefficient of determination
r2 is the square of r: with r = −0.9, r2 = 0.81. It is never negative and gives the share of variation explained.
Quick check: 3 questions
Question 1 of 3
Which value could NOT be a correlation coefficient?
Show answer and explanation
Answer: C.r is always between −1 and 1, so 1.2 is impossible. −0.99 is a very strong negative correlation, 0 is no linear pattern, and 0.45 is moderate positive.
Question 2 of 3
As outdoor temperature rises, heating bills fall, and the points lie close to a line. Which r fits best?
Show answer and explanation
Answer: B. One variable goes up as the other goes down, so r is negative, and points close to a line make it near −1. 0.92 has the wrong sign, and −0.15 and 0 describe weak or no patterns.
Question 3 of 3
A data set has r = −0.6. What is r2?
Show answer and explanation
Answer: D.(−0.6)2 = 0.36: squaring a negative gives a positive. −0.36 kept the sign, 1.2 doubled instead of squaring, and 0.6 only dropped the sign.
It depends on the field, but a common rule of thumb is that |r| of 0.7 or more is strong, 0.3 to 0.7 is moderate, and below 0.3 is weak. In physics, data often have r above 0.99; in social science, 0.5 can be notable. Strong does not mean useful or causal, so always look at the scatter plot and the context.
Can the correlation coefficient be greater than 1?
No. The formula guarantees −1 ≤ r ≤ 1. It reaches exactly 1 or −1 only when every point lies on one straight line that is not horizontal. If a calculation gives a value outside that range, there is an arithmetic error, often a missing square root in the denominator.
What is the difference between r and r squared?
r shows direction and strength of a linear relationship and can be negative. r2, the coefficient of determination, is always between 0 and 1 and tells what fraction of the variation in y the line explains. With r = 0.7, r2 = 0.49, so about 49% of the variation is explained by x.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-ID.C.8
Cite this entry
Effortless Math Team. “Correlation Coefficient.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/correlation-coefficient/
Flashcard
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