Correlation Coefficient

Learn how to measure the strength and direction of a linear relationship between two variables using the correlation coefficient r.

Advanced25 minLesson
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Definition

The correlation coefficient (denoted rr) measures the strength and direction of a linear relationship between two variables.
r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2⋅∑(yi−yˉ)2r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \cdot \sum(y_i - \bar{y})^2}}
**Key properties of rr:**
  • rr is always between −1-1 and 11: −1≤r≤1-1 \le r \le 1
  • r=1r = 1: Perfect positive correlation
  • r=−1r = -1: Perfect negative correlation
  • r=0r = 0: No linear correlation
**Interpreting ∣r∣|r|:**
Value of $rStrength
0.00−0.190.00 - 0.19Very weak
0.20−0.390.20 - 0.39Weak
0.40−0.590.40 - 0.59Moderate
0.60−0.790.60 - 0.79Strong
0.80−1.000.80 - 1.00Very strong

Try it now

What does a correlation coefficient of r=1r = 1 indicate?

Worked Examples

Find the correlation coefficient for the data: (1,2),(2,4),(3,5),(4,4),(5,5)(1, 2), (2, 4), (3, 5), (4, 4), (5, 5)
1

Find the means xˉ\bar{x} and yˉ\bar{y}

xˉ=1+2+3+4+55=3\bar{x} = \frac{1+2+3+4+5}{5} = 3, yˉ=2+4+5+4+55=4\bar{y} = \frac{2+4+5+4+5}{5} = 4 → xˉ=3\bar{x} = 3, yˉ=4\bar{y} = 4
2

Calculate deviations (xi−xˉ)(x_i - \bar{x}) and (yi−yˉ)(y_i - \bar{y})

Point (1,2)(1,2): (−2,−2)(-2, -2) Point (2,4)(2,4): (−1,0)(-1, 0) Point (3,5)(3,5): (0,1)(0, 1) Point (4,4)(4,4): (1,0)(1, 0) Point (5,5)(5,5): (2,1)(2, 1) → Deviations calculated
3

Calculate products (xi−xˉ)(yi−yˉ)(x_i - \bar{x})(y_i - \bar{y})

(−2)(−2)=4(-2)(-2) = 4 (−1)(0)=0(-1)(0) = 0 (0)(1)=0(0)(1) = 0 (1)(0)=0(1)(0) = 0 (2)(1)=2(2)(1) = 2 Sum =4+0+0+0+2=6= 4 + 0 + 0 + 0 + 2 = 6 → ∑(xi−xˉ)(yi−yˉ)=6\sum(x_i - \bar{x})(y_i - \bar{y}) = 6
4

Calculate ∑(xi−xˉ)2\sum(x_i - \bar{x})^2

(−2)2+(−1)2+02+12+22=4+1+0+1+4=10(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10 → ∑(xi−xˉ)2=10\sum(x_i - \bar{x})^2 = 10
5

Calculate ∑(yi−yˉ)2\sum(y_i - \bar{y})^2

(−2)2+02+12+02+12=4+0+1+0+1=6(-2)^2 + 0^2 + 1^2 + 0^2 + 1^2 = 4 + 0 + 1 + 0 + 1 = 6 → ∑(yi−yˉ)2=6\sum(y_i - \bar{y})^2 = 6
6

Apply the formula

r=610⋅6=660=67.746≈0.775r = \frac{6}{\sqrt{10 \cdot 6}} = \frac{6}{\sqrt{60}} = \frac{6}{7.746} \approx 0.775 → r≈0.77r \approx 0.77

Common Mistakes

Assuming correlation implies causation

Why it's wrong: A strong correlation between AA and BB doesn't mean AA causes BB. There could be a third variable affecting both, or the relationship could be coincidental.

Correct: Always say "associated with" or "correlated with" rather than "causes." Look for experimental evidence to establish causation.

Ignoring outliers when interpreting rr

Why it's wrong: A single outlier can dramatically change the correlation coefficient. rr is sensitive to extreme values.

Correct: Always plot the data first. Look for outliers and consider calculating rr with and without them.

Using rr for non-linear relationships

Why it's wrong: rr only measures linear relationships. A perfect curved pattern could have r=0r = 0.

Correct: Examine the scatter plot. If the pattern is curved, rr is not the right measure. Consider transformations or non-linear models.

Interactive Visual

Scatter Plot

Data: (1, 2), (2, 4), (3, 5), (4, 4), (5, 5)

123452345xy

ŷ = 0.6x + 2.2

r = 0.775 · strong positive correlation

Tap a point to see its coordinates.

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Practice Problems

16 problems
Problem 1 of 16
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What does a correlation coefficient of r=1r = 1 indicate?

Why It Matters

The correlation coefficient is fundamental in data analysis:
  • Science: Researchers measure how strongly variables like temperature and ice cream sales relate
  • Finance: Analysts track how stock prices move together (portfolio diversification)
  • Medicine: Doctors study relationships between risk factors and health outcomes
  • Sports: Coaches analyze which statistics best predict team success
Understanding rr helps you:
  • Quantify relationships that "look" linear in scatter plots
  • Make predictions using regression analysis
  • Evaluate the reliability of statistical models
  • Communicate findings precisely ("strong positive correlation" vs "weak negative correlation")

Real World Applications

Stock Market Diversification

Investment analysts use correlation to build diversified portfolios. Stocks with low or negative correlation balance risk.

Example:

If Stock A and Stock B have r=−0.3r = -0.3, when A drops, B tends to rise slightly, reducing portfolio volatility.

1Try It Yourself

Two tech stocks have r=0.92r = 0.92. Two stocks from different sectors have r=0.15r = 0.15.

Which pair provides better diversification?

Step 1: Write the mathematical expression

Compare: ∣0.92∣|0.92| vs ∣0.15∣|0.15|

Medical Research

Researchers use correlation to identify potential risk factors for diseases.

Example:

A study finds r=0.68r = 0.68 between blood pressure and risk of heart disease, indicating a strong positive relationship.

2Try It Yourself

Three health factors show correlations with diabetes risk: Exercise (r=−0.45r = -0.45), Sugar intake (r=0.52r = 0.52), Sleep quality (r=−0.31r = -0.31).

Which factor has the strongest relationship with diabetes risk?

Step 1: Write the mathematical expression

Compare absolute values: ∣−0.45∣|-0.45|, ∣0.52∣|0.52|, ∣−0.31∣|-0.31|

Key Takeaways

  • 1The correlation coefficient rr measures the strength and direction of a linear relationship
  • 2rr ranges from −1-1 (perfect negative) to +1+1 (perfect positive), with 00 indicating no linear relationship
  • 3The absolute value ∣r∣|r| indicates strength: closer to 11 is stronger, closer to 00 is weaker
  • 4Always visualize data with a scatter plot before interpreting rr
  • 5Correlation does not imply causation - association is not the same as cause-and-effect

Frequently Asked Questions

Correlation shows that two variables move together, but causation means one variable directly causes changes in another. Ice cream sales and drowning rates are correlated (both increase in summer), but ice cream doesn't cause drowning - hot weather affects both.
Correlation shows that two variables move together, but causation means one variable directly causes changes in another. Ice cream sales and drowning rates are correlated (both increase in summer), but ice cream doesn't cause drowning - hot weather affects both.
No. By mathematical definition, rr is always between −1-1 and 11. If you calculate a value outside this range, there's an error in your calculations.
r=0r = 0 means there is no linear relationship between the variables. However, there could still be a non-linear relationship (like a parabola or sine curve).

Glossary

Correlation coefficient (rr)
A numerical measure from −1-1 to +1+1 that quantifies the strength and direction of a linear relationship between two variables
Positive correlation
When r>0r > 0: as one variable increases, the other tends to increase
Negative correlation
When r<0r < 0: as one variable increases, the other tends to decrease
Pearson correlation
The most common type of correlation coefficient, measuring linear relationships (also called Pearson's rr)
Scatter plot
A graph showing points for paired data, used to visualize relationships between variables

Formula Card

Correlation Coefficient Formula

r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2⋅∑(yi−yˉ)2r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \cdot \sum(y_i - \bar{y})^2}}

Alternative form: $r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$

Today's challengeOne problem, three tries

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