Correlation and linear regression

IB Mathematics Applications and Interpretation HL· 20 min read

1. Bivariate Data and Correlation★★☆☆☆⏱ 15 min

Bivariate data describes pairs of values for two different variables collected from the same sample. We use scatter graphs to visualize relationships between variables, and Pearson's correlation coefficient to quantify the strength of linear relationships.

📘 Definition

Pearson's Correlation Coefficient

$r$

A numerical measure of the strength and direction of the linear relationship between two continuous variables, ranging from $- 1$ to $+ 1$ .

Example:

$r = 1$ = perfect positive linear correlation, $r = - 1$ = perfect negative linear correlation, $r = 0$ = no linear correlation.

General interpretation guidelines: $∣ r ∣ > 0.75$ = strong correlation, $0.25 < ∣ r ∣ < 0.75$ = moderate correlation, $∣ r ∣ < 0.25$ = weak/no linear correlation.

📐 Worked Example

For the data $x = [1, 2, 3, 4, 5]$ , $y = [2, 4, 5, 7, 9]$ , calculate $r$ and interpret the result.

1
1. Calculate sample means:
2
$\bar{x} = 3, \quad \bar{y} = 5.4$
3
1. Calculate sample standard deviations:
4
$s_x \approx 1.581, \quad s_y \approx 2.702$
5
1. Calculate covariance: $Cov (x, y) \approx 4.2$
6
1. Use the formula $r = \frac{Cov ( x , y )}{s _{x} s _{y}}$ :
7
$r \approx \frac{4.2}{(1.581)(2.702)} = 0.984$
8
Interpretation: There is a very strong positive linear relationship between $x$ and $y$ .

Exam tip:

Always use your GDC to calculate $r$ in exams, manual calculation is rarely required.

2. Least Squares Regression Line★★★☆☆⏱ 20 min

A regression line models the linear relationship between two variables. The least squares method fits the line by minimizing the sum of squared vertical distances (residuals) between observed and predicted $y$ values.

📘 Definition

y on x Regression Line

$y = a + b x$

A line used to predict the value of the response variable $y$ from the explanatory variable $x$ . It always passes through the point $(ar{x}, ar{y})$ .

Example:

The gradient $b = r \frac{s _{y}}{s _{x}}$ and the intercept $a = \overset{y}{ˉ} - b \overset{x}{ˉ}$ .

We use different lines for different prediction goals: $y$ on $x$ predicts $y$ from $x$ , and $x$ on $y$ predicts $x$ from $y$ . The two lines are not the same.

📐 Worked Example

Using the previous data ( $r \approx 0.984$ , $ar{x}=3$ , $ar{y}=5.4$ , $s_{x} \approx 1.581$ , $s_{y} \approx 2.702$ ), find the $y$ on $x$ regression line and interpret the gradient.

1
1. Calculate the gradient $b$ :
2
$b = 0.984 \times \frac{2.702}{1.581} \approx 1.68$
3
1. Calculate the intercept $a$ :
4
$a = 5.4 - (1.68)(3) = 0.36$
5
1. Final equation:
6
$y = 0.36 + 1.68x$
7
Interpretation: For every 1 unit increase in $x$ , $y$ is predicted to increase by 1.68 units on average.

Exam tip:

Use the fact that the line passes through $(ar{x}, ar{y})$ to check your answer in exams.

3. Prediction: Interpolation vs Extrapolation★★★☆☆⏱ 15 min

Regression lines are used to predict unknown values of the response variable. The reliability of a prediction depends on whether the input value falls inside or outside the range of the original collected data.

Interpolation is generally reliable because we know the linear relationship holds for this range. Extrapolation is unreliable because we have no evidence the linear relationship continues outside the observed range.

📐 Worked Example

Using the regression line $y = 0.36 + 1.68 x$ (original $x$ ranges from 1 to 5), predict $y$ for $x = 3.5$ and $x = 10$ , and comment on reliability.

1
For $x = 3.5$ :
2
$y = 0.36 + 1.68(3.5) = 6.24$
3
Comment: $x = 3.5$ is inside the original range of 1 to 5, so this is interpolation. The prediction is reliable.
4
For $x = 10$ :
5
$y = 0.36 + 1.68(10) = 17.16$
6
Comment: $x = 10$ is outside the original range, so this is extrapolation. The prediction is unreliable, as we do not know if the linear relationship continues to $x = 10$ .

4. Correlation vs Causation★★☆☆☆⏱ 10 min

A common misconception is that a strong correlation between two variables proves that one causes the other. This is not true, and correlation can arise from multiple non-causal scenarios.

📐 Worked Example

A study finds a strong positive correlation between monthly ice cream sales and monthly drowning deaths. Does ice cream sales cause drowning?

1
1. Observe that there is a strong positive correlation, but no obvious direct link.
2
1. Identify the lurking variable: temperature. Higher temperatures increase both ice cream sales and the number of people swimming, leading to more drownings.
3
1. Conclusion: There is no causal relationship between ice cream sales and drowning deaths. The correlation is explained by the lurking variable.

Exam tip:

Always explicitly state that correlation does not imply causation when asked to interpret results in exams.

5. Common Pitfalls

Wrong move:

Claiming a strong correlation proves causation

Why:

Correlation only measures association, not a directional causal link, and can be explained by lurking variables

Correct move:

Always explicitly state that correlation does not imply causation unless experimental evidence confirms a causal link

Wrong move:

Treating extrapolated predictions as equally reliable as interpolated predictions

Why:

The linear relationship is only validated for the range of observed data

Correct move:

Always flag extrapolated predictions as unreliable when discussing results

Wrong move:

Using an x on y regression line to predict y from x

Why:

Regression lines are fitted to minimize error for a specific response variable, so the wrong line gives biased predictions

Correct move:

Use the y on x line to predict y from x, and the x on y line to predict x from y

Wrong move:

Assuming a near-zero correlation means no relationship exists

Why:

Pearson's r only measures linear correlation, and strong non-linear relationships can give a near-zero r

Correct move:

Always plot a scatter graph to check for non-linear patterns before concluding no relationship exists

Wrong move:

Rounding regression coefficients to 2 decimal places early in calculations

Why:

Early rounding introduces significant error into final predictions

Correct move:

Keep full unrounded coefficients stored in your GDC until the final step of your calculation

6. Quick Reference Cheatsheet

Concept	Key Value/Formula	Note
Pearson's $r$	$- 1 \leq r \leq 1$	Measures strength of linear correlation
Strong correlation	$∣ r ∣ > 0.75$	Clear linear trend
Weak correlation	$∣ r ∣ < 0.25$	No clear linear trend
y on x line	$y = a + b x$	Predict $y$ from $x$
Gradient $b$	$b = r \frac{s _{y}}{s _{x}}$	Change in $y$ per unit $x$
Intercept $a$	$a = \overset{y}{ˉ} - b \overset{x}{ˉ}$	Predicted $y$ when $x = 0$
Interpolation	Predict within data range	Generally reliable
Extrapolation	Predict outside data range	Usually unreliable
Core rule	Correlation $\neq =$ Causation	Always mention lurking variables

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Calculate r and regression line
2022 · 2
Interpolation vs extrapolation
2023 · 1
Correlation vs causation discussion

Going deeper

syllabusIB AI HL Syllabus: Bivariate StatisticsOfficial syllabus reference

What's Next

Correlation and linear regression form the foundation of all statistical modeling for IB AI HL. These skills are directly applicable to your internal assessment, where you will be expected to analyze relationships between variables in your own dataset. Next, you will extend these ideas to non-linear regression, where you transform non-linear relationships to fit a linear model, and learn how to test if an observed correlation is statistically significant. Mastery of this subtopic is critical for all further statistical work in the course.