Topic 2.5 Notes – Correlation

What Correlation r Is

When you look at a scatterplot, you judge two things:

• Do the points trend up or down?
• How tightly do they cluster around a line?

Correlation $r$ turns that visual judgment into a number.

Here’s what $r$ tells you:

• It measures the direction and strength of a linear association between two quantitative variables.
• It is always between −1 and 1, inclusive.
• $r > 0$ → positive association (as x increases, y tends to increase).
• $r < 0$ → negative association (as x increases, y tends to decrease).
• $r = 0$ → no linear association.
• $r = 1$ or $r = -1$ → perfect linear relationship (all points lie exactly on a line).
• It is unit-free (because it’s based on standardized values).
• It is not resistant to outliers.

Think visually. The sign matches the slope direction. The closer the points are to a straight line, the closer $|r|$ is to 1.

Here’s a quick visual comparison of three different correlations:

Scatterplots with strong positive, weak positive, and strong negative correlation

On the left, the points hug an upward-sloping line, so $r$ is close to 1. In the middle, the trend is positive but scattered, so $r$ is closer to 0. On the right, the points tightly follow a downward-sloping line, so $r$ is close to −1.

One important limitation shows up a lot on tests: r only measures linear relationships. A curved pattern can have a strong relationship but a small $r$.

How to Determine Correlation

The formula

$$r = \frac{1}{n - 1} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s_x} \right)\left( \frac{y_i - \bar{y}}{s_y} \right)$$

What’s happening conceptually:

1. Each value is converted to a z-score (standardized using its mean and standard deviation).
2. Pair the standardized x and y values.
3. Multiply each pair.
4. Average those products (divide by $n-1$).

Why this works:

• If x and y are often both above their means or both below, the products are positive → $r$ positive.
• If one tends to be above its mean when the other is below, products are negative → $r$ negative.
• If there’s no consistent pattern, positives and negatives cancel → $r$ near 0.

You almost never compute this by hand. On a TI-84:

• Enter x-values in L1, y-values in L2.
• Run LinReg(ax + b).
• Make sure Diagnostics On so you see $r$.

Understanding what the formula represents matters more than memorizing the mechanics.

How to Interpret r in Context

An interpretation must include:

1. Strength
2. Direction
3. The word linear
4. The actual variables in context

Suppose $r = -0.82$ for hours of sleep and reaction time.

A correct interpretation would sound like:

There is a strong negative linear association between hours of sleep and reaction time for the students in this study.

Notice:

• We didn’t say “r%.”
• We didn’t mention slope.
• We didn’t claim sleep causes reaction time changes.

Some rough guidelines for strength (no hard cutoffs):

• $|r|$ around 0.2 → weak
• $|r|$ around 0.5 → moderate
• $|r|$ 0.7 or higher → strong

On FRQs, leaving out “linear” or failing to name both variables can cost you.

What Correlation Does Not Tell You

Correlation is not causation

Even if $r$ is strong:

• There could be a lurking variable.
• The direction of influence could be reversed.
• The relationship could be coincidental.

Unless the data come from a well-designed experiment, you cannot say one variable causes the other.

r only measures linear association

You must look at the scatterplot.

In the example below, the points clearly follow a curved, U-shaped pattern. The linear correlation is about $r \approx 0.05$, which suggests almost no linear association, even though there is a very strong nonlinear relationship.

A strong curved (U-shaped) relationship whose least-squares line is almost flat, so $r \approx 0.05$

This is why AP questions often give you both a scatterplot and $r$. Correlation only measures linear association. If the pattern is curved, a linear model is not appropriate, even if the relationship between the variables is clearly strong.

The reverse also happens: curved data, like steady exponential growth, can give $r \approx 0.9$ or higher. A large $|r|$ alone does not prove the pattern is linear, so check the scatterplot (or a residual plot) before choosing a linear model.

r is not resistant

One extreme point, especially far in the x-direction, can dramatically change $r$. If you see an outlier, mentally question how much it might be influencing the correlation.