Scatter Plots and Correlation: Quick Revision

A brief note on Scatter Plots and Correlation. Read it before a test. Try each question at the end before you open its answer.

Scatter plots

• One dot for each individual, the explanatory variable $x$ across and the response $y$ up.
• Four words: direction (positive or negative), form (linear, curved, clusters), strength (strong or weak), outliers.
• Centre: the point of the means $(\bar{x}, \bar{y})$.

The correlation coefficient

• Sums: $S_{xx} = \sum (x - \bar{x})^2$, $S_{yy} = \sum (y - \bar{y})^2$, $S_{xy} = \sum (x - \bar{x})(y - \bar{y})$.
• Formula: $r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$.
• With z-scores: $r = \dfrac{1}{n - 1}\sum z_x z_y$ (sample standard deviations).
• Range: $-1 \le r \le 1$. The sign is the direction and the size is the strength.
• No unit. A new unit, or adding a number, leaves $r$ alone. A negative factor changes its sign.

What r cannot see

• Curves: a strong curve can give $r$ near $0$.
• Outliers: one point can change $r$ a lot.
• Slope and cause: $r$ gives neither.
• Range: a narrow range of $x$ usually lowers $r$.

A worked example

A made-up lemonade stall records the noon temperature and the cups sold on four days: $(20, 15)$, $(24, 21)$, $(28, 22)$ and $(32, 30)$, temperature in $^\circ\text{C}$ first.

1. Means. $\bar{x} = 26\ ^\circ\text{C}$ and $\bar{y} = 22$ cups.
2. Sums. $S_{xx} = 80$, $S_{yy} = 114$ and $S_{xy} = 92$.
3. r. $r = \dfrac{92}{\sqrt{80 \times 114}} = 0.9634$, a strong positive pattern.

Practice with new numbers

A program to try

The program finds the sums and $r$ of the worked example, and checks $r$ with the statistics module.

Mistakes to avoid

• Dividing by $n$ with sample standard deviations. Use $n - 1$.
• Calling a negative $r$ weak. The size gives the strength.
• Saying $r = 0$ means no relationship. It means no straight-line relationship.
• Trusting $r$ without the plot. Draw the scatter plot first.

Check yourself

1. Which shows the stronger straight-line pattern, $r = -0.7$ or $r = 0.5$?

2. Three points have $(x - \bar{x}, y - \bar{y})$ of $(-2, -1)$, $(0, 0)$ and $(2, 1)$. Find $r$.

3. A data set gives $r = 0.85$. Every $x$ is multiplied by $-2$. What is the new $r$?

4. Points follow a hill shape: they rise and then fall. What will $r$ be close to?

Sources

• OpenStax, Introductory Statistics, Chapter 12 (Linear Regression and Correlation).