Scatter plots and covariance

Two variables, one picture

When each unit carries two quantitative measurements, the first move is a scatter plot: the explanatory variable $x$ across, the response $y$ up, one point per unit. Describe it in three words.

• Direction — positive (the cloud rises left to right), negative (it falls), or none.
• Form — linear (a straight band), curved, or clustered.
• Strength — how tightly the points hug the form: strong, moderate, weak.

And always: any point that sits away from the pattern.

Worked example — Northfield Credit's eight clients

Northfield Credit, a Waterloo lender, pulled eight clients' annual income ($000) and credit score.

Plotted, the points climb steadily from (35, 610) to (110, 760) in a narrow band. Positive, linear, strong. One small wrinkle — client 3 earns more than client 2 but scores slightly lower — and nothing that leaves the band.

The sums table, extended

Module 3's habit was $\sum x$ and $\sum x^2$. For two variables the table grows three columns — $y$, $y^2$ and the cross-product $xy$ — and every quantity in this module and Module 13 reads off its totals:

$$n=8,\sum x=555,\sum y=5,500,\sum x^2=43,035,\sum y^2=3,801,150,\sum xy=390,925$$

so $\bar{x}=555/8=69.375$ and $\bar{y}=5,500/8=687.5$.

Covariance — the direction, in a number

The sample covariance averages the product of the two deviations:

$$s_{xy}=\frac{\sum (x-\bar{x})(y-\bar{y})}{n-1}=\frac{\sum xy-n\bar{x}\bar{y}}{n-1}$$

When $x$ and $y$ are above their means together (or below together) the products are positive; when one is above and the other below, negative. For Northfield:

$$s_{xy}=\frac{390,925-8\times 69.375\times 687.5}{7}=\frac{390,925-381,562.5}{7}=\frac{9,362.5}{7}=1,337.5$$

Positive — the direction the plot showed. But 1,337.5 what? Thousand-dollars-times-points. Re-express income in dollars and the covariance becomes 1,337,500 with nothing about the relationship having changed. Covariance carries the units of both variables, so its size means nothing by itself; only its sign is interpretable. Lesson 4.2 divides the units out.

Reading a flat cloud

A horizontal, shapeless cloud has covariance near zero and is described as showing no linear association. It does not say the variables are unrelated: a U-shaped pattern — cost against batch size, say, high at both extremes — has a covariance near zero and a very real relationship. The plot tells you what the number cannot.