Question

In the data set below, the two numbers in each row add to 100. That is:
x1+ y1= 100, x2+ y2 = 100, …, xn + yn = 100
The average of the x-column is 60 and the SD is 10.
x y
x1 y1
x2 y2
. .
. .
. .
xn yn

Find average of the y column.
Find the correlation between x and y (and is this the same as cov(x,y)?

Answer 1

Let \(\bar x\) and \(\bar y\) denote the averages for the \(x\) and \(y\) columns respectively. Then
\[\begin{align*}
\bar x+\bar y&=\frac{x_1+\cdots+x_n}{n}+\frac{y_1+\cdots+y_n}{n}\\[1ex]
&=\frac{(x_1+y_1)+\cdots(x_n+y_n)}{n}\\[1ex]
&=\frac{100n}{n}\\[1ex]
&=100
\end{align*}\]You know \(\bar x\), so you can easily find \(\bar y\).

By correlation, I assume you're asked to find the correlation coefficient, which would be given by
\[\rho=\frac{\mathrm{cov}(x,y)}{\mathrm{var}(x)\mathrm{var}(y)}\]so this is equal to the covariance if \(\mathrm{var}(x)\mathrm{var}(y)=1\).