Question

If x¯=y¯=0, ∑xiyi=12, σx=2, σy=3 and n=10, then the coefficient of correlation is

Answer 1

0.2

Answer 2

The coefficient of correlation (Pearson's correlation coefficient), denoted by $r$, is given by the formula: $r = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y}$ where $\text{Cov}(x, y)$ is the covariance between $x$ and $y$, $\sigma_x$ is the standard deviation of $x$, and $\sigma_y$ is the standard deviation of $y$.

The formula for covariance is: $\text{Cov}(x, y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n}$ Given that $\bar{x} = 0$ and $\bar{y} = 0$, the covariance formula simplifies to: $\text{Cov}(x, y) = \frac{\sum (x_i - 0)(y_i - 0)}{n} = \frac{\sum x_i y_i}{n}$

We are given the following values: $\sum x_i y_i = 12$ $n = 10$ $\sigma_x = 2$ $\sigma_y = 3$

First, calculate the covariance: $\text{Cov}(x, y) = \frac{12}{10} = 1.2$

Next, substitute the covariance and standard deviations into the correlation coefficient formula: $r = \frac{1.2}{2 \times 3}$ $r = \frac{1.2}{6}$ $r = 0.2$

The coefficient of correlation is 0.2.