To compute the sample covariance, we use the following formula:

$\text{Cov}(x, y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$

Where:

• $n$ is the number of observations.
• $x_i$ and $y_i$ are the individual data points for the variables $x$ and $y$.
• $\bar{x}$ and $\bar{y}$ are the sample means of $x$ and $y$, respectively.

Step 1: Find the means of $x_i$ and $y_i$

The values of $x_i$ are: 4, 6, 11, 3, 16.

The values of $y_i$ are: 50, 60, 40, 60, 30.

$\bar{x} = \frac{4 + 6 + 11 + 3 + 16}{5} = \frac{40}{5} = 8$ $\bar{y} = \frac{50 + 60 + 40 + 60 + 30}{5} = \frac{240}{5} = 48$

Step 2: Compute the deviations $(x_i - \bar{x})$ and $(y_i - \bar{y})$

$x_i$	$y_i$	$x_i - \bar{x}$	$y_i - \bar{y}$	$(x_i - \bar{x})(y_i - \bar{y})$
4	50	4 - 8 = -4	50 - 48 = 2	(-4)(2) = -8
6	60	6 - 8 = -2	60 - 48 = 12	(-2)(12) = -24
11	40	11 - 8 = 3	40 - 48 = -8	(3)(-8) = -24
3	60	3 - 8 = -5	60 - 48 = 12	(-5)(12) = -60
16	30	16 - 8 = 8	30 - 48 = -18	(8)(-18) = -144

Step 3: Sum of the products of deviations

$\sum (x_i - \bar{x})(y_i - \bar{y}) = -8 + (-24) + (-24) + (-60) + (-144) = -260$

Step 4: Compute the sample covariance

$\text{Cov}(x, y) = \frac{-260}{5 - 1} = \frac{-260}{4} = -65$

Final Answer:

The sample covariance is $-65$.

Would you like more details or have any questions?

Here are some related questions for further practice:

1. How do we interpret a negative covariance?
2. What is the formula for the population covariance, and how does it differ from sample covariance?
3. Can the covariance be larger than the variance of individual variables?
4. How would you compute the correlation coefficient from the covariance?
5. How would the covariance change if all $x_i$ values were multiplied by 2?

Tip: Covariance indicates the direction of the linear relationship between two variables. A negative value suggests they move in opposite directions.