To compute the sample covariance between two variables $x$ and $y$, we use the following formula:

$s_{xy} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$

where:

• $n$ is the number of data points (in this case, $n = 5$),
• $\bar{x}$ is the mean of the $x$-values,
• $\bar{y}$ is the mean of the $y$-values,
• $x_i$ and $y_i$ are the individual data points for the two variables.

Step 1: Calculate the means of $x$ and $y$.

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{5 + 7 + 10 + 4 + 14}{5} = \frac{40}{5} = 8$

$\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{50 + 50 + 40 + 60 + 30}{5} = \frac{230}{5} = 46$

Step 2: Compute the covariance using the formula.

The deviations from the mean for $x$ and $y$ are:

$(x_1 - \bar{x}) = 5 - 8 = -3, \quad (y_1 - \bar{y}) = 50 - 46 = 4$ $(x_2 - \bar{x}) = 7 - 8 = -1, \quad (y_2 - \bar{y}) = 50 - 46 = 4$ $(x_3 - \bar{x}) = 10 - 8 = 2, \quad (y_3 - \bar{y}) = 40 - 46 = -6$ $(x_4 - \bar{x}) = 4 - 8 = -4, \quad (y_4 - \bar{y}) = 60 - 46 = 14$ $(x_5 - \bar{x}) = 14 - 8 = 6, \quad (y_5 - \bar{y}) = 30 - 46 = -16$

Now, compute the products $(x_i - \bar{x})(y_i - \bar{y})$:

$(-3)(4) = -12, \quad (-1)(4) = -4, \quad (2)(-6) = -12, \quad (-4)(14) = -56, \quad (6)(-16) = -96$

Sum the products:

$\sum (x_i - \bar{x})(y_i - \bar{y}) = -12 - 4 - 12 - 56 - 96 = -180$

Step 3: Compute the sample covariance:

$s_{xy} = \frac{1}{5 - 1} \times (-180) = \frac{-180}{4} = -45$

Final Answer:

The sample covariance is $-45$.

---

Would you like further details on covariance, or have any questions? Here are 5 related questions:

1. What is the difference between sample covariance and population covariance?
2. How do you interpret the value of covariance?
3. How do you calculate the correlation coefficient from covariance?
4. What effect does scaling the variables have on covariance?
5. How can covariance be used in multivariate analysis?

Tip: Always check the sign of the covariance. A negative covariance suggests that as one variable increases, the other tends to decrease.