To compute the z-score for each observation, we use the following formula:

$z = \frac{x - \mu}{\sigma}$

Where:

• $x$ is an individual data value.
• $\mu$ is the mean of the data set.
• $\sigma$ is the standard deviation of the data set.

Step 1: Calculate the mean $\mu$

$\mu = \frac{10 + 20 + 12 + 17 + 16}{5} = \frac{75}{5} = 15$

Step 2: Calculate the standard deviation $\sigma$

First, find the squared differences from the mean for each data point:

$(10 - 15)^2 = (-5)^2 = 25$ $(20 - 15)^2 = (5)^2 = 25$ $(12 - 15)^2 = (-3)^2 = 9$ $(17 - 15)^2 = (2)^2 = 4$ $(16 - 15)^2 = (1)^2 = 1$

Now, sum these squared differences:

$25 + 25 + 9 + 4 + 1 = 64$

Next, divide the sum by the number of data points minus 1 (this is the sample variance):

$\text{Variance} = \frac{64}{5 - 1} = \frac{64}{4} = 16$

Finally, take the square root of the variance to get the standard deviation:

$\sigma = \sqrt{16} = 4$

Step 3: Calculate the z-scores

Now, compute the z-score for each observation using the formula $z = \frac{x - \mu}{\sigma}$.

For $x = 10$:

$z = \frac{10 - 15}{4} = \frac{-5}{4} = -1.25$

For $x = 20$:

$z = \frac{20 - 15}{4} = \frac{5}{4} = 1.25$

For $x = 12$:

$z = \frac{12 - 15}{4} = \frac{-3}{4} = -0.75$

For $x = 17$:

$z = \frac{17 - 15}{4} = \frac{2}{4} = 0.5$

For $x = 16$:

$z = \frac{16 - 15}{4} = \frac{1}{4} = 0.25$

Final z-scores (rounded to 2 decimals):

• $z_{10} = -1.25$
• $z_{20} = 1.25$
• $z_{12} = -0.75$
• $z_{17} = 0.50$
• $z_{16} = 0.25$

Would you like further clarification or additional details?

Here are 5 related questions:

1. What does a z-score represent in statistics?
2. How do z-scores relate to standard deviation?
3. Why is it important to calculate z-scores in data analysis?
4. Can you explain the difference between population and sample standard deviation?
5. What does it mean if a z-score is negative?

Tip: Z-scores allow you to compare data points from different distributions by standardizing them.