Consider a sample with data values of 10,20 ,12 ,17 , and 16 . Compute the z-score for each of the five observations (to 2 decimals). Enter negative values as negative numbers.

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.

Consider a sample with data values of 10, 20, 12, 17, and 16. Compute the z-score for each of the five observations (to 2 decimals). Enter negative values as negative numbers.

Learn how to calculate z-scores for the data set {10, 20, 12, 17, 16} using the mean and standard deviation formulas. Follow a step-by-step solution to compute each z-score, with results rounded to 2 decimals. Perfect for high school statistics students (Grades 9-12).

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