Consider the following data set with a mean of 12:

9, 11, 12, 16

**Using the equation below or the standard deviation formula in Excel, calculate the standard deviation for this data set. Answer choices are rounded to the hundredths place.**



A.)

1.54

B.)

2.55

C.)

2.94

D.)

5.09

To calculate the standard deviation of a data set, we first need to understand the formula for standard deviation for a sample:

\[
\text{Standard Deviation} (s) = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]

where:
- \( x_i \) represents each data point.
- \( \bar{x} \) is the mean of the data set.
- \( n \) is the number of data points.

### Step-by-Step Calculation

Given the data set: \( 9, 11, 12, 16 \) with a mean (\( \bar{x} \)) of 12.

1. **Calculate the differences between each data point and the mean:**
   - \( 9 - 12 = -3 \)
   - \( 11 - 12 = -1 \)
   - \( 12 - 12 = 0 \)
   - \( 16 - 12 = 4 \)

2. **Square each difference:**
   - \( (-3)^2 = 9 \)
   - \( (-1)^2 = 1 \)
   - \( (0)^2 = 0 \)
   - \( (4)^2 = 16 \)

3. **Sum of squared differences:**
   \[
   9 + 1 + 0 + 16 = 26
   \]

4. **Calculate the variance:**
   Since we are calculating the standard deviation for a sample, divide by \( n - 1 = 4 - 1 = 3 \):

   \[
   \text{Variance} = \frac{26}{3} \approx 8.67
   \]

5. **Calculate the standard deviation:**
   \[
   s = \sqrt{8.67} \approx 2.94
   \]

### Conclusion

The correct answer is:

**C.) 2.94**

Would you like more details or have any questions?

### Related Questions:

1. What is the difference between population and sample standard deviation?
2. How does the standard deviation measure data spread?
3. How would the standard deviation change if an additional value were added to the data set?
4. What are some practical examples where standard deviation is useful?
5. How do you calculate variance in Excel?

**Tip:** Remember that the standard deviation provides insight into the variability or spread of your data set.

Learn how to calculate the standard deviation for a data set with a mean of 12: 9, 11, 12, 16. Step-by-step solution provided using the standard deviation formula for a sample. Answer choices are rounded to the hundredths place.

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