To calculate the standard deviation of the data set given, we need to follow these steps:
- Find the mean (average) of the data set.
- Calculate the differences from the mean for each data point.
- Square each of these differences.
- Find the average of these squared differences (this is the variance).
- Take the square root of the variance to get the standard deviation.
Step 1: Find the Mean
The data points are: 15, 18, 11, 17, and 14.
Mean \( \mu \) is calculated as follows:
\[ \mu = \frac{15 + 18 + 11 + 17 + 14}{5} = \frac{75}{5} = 15 \]
Step 2: Calculate the Differences from the Mean
Now we calculate the difference of each data point from the mean (15):
- For 15: \( 15 - 15 = 0 \)
- For 18: \( 18 - 15 = 3 \)
- For 11: \( 11 - 15 = -4 \)
- For 17: \( 17 - 15 = 2 \)
- For 14: \( 14 - 15 = -1 \)
Step 3: Square Each Difference
Now we square each difference:
- For 0: \( 0^2 = 0 \)
- For 3: \( 3^2 = 9 \)
- For -4: \( (-4)^2 = 16 \)
- For 2: \( 2^2 = 4 \)
- For -1: \( (-1)^2 = 1 \)
Step 4: Find the Variance
Now we find the average of the squared differences:
\[ \text{Variance} = \frac{0 + 9 + 16 + 4 + 1}{5} = \frac{30}{5} = 6 \]
Step 5: Calculate the Standard Deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{6} \approx 2.45 \]
Final Result
Thus, the standard deviation of the data set, rounded to the nearest hundredth, is:
\[ \boxed{2.45} \]