To find the standard deviation of the given set of observations, we'll follow these steps:

1. Find the mean (average) of the observations:

   $\text{Mean} = \frac{\sum \text{observations}}{\text{number of observations}}$

2. Calculate the variance:

   $\text{Variance} = \frac{\sum (\text{each observation} - \text{mean})^2}{\text{number of observations}}$

3. Find the standard deviation:

   $\text{Standard Deviation} = \sqrt{\text{Variance}}$

Let's go through these steps:

Step 1: Find the Mean

Given observations: $4, 8, 11, 17, 20, 24, 32$

$\text{Mean} = \frac{4 + 8 + 11 + 17 + 20 + 24 + 32}{7} = \frac{116}{7} \approx 16.57$

Step 2: Calculate the Variance

Now, we'll calculate the squared difference from the mean for each observation:

(4 - 16.57)^2 &\approx 158.27 \\ (8 - 16.57)^2 &\approx 73.93 \\ (11 - 16.57)^2 &\approx 31.06 \\ (17 - 16.57)^2 &\approx 0.19 \\ (20 - 16.57)^2 &\approx 11.83 \\ (24 - 16.57)^2 &\approx 55.10 \\ (32 - 16.57)^2 &\approx 237.79 \\ \end{align*}$$ Summing these: $$\text{Sum of squared differences} \approx 568.17$$ Now, divide by the number of observations (7): $$\text{Variance} \approx \frac{568.17}{7} \approx 81.17$$ ### Step 3: Find the Standard Deviation $$\text{Standard Deviation} = \sqrt{81.17} \approx 9.01$$ So, the standard deviation of the given set of observations is approximately **9.01**. Would you like further details or have any questions? ### Related Questions: 1. How is the standard deviation used in real-world data analysis? 2. What is the difference between population and sample standard deviation? 3. How would the standard deviation change if another observation was added? 4. Can standard deviation ever be negative? 5. What does a low standard deviation indicate about a data set? ### Tip: Always ensure that you're distinguishing between population and sample when calculating the standard deviation, as the formula differs slightly!