Let's break down the given formula step-by-step to compute the standard deviation.
### Step-by-Step Solution
Given:
- [tex]\(\sum d_i^2 = 1141.17\)[/tex]
- [tex]\(\sum d_i = -108.5\)[/tex]
- [tex]\(n = 17\)[/tex]
We need to compute the standard deviation using the formula:
[tex]\[ s_d = \sqrt{\frac{\sum d_i^2 - \left(\frac{(\sum d_i)^2}{n}\right)}{n-1}} \][/tex]
#### Step 1: Compute the numerator
First, let's compute the part inside the square root, starting with the numerator:
[tex]\[ \text{Numerator} = \sum d_i^2 - \left(\frac{(\sum d_i)^2}{n}\right) \][/tex]
Substitute the given values into the equation:
[tex]\[ \text{Numerator} = 1141.17 - \left(\frac{(-108.5)^2}{17}\right) \][/tex]
Calculate [tex]\(\left(\sum d_i\right)^2\)[/tex]:
[tex]\[ (-108.5)^2 = 11766.25 \][/tex]
Now, divide by [tex]\(n\)[/tex]:
[tex]\[ \frac{11766.25}{17} = 692.485294117647 \][/tex]
Now, subtract this from [tex]\(\sum d_i^2\)[/tex]:
[tex]\[ \text{Numerator} = 1141.17 - 692.485294117647 = 448.684705882353 \][/tex]
#### Step 2: Compute the denominator
The denominator is simply [tex]\(n - 1\)[/tex]:
[tex]\[ \text{Denominator} = 17 - 1 = 16 \][/tex]
#### Step 3: Compute the ratio
Next, compute the ratio of the numerator to the denominator:
[tex]\[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{448.684705882353}{16} \][/tex]
#### Step 4: Compute the standard deviation
Finally, take the square root of the ratio to find the standard deviation:
[tex]\[ s_d = \sqrt{\frac{448.684705882353}{16}} \][/tex]
[tex]\[ s_d = \sqrt{28.042794117647} \][/tex]
[tex]\[ s_d \approx 5.295544742294891 \][/tex]
### Conclusion
Therefore, the steps lead to:
- Numerator: [tex]\(448.684705882353\)[/tex]
- Denominator: [tex]\(16\)[/tex]
- Standard Deviation: [tex]\(\approx 5.295544742294891\)[/tex]
These are the values obtained through our computations.
