Sure, let's find the standard deviation of the given sample of distances: 5, 40, 10, 20, 40.
Here is a step-by-step solution:
### Step 1: Calculate the Mean Distance
First, we find the mean (average) of the distances.
[tex]\[ \text{Mean distance} = \frac{5 + 40 + 10 + 20 + 40}{5} = \frac{115}{5} = 23.0 \][/tex]
### Step 2: Subtract the Mean and Square the Result
Next, we subtract the mean from each distance and square the result:
- For 5: [tex]\( (5 - 23.0)^2 = (-18.0)^2 = 324.0 \)[/tex]
- For 40: [tex]\( (40 - 23.0)^2 = 17.0^2 = 289.0 \)[/tex]
- For 10: [tex]\( (10 - 23.0)^2 = (-13.0)^2 = 169.0 \)[/tex]
- For 20: [tex]\( (20 - 23.0)^2 = (-3.0)^2 = 9.0 \)[/tex]
- For 40: [tex]\( (40 - 23.0)^2 = 17.0^2 = 289.0 \)[/tex]
### Step 3: Sum of Squared Differences
We sum all the squared differences:
[tex]\[ 324.0 + 289.0 + 169.0 + 9.0 + 289.0 = 1080.0 \][/tex]
### Step 4: Calculate the Sample Variance
To find the variance, we divide the sum of squared differences by the number of data points minus one (degrees of freedom, [tex]\( n-1 \)[/tex]):
[tex]\[ \text{Variance} = \frac{\sum (x - \text{mean})^2 }{n-1} = \frac{1080.0}{5-1} = \frac{1080.0}{4} = 270.0 \][/tex]
### Step 5: Calculate the Standard Deviation
Finally, to find the standard deviation, we take the square root of the variance:
[tex]\[ \text{Standard Deviation} = \sqrt{270.0} \approx 16.431676725154983 \][/tex]
### Step 6: Round the Standard Deviation
We round the standard deviation to two decimal places:
[tex]\[ \text{Standard Deviation (rounded)} \approx 16.43 \][/tex]
So, the standard deviation of this sample of distances, rounded to two decimal places, is [tex]\( 16.43 \)[/tex].
