Answer :
To solve this problem, we need to find the range and the standard deviation of the given data set.
Step 1: Calculate the Range
The range of a data set is found by subtracting the smallest value from the largest value in the set.
Given data:
[tex]\[ 75.7, 38.6, 75.7, 56.8, 54.9, 44.5, 66.3, 58.2, 75.7, 40.2 \][/tex]
1. Find the Maximum Value: The largest value is [tex]\( 75.7 \)[/tex].
2. Find the Minimum Value: The smallest value is [tex]\( 38.6 \)[/tex].
3. Calculate the Range:
[tex]\[
\text{Range} = \text{Max} - \text{Min} = 75.7 - 38.6 = 37.1
\][/tex]
Thus, the range of the data set is [tex]\( 37.1 \)[/tex].
Step 2: Calculate the Standard Deviation
The standard deviation measures how much the values in a data set deviate from the mean. Follow these steps:
1. Calculate the Mean:
[tex]\[
\text{Mean} = \frac{\sum \text{data values}}{n} = \frac{75.7 + 38.6 + 75.7 + 56.8 + 54.9 + 44.5 + 66.3 + 58.2 + 75.7 + 40.2}{10}
\][/tex]
[tex]\[
\text{Mean} = \frac{586.6}{10} = 58.66
\][/tex]
2. Calculate Each Deviation from the Mean and Square It:
[tex]\[
\text{For each } x, (x - \text{Mean})^2
\][/tex]
3. Calculate the Variance (Average of Squared Deviations):
[tex]\[
\text{Variance} = \frac{1}{n} \sum (x - \text{Mean})^2
\][/tex]
4. Standard Deviation (Square Root of Variance):
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\][/tex]
By calculating each step carefully, we find that the standard deviation is approximately [tex]\( 13.751 \)[/tex] when rounded to three decimal places.
In conclusion, for the given data set:
- The range is [tex]\( 37.1 \)[/tex].
- The standard deviation is [tex]\( 13.751 \)[/tex].
Step 1: Calculate the Range
The range of a data set is found by subtracting the smallest value from the largest value in the set.
Given data:
[tex]\[ 75.7, 38.6, 75.7, 56.8, 54.9, 44.5, 66.3, 58.2, 75.7, 40.2 \][/tex]
1. Find the Maximum Value: The largest value is [tex]\( 75.7 \)[/tex].
2. Find the Minimum Value: The smallest value is [tex]\( 38.6 \)[/tex].
3. Calculate the Range:
[tex]\[
\text{Range} = \text{Max} - \text{Min} = 75.7 - 38.6 = 37.1
\][/tex]
Thus, the range of the data set is [tex]\( 37.1 \)[/tex].
Step 2: Calculate the Standard Deviation
The standard deviation measures how much the values in a data set deviate from the mean. Follow these steps:
1. Calculate the Mean:
[tex]\[
\text{Mean} = \frac{\sum \text{data values}}{n} = \frac{75.7 + 38.6 + 75.7 + 56.8 + 54.9 + 44.5 + 66.3 + 58.2 + 75.7 + 40.2}{10}
\][/tex]
[tex]\[
\text{Mean} = \frac{586.6}{10} = 58.66
\][/tex]
2. Calculate Each Deviation from the Mean and Square It:
[tex]\[
\text{For each } x, (x - \text{Mean})^2
\][/tex]
3. Calculate the Variance (Average of Squared Deviations):
[tex]\[
\text{Variance} = \frac{1}{n} \sum (x - \text{Mean})^2
\][/tex]
4. Standard Deviation (Square Root of Variance):
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\][/tex]
By calculating each step carefully, we find that the standard deviation is approximately [tex]\( 13.751 \)[/tex] when rounded to three decimal places.
In conclusion, for the given data set:
- The range is [tex]\( 37.1 \)[/tex].
- The standard deviation is [tex]\( 13.751 \)[/tex].
