Answer :
To find the sample standard deviation of the given data set, follow these steps:
1. List the Data:
The data set is: 129, 131, 128, 123, 124, 129, 139, 139.
2. Calculate the Mean:
Add up all the numbers and divide by the amount of numbers.
[tex]\[
\text{Mean} = \frac{129 + 131 + 128 + 123 + 124 + 129 + 139 + 139}{8} = \frac{1042}{8} = 130.25
\][/tex]
3. Calculate the Deviations from the Mean:
Subtract the mean from each number in the data set:
- [tex]\(129 - 130.25 = -1.25\)[/tex]
- [tex]\(131 - 130.25 = 0.75\)[/tex]
- [tex]\(128 - 130.25 = -2.25\)[/tex]
- [tex]\(123 - 130.25 = -7.25\)[/tex]
- [tex]\(124 - 130.25 = -6.25\)[/tex]
- [tex]\(129 - 130.25 = -1.25\)[/tex]
- [tex]\(139 - 130.25 = 8.75\)[/tex]
- [tex]\(139 - 130.25 = 8.75\)[/tex]
4. Calculate the Squared Deviations:
Square each deviation:
- [tex]\((-1.25)^2 = 1.5625\)[/tex]
- [tex]\(0.75^2 = 0.5625\)[/tex]
- [tex]\((-2.25)^2 = 5.0625\)[/tex]
- [tex]\((-7.25)^2 = 52.5625\)[/tex]
- [tex]\((-6.25)^2 = 39.0625\)[/tex]
- [tex]\((-1.25)^2 = 1.5625\)[/tex]
- [tex]\(8.75^2 = 76.5625\)[/tex]
- [tex]\(8.75^2 = 76.5625\)[/tex]
5. Calculate the Sample Variance:
Divide the sum of squared deviations by the number of observations minus one (n-1):
[tex]\[
\text{Sum of squared deviations} = 1.5625 + 0.5625 + 5.0625 + 52.5625 + 39.0625 + 1.5625 + 76.5625 + 76.5625 = 253.5
\][/tex]
[tex]\[
\text{Sample Variance} = \frac{253.5}{7} \approx 36.21
\][/tex]
6. Calculate the Sample Standard Deviation:
Take the square root of the sample variance:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{36.21} \approx 6.02
\][/tex]
So, the sample standard deviation of the data set, to the nearest hundredth, is 6.02.
1. List the Data:
The data set is: 129, 131, 128, 123, 124, 129, 139, 139.
2. Calculate the Mean:
Add up all the numbers and divide by the amount of numbers.
[tex]\[
\text{Mean} = \frac{129 + 131 + 128 + 123 + 124 + 129 + 139 + 139}{8} = \frac{1042}{8} = 130.25
\][/tex]
3. Calculate the Deviations from the Mean:
Subtract the mean from each number in the data set:
- [tex]\(129 - 130.25 = -1.25\)[/tex]
- [tex]\(131 - 130.25 = 0.75\)[/tex]
- [tex]\(128 - 130.25 = -2.25\)[/tex]
- [tex]\(123 - 130.25 = -7.25\)[/tex]
- [tex]\(124 - 130.25 = -6.25\)[/tex]
- [tex]\(129 - 130.25 = -1.25\)[/tex]
- [tex]\(139 - 130.25 = 8.75\)[/tex]
- [tex]\(139 - 130.25 = 8.75\)[/tex]
4. Calculate the Squared Deviations:
Square each deviation:
- [tex]\((-1.25)^2 = 1.5625\)[/tex]
- [tex]\(0.75^2 = 0.5625\)[/tex]
- [tex]\((-2.25)^2 = 5.0625\)[/tex]
- [tex]\((-7.25)^2 = 52.5625\)[/tex]
- [tex]\((-6.25)^2 = 39.0625\)[/tex]
- [tex]\((-1.25)^2 = 1.5625\)[/tex]
- [tex]\(8.75^2 = 76.5625\)[/tex]
- [tex]\(8.75^2 = 76.5625\)[/tex]
5. Calculate the Sample Variance:
Divide the sum of squared deviations by the number of observations minus one (n-1):
[tex]\[
\text{Sum of squared deviations} = 1.5625 + 0.5625 + 5.0625 + 52.5625 + 39.0625 + 1.5625 + 76.5625 + 76.5625 = 253.5
\][/tex]
[tex]\[
\text{Sample Variance} = \frac{253.5}{7} \approx 36.21
\][/tex]
6. Calculate the Sample Standard Deviation:
Take the square root of the sample variance:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{36.21} \approx 6.02
\][/tex]
So, the sample standard deviation of the data set, to the nearest hundredth, is 6.02.
