Answer :
To find the sample standard deviation for the data set [tex]\(23, 27, 12, 17, 16\)[/tex], you can follow these steps:
1. Calculate the mean (average) of the data set:
[tex]\[
\text{Mean} = \frac{23 + 27 + 12 + 17 + 16}{5} = \frac{95}{5} = 19
\][/tex]
2. Find the squared differences from the mean:
- For [tex]\(23\)[/tex]: [tex]\((23 - 19)^2 = 4^2 = 16\)[/tex]
- For [tex]\(27\)[/tex]: [tex]\((27 - 19)^2 = 8^2 = 64\)[/tex]
- For [tex]\(12\)[/tex]: [tex]\((12 - 19)^2 = (-7)^2 = 49\)[/tex]
- For [tex]\(17\)[/tex]: [tex]\((17 - 19)^2 = (-2)^2 = 4\)[/tex]
- For [tex]\(16\)[/tex]: [tex]\((16 - 19)^2 = (-3)^2 = 9\)[/tex]
3. Calculate the variance of the data set:
- Sum the squared differences: [tex]\(16 + 64 + 49 + 4 + 9 = 142\)[/tex]
- Divide by the number of data points minus 1 (because it is a sample, not a whole population):
[tex]\[
\text{Variance} = \frac{142}{5 - 1} = \frac{142}{4} = 35.5
\][/tex]
4. Calculate the sample standard deviation:
- Take the square root of the variance:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{35.5} \approx 5.96
\][/tex]
Therefore, the sample standard deviation of the given data set is approximately 5.96, which matches the choice of 6.0 when rounded to the nearest value from the options.
1. Calculate the mean (average) of the data set:
[tex]\[
\text{Mean} = \frac{23 + 27 + 12 + 17 + 16}{5} = \frac{95}{5} = 19
\][/tex]
2. Find the squared differences from the mean:
- For [tex]\(23\)[/tex]: [tex]\((23 - 19)^2 = 4^2 = 16\)[/tex]
- For [tex]\(27\)[/tex]: [tex]\((27 - 19)^2 = 8^2 = 64\)[/tex]
- For [tex]\(12\)[/tex]: [tex]\((12 - 19)^2 = (-7)^2 = 49\)[/tex]
- For [tex]\(17\)[/tex]: [tex]\((17 - 19)^2 = (-2)^2 = 4\)[/tex]
- For [tex]\(16\)[/tex]: [tex]\((16 - 19)^2 = (-3)^2 = 9\)[/tex]
3. Calculate the variance of the data set:
- Sum the squared differences: [tex]\(16 + 64 + 49 + 4 + 9 = 142\)[/tex]
- Divide by the number of data points minus 1 (because it is a sample, not a whole population):
[tex]\[
\text{Variance} = \frac{142}{5 - 1} = \frac{142}{4} = 35.5
\][/tex]
4. Calculate the sample standard deviation:
- Take the square root of the variance:
[tex]\[
\text{Sample Standard Deviation} = \sqrt{35.5} \approx 5.96
\][/tex]
Therefore, the sample standard deviation of the given data set is approximately 5.96, which matches the choice of 6.0 when rounded to the nearest value from the options.
