To determine the standard deviation of the given sample [tex]\(1, 1, 1, 2, 3, 3, 3\)[/tex], let's go through the necessary steps:
1. Calculate the Mean (Average):
- Add all the values together and divide by the number of values:
[tex]\[
\text{Mean} = \frac{1 + 1 + 1 + 2 + 3 + 3 + 3}{7} = \frac{14}{7} = 2.0
\][/tex]
2. Calculate the Variance:
- Variance is the average of the squared differences from the mean:
[tex]\[
\text{Variance} = \frac{(1-2)^2 + (1-2)^2 + (1-2)^2 + (2-2)^2 + (3-2)^2 + (3-2)^2 + (3-2)^2}{7}
\][/tex]
- Calculate each squared difference:
[tex]\[
(1-2)^2 = 1, \quad (1-2)^2 = 1, \quad (1-2)^2 = 1, \quad (2-2)^2 = 0, \quad (3-2)^2 = 1, \quad (3-2)^2 = 1, \quad (3-2)^2 = 1
\][/tex]
- Sum these squared differences:
[tex]\[
1 + 1 + 1 + 0 + 1 + 1 + 1 = 6
\][/tex]
- Divide by the number of values (n):
[tex]\[
\text{Variance} = \frac{6}{7} \approx 0.8571428571428571
\][/tex]
3. Calculate the Standard Deviation:
- The standard deviation is the square root of the variance:
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.8571428571428571} \approx 0.9258200997725514
\][/tex]
Thus, the standard deviation of the sample [tex]\(1, 1, 1, 2, 3, 3, 3\)[/tex] is approximately [tex]\(0.93\)[/tex] (to two decimal places).
Given the answer choices:
1. 1
2. 2
3. 3
4. 0
The correct answer is none of the provided options exactly match [tex]\(0.93\)[/tex]. However, the closest standard choice not given in the options can lead to identifying any numerical close value in further options if possible.
