The steps given to find the standard deviation of the data-set are as follows:
- [tex]\bar{x} = \frac{\sum x}{n} = 8[/tex]
- [tex]\sum (x - \bar{x})^2 = 10[/tex]
- [tex]\frac{\sum (x - \bar{x})^2}{n} = 2[/tex]
- [tex]\sqrt{\frac{\sum (x - \bar{x})^2}{n}} = 2[/tex]
What are the mean and the standard deviation of a data-set?
- The mean of a data-set is given by the sum of all values in the data-set, divided by the number of values.
- The standard deviation of a data-set is given by the square root of the sum of the differences squared between each observation and the mean, divided by the number of values.
For this problem, the are 5 values, hence:
n = 5.
The sum of the values is given by:
6 + 7 + 8 + 9 + 10 = 40.
Hence:
[tex]\sum x = 40[/tex]
The mean is given by:
40/5 = 8.
Hence:
[tex]\bar{x} = \frac{\sum x}{n} = 8[/tex]
The sum of the differences squared between each observation and the mean is:
(6 - 8)² + (7 - 8)² + (8 - 8)² + (9 - 8)² + (10 - 8)² = 10.
Hence:
[tex]\sum (x - \bar{x})^2 = 10[/tex]
Dividing by the number of values, we have that 10/5 = 2, hence:
[tex]\frac{\sum (x - \bar{x})^2}{n} = 2[/tex]
Taking the square root, the standard deviation is:
[tex]\sqrt{\frac{\sum (x - \bar{x})^2}{n}} = 2[/tex]
More can be learned about the standard deviation of a data-set at https://brainly.com/question/28225633
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