Answer :
The sample standard deviation, to the nearest thousandth, is [tex]7.246[/tex].
To find the sample standard deviation of the given data set (86, 83, 76, 82, 67, 69, 82, 82, 69), follow these steps:
- Calculate the mean (average):
[tex]\bar{x} = \frac{\sum x}{n} = \frac{86 + 83 + 76 + 82 + 67 + 69 + 82 + 82 + 69}{9} = \frac{696}{9} = 77.33[/tex]
- Calculate each deviation from the mean and square it:
[tex](86 - 77.33)^2 = 75.17[/tex]
[tex](83 - 77.33)^2 = 32.15[/tex]
[tex](76 - 77.33)^2 = 1.77[/tex]
[tex](82 - 77.33)^2 = 21.81[/tex]
[tex](67 - 77.33)^2 = 106.71[/tex]
[tex](69 - 77.33)^2 = 69.39[/tex]
[tex](82 - 77.33)^2 = 21.81[/tex]
[tex](82 - 77.33)^2 = 21.81[/tex]
[tex](69 - 77.33)^2 = 69.39[/tex]
- Sum these squared deviations:
[tex]\sum (x - \bar{x})^2 = 75.17 + 32.15 + 1.77 + 21.81 + 106.71 + 69.39 + 21.81 + 21.81 + 69.39 = 420.01[/tex]
- Divide by the number of data points minus 1 (n-1) to get the sample variance:
[tex]s^2 = \frac{\sum (x - \bar{x})^2}{n-1} = \frac{420.01}{8} = 52.50[/tex]
- Take the square root of the variance to get the sample standard deviation:
[tex]s = \sqrt{s^2} = \sqrt{52.50} \approx 7.246[/tex]
