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We appreciate your visit to For the following set of data find the sample standard deviation to the nearest thousandth 86 83 76 82 67 69 82 82 69. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

For the following set of data, find the sample standard deviation to the nearest thousandth:

86, 83, 76, 82, 67, 69, 82, 82, 69

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]

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