Question

we are going to calculate the standard deviation for the following set of sample data. 2 11 14 6 8 1) calculate the mean. \\(\bar{x} = \\) 2) fill in the table below: fill in the differences of each data value from the mean, then the squared differences. \\(\

$$\begin{array}{|l|l|l|} \\hline x & x - \\bar{x} & (x - \\bar{x})^2 \\\\ \\hline 2 & -6.2 & 38.44 \\\\ \\hline 11 & 2.8 & 7.84 \\\\ \\hline 14 & 5.8 & 33.64 \\\\ \\hline 6 & -2.2 & 4.84 \\\\ \\hline 8 & -0.2 &.04 \\\\ \\hline & \\sum (x - \\bar{x})^2 = & 84.8 \\\\ \\hline \\end{array}$$

\\) enter an integer or decimal number more... 3) calculate the sample standard deviation (s). \\( s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} = \\) 4.60 (please round your answer to two decimal places)

Explanation:

1) Calculate the mean

Step1: Sum the data values

The data values are \(2, 11, 14, 6, 8\). The sum is \(2 + 11 + 14 + 6 + 8 = 41\).

Step2: Divide by the number of data points

There are \(n = 5\) data points. The mean \(\bar{x}=\frac{41}{5}=8.2\).

Step1: List the squared differences

We have the squared differences: \(38.44, 7.84, 33.64, 4.84, 0.04\).

Step2: Sum the squared differences

\(38.44+7.84 + 33.64+4.84 + 0.04=84.8\) (matches the given sum).

Step1: Identify \(n\) and \(\sum(x - \bar{x})^2\)

We know \(n = 5\) and \(\sum(x - \bar{x})^2=84.8\).

Step2: Use the sample standard deviation formula

The formula for sample standard deviation is \(s=\sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}\). Substitute the values: \(\frac{84.8}{5 - 1}=\frac{84.8}{4}=21.2\). Then take the square root: \(s=\sqrt{21.2}\approx4.60\) (rounded to two decimal places).

Answer:

\(8.2\)

2) Fill in the table (already partially filled, but for completeness, let's verify the sum of squared differences)