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\begin{tabular}{|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$x-\bar{x}$[/tex] & [tex]$(x-\bar{x})^2$[/tex] \\
\hline
6 & -3.4 & 11.56 \\
\hline
15 & 5.6 & 31.36 \\
\hline
13 & 3.6 & 12.96 \\
\hline
2 & -7.4 & 54.76 \\
\hline
11 & 1.6 & 2.56 \\
\hline
& Total & 113.2 \\
\hline
\end{tabular}

Standard deviation:
[tex]\[ s = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}} = \square \][/tex]
Round to two decimal places. Enter an integer or decimal number.


Sagot :

Let's solve this step by step.

### Step 1: Calculate the Mean ([tex]\(\bar{x}\)[/tex])
The given [tex]\( x \)[/tex] values are:
[tex]\[ 6, 15, 13, 2, 11 \][/tex]

The mean ([tex]\(\bar{x}\)[/tex]) is calculated as follows:
[tex]\[ \bar{x} = \frac{\sum x_i}{n} \][/tex]
Where [tex]\( n \)[/tex] is the number of values.

[tex]\[ \bar{x} = \frac{6 + 15 + 13 + 2 + 11}{5} = \frac{47}{5} = 9.4 \][/tex]

### Step 2: Calculate [tex]\((x - \bar{x})\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[ x - \bar{x} = \begin{cases} 6 - 9.4 = -3.4 \\ 15 - 9.4 = 5.6 \\ 13 - 9.4 = 3.6 \\ 2 - 9.4 = -7.4 \\ 11 - 9.4 = 1.6 \\ \end{cases} \][/tex]

### Step 3: Calculate [tex]\((x - \bar{x})^2\)[/tex] for each [tex]\(x\)[/tex] value
[tex]\[ (x - \bar{x})^2 = \begin{cases} (-3.4)^2 = 11.56 \\ 5.6^2 = 31.36 \\ 3.6^2 = 12.96 \\ (-7.4)^2 = 54.76 \\ 1.6^2 = 2.56 \\ \end{cases} \][/tex]

### Step 4: Calculate [tex]\(\sum (x - \bar{x})^2\)[/tex]
[tex]\[ \sum (x - \bar{x})^2 = 11.56 + 31.36 + 12.96 + 54.76 + 2.56 = 113.2 \][/tex]

### Step 5: Calculate the Sample Standard Deviation [tex]\(s\)[/tex]
The standard deviation for a sample is calculated as follows:
[tex]\[ s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \][/tex]
Where [tex]\( n - 1 \)[/tex] is the degrees of freedom, in this case, [tex]\( 5 - 1 = 4 \)[/tex].

[tex]\[ s = \sqrt{\frac{113.2}{4}} = \sqrt{28.3} \approx 5.319774431308154 \][/tex]

### Step 6: Round to Two Decimal Places
The rounded standard deviation [tex]\( s \)[/tex] is:
[tex]\[ s \approx 5.32 \][/tex]

### Summary of Results
[tex]\[ \bar{x} = 9.4 \\ (x - \bar{x}) = [-3.4, 5.6, 3.6, -7.4, 1.6] \\ (x - \bar{x})^2 = [11.56, 31.36, 12.96, 54.76, 2.56] \\ \sum (x - \bar{x})^2 = 113.2 \\ s = 5.32 \][/tex]

### Final Answer:
The sample standard deviation is approximately 5.32.