Given the dataset
[tex]x = \{21,\ 31,\ 26,\ 24,\ 28,\ 26\}[/tex]
We start by computing the average:
[tex]\overline{x} = \dfrac{21+31+26+24+28+26}{6}=\dfrac{156}{6}=26[/tex]
We compute the difference bewteen each element and the average:
[tex]x-\overline{x} = \{-6,\ 5,\ 0,\ -2,\ 2,\ 0\}[/tex]
We square those differences:
[tex](x-\overline{x})^2 = \{36,\ 25,\ 0,\ 4,\ 4,\ 0\}[/tex]
And take the average of those squared differences: we sum them
[tex]\displaystyle \sum_{i=1}^n (x-\overline{x})^2=36+25+4+4+0+0=69[/tex]
And we divide by the number of elements:
[tex]\displaystyle \sigma^2=\dfrac{\sum_{i=1}^n (x-\overline{x})^2}{n} = \dfrac{69}{6} = 11.5[/tex]
Finally, we take the square root of this quantity and we have the standard deviation:
[tex]\displaystyle\sigma = \sqrt{\dfrac{\sum_{i=1}^n (x-\overline{x})^2}{n}} = \sqrt{11.5}\approx 3.39[/tex]
