The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is

First, find the mean $(\mu)$ of the data:

$\mu=\frac{6+5+9+13+12+8+10}{7}=\frac{63}{7}=9$

Find the Deviations from the Mean

Subtract the mean from each data point to get the deviations:

$\begin{aligned} 6-9 & =-3 \\ 5-9 & =-4 \\ 9-9 & =0 \\ 13-9 & =4 \\ 12-9 & =3 \\ 8-9 & =-1 \\ 10-9 & =1 \end{aligned}$

Square the Deviations

Square each deviation to eliminate negative values:

$\begin{aligned} (-3)^{2} & =9 \\ (-4)^{2} & =16 \\ 0^{2} & =0 \\ 4^{2} & =16 \\ 3^{2} & =9 \end{aligned}$

$\begin{aligned} (-1)^{2} & =1 \\ 1^{2} & =1 \end{aligned}$

Calculate the Variance

Find the average of these squared deviations (this is the variance, $\sigma^{2}$ ):

$\sigma^{2}=\frac{9+16+0+16+9+1+1}{7}=\frac{52}{7} \approx 7.4286$

Compute the Standard Deviation

Take the square root of the variance to get the standard deviation ( $\sigma$ ):

$\sigma=\sqrt{\frac{52}{7}} \approx \sqrt{7.4286} \approx 2.7255$