Find the standard deviation of 2.9, 2.5, 2.9.

Step-by-step solution

1. 1

   Compute the mean, keeping it as a fraction. Two of the three values are identical:

   $\bar x=\frac{2.9+2.5+2.9}{3}=\frac{8.3}{3}=2.7\overline{6}\approx 2.76667$

   Because the mean repeats, rounding it to $2.77$ before squaring would visibly distort the answer — keep at least five digits, or work with $\tfrac{83}{30}$.

2. 2

   Find the deviations.

   $2.9-\frac{83}{30}=\frac{87-83}{30}=\frac{4}{30}=0.13\overline{3}\quad(\text{twice}),\qquad 2.5-\frac{83}{30}=\frac{75-83}{30}=-\frac{8}{30}=-0.26\overline{6}$

   They sum to $\tfrac{4+4-8}{30}=0\;\checkmark$.

3. 3

   Square and add, using the fractions.

   $2\left(\frac{4}{30}\right)^2+\left(\frac{8}{30}\right)^2=\frac{32+64}{900}=\frac{96}{900}=\frac{8}{75}\approx 0.106667$

4. 4

   Divide by $n-1$ or by $n$.

   $s^2=\frac{8/75}{2}=\frac{4}{75}\approx 0.053333,\qquad \sigma^2=\frac{8/75}{3}=\frac{8}{225}\approx 0.035556$

5. 5

   Take the square roots.

   $s=\sqrt{\frac{4}{75}}=\frac{2}{\sqrt{75}}\approx 0.23094,\qquad \sigma=\sqrt{\frac{8}{225}}=\frac{2\sqrt2}{15}\approx 0.18856$

   Report $s\approx 0.231$ for a sample and $\sigma\approx 0.189$ for a complete population. As a check, the range is $2.9-2.5=0.4$, and both values are comfortably below it $\checkmark$.