We have to calculate the standard deviation of this set of numbers.
We start by calculating the mean M as:
[tex]\begin{gathered} M=\dfrac{1}{n}\sum ^n_{i=1}\, x_i \\ M=\dfrac{1}{16}(6+6+6+6+6+5+4+4+3+3+3+3+2+2+1+0) \\ M=\dfrac{60}{16} \\ M=3.75 \end{gathered}[/tex]We can now calculate the standard deviation as:
[tex]\begin{gathered} s=\sqrt{\dfrac{1}{n-1}\sum_{i=1}^n\,(x_i-M)^2} \\ s=\sqrt[]{\dfrac{1}{15}(5\cdot(6-3.75)^2+(5-3.75)^2+2\cdot(4-3.75)^2+4\cdot(3-3.75)^2+2\cdot(2-3.75)^2+(1-3.75)^2+(0-3.75)^2)} \end{gathered}[/tex]NOTE: When a data is repeated we can group them in the same term with a factor that shows how many times the item repeats in the data set, like "6", that is present 5 times.
We can continue with the calculation as:
[tex]\begin{gathered} s=\sqrt[]{\dfrac{1}{15}(5\cdot2.25^2+1.25^2+2\cdot0.25^2+4\cdot(-0.75)^2+2\cdot(-1.75)^2+(-2.75)^2+(-3.75)^2)} \\ s=\sqrt[]{\dfrac{1}{15}(5\cdot5.0625+1.5625+2\cdot0.0625+4\cdot0.5625+2\cdot3.0625+7.5625+14.0625)} \\ s=\sqrt[]{\frac{1}{15}(25.3125+1.5625+0.125+2.25+6.125+7.5625+14.0625)} \\ s=\sqrt[]{\frac{57}{15}} \\ s=\sqrt[]{3.8} \\ s\approx1.95 \end{gathered}[/tex]Answer: The standard deviation is 1.95.
