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Msalafia · Answer 1 · 2023-11-11T21:32:23+0000

We have to calculate the standard deviation of this set of numbers.

We start by calculating the mean M as:

$\begin{gathered} M=(1)/(n)\sum ^n_(i=1)\, x_i \\ M=(1)/(16)(6+6+6+6+6+5+4+4+3+3+3+3+2+2+1+0) \\ M=(60)/(16) \\ M=3.75 \end{gathered}$

We can now calculate the standard deviation as:

$\begin{gathered} s=\sqrt{(1)/(n-1)\sum_(i=1)^n\,(x_i-M)^2} \\ s=\sqrt[]{(1)/(15)(5\cdot(6-3.75)^2+(5-3.75)^2+2\cdot(4-3.75)^2+4\cdot(3-3.75)^2+2\cdot(2-3.75)^2+(1-3.75)^2+(0-3.75)^2)} \end{gathered}$

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:

$\begin{gathered} s=\sqrt[]{(1)/(15)(5\cdot2.25^2+1.25^2+2\cdot0.25^2+4\cdot(-0.75)^2+2\cdot(-1.75)^2+(-2.75)^2+(-3.75)^2)} \\ s=\sqrt[]{(1)/(15)(5\cdot5.0625+1.5625+2\cdot0.0625+4\cdot0.5625+2\cdot3.0625+7.5625+14.0625)} \\ s=\sqrt[]{(1)/(15)(25.3125+1.5625+0.125+2.25+6.125+7.5625+14.0625)} \\ s=\sqrt[]{(57)/(15)} \\ s=\sqrt[]{3.8} \\ s\approx1.95 \end{gathered}$

Answer: The standard deviation is 1.95.

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