Question

What is the standard deviation of 90, 96, 147, 1371, and 49258?

Answer 1

The sample standard deviation is given by the formula

`\begin{gathered} s=\sqrt[]{(\sum ^n_(i\mathop=1)(x_i-\mu)^2)/(n)} \\ \text{Where n is the number of values, }x_i,i=1\ldots.n\text{ are the values } \\ \mu\text{ is the mean of the values } \end{gathered}`

The mean of the data is

`\begin{gathered} \mu=(90+96+147+1371+49258)/(5) \\ \mu=(509625)/(5) \end{gathered}`

Also

`\begin{gathered} \text{The summation of }(x_i-\mu)^2\text{ becomes } \\ (90-(509625)/(5))^2+(96-(509625)/(5))^2+(147-(509625)/(5))^2 \\ +(1371-(509625)/(5))^2+(49258-(509625)/(5))^2 \\ =(9544220206)/(5) \end{gathered}`

Hence

`\begin{gathered} \frac{\sum ^n_{i\mathop{=}1}(x_i-\mu)^2}{n}=((9544220206)/(5))/(5)=((9544220206)/(5))/(5) \\ =381768808.2 \end{gathered}`

So we have

`\begin{gathered} s=\sqrt[]{\frac{\sum ^n_{i\mathop{=}1}(x_i-\mu)^2}{n}}=\sqrt[]{381768808.2}=19538.90499 \\ =19538.90\text{ to the nearest hundredth} \end{gathered}`

Hence the Answer is 19538.90 to the nearest hundredth