Question

Calculate the samples standard deviation for the following data set if necessary around to one more decimal place than the larger number of decimal places given in the data

Answer 1

Here are the steps in calculating the sample standard deviation of the given data.

1. First, calculate the mean or the average of the data. To find the average, simply add the data and then, divide the result by the number of data.

`\begin{gathered} 0.428+0.391+0.435+0.395=1.649 \\ 0.386+0.467+0.414+0.448=1.715 \\ 0.429+0.414+0.393+0.457=1.693 \\ 0.391+0.447+0.406+0.460=1.704 \\ 0.453+0.424+0.424+0.464=1.765 \end{gathered}`
`1.649+1.715+1.693+1.704+1.765=8.526`
`8.526/20=0.4263`

The sample mean is 0.4263.

2. Subtract the mean from each data and square the difference.

`\begin{gathered} 0.428-0.4263=0.0017\Rightarrow(0.0017^2)=0.00000289 \\ 0.391-0.4263=-0.0353\Rightarrow(-0.353^2)=0.00124609 \\ 0.435-0.4263=0.0087\Rightarrow(0.0087^2)=0.00007569 \\ 0.395-0.4263=-0.0313\Rightarrow(-0.0313^2)=0.00097969 \end{gathered}`
`\begin{gathered} 0.386-0.4263=-0.0403\Rightarrow(-0.0403^2)=0.00162409 \\ 0.467-0.4263=0.0407\Rightarrow(0.0407^2)=0.00165649 \\ 0.414-0.4263=-0.0123\Rightarrow(-0.0123^2)=0.00015129 \\ 0.448-0.4263=0.0217\Rightarrow(0.0217^2)=0.00047089 \end{gathered}`

The third to the fifth column is shown below:

3. Add the squared difference. The sum is 0.0137842.

4. Divide the sum by the number of data minus 1.

`0.0137842/(20-1)`
`0.0137842/19=0.000725484`

5. Get the square root of the quotient.

`√(0.000725484)=0.0269348\approx0.0269`

Therefore, the standard deviation of the given data set is approximately 0.0269.