Question

Consider the following data set that has a mean of 4:2, 3, 4, 4,7Using the equation below or the standard deviation formula in Excel,calculate the standard deviation for this data set. Answer choices arerounded to the hundredths place.S=n-Σ(x₁-x)²n-1=1

Answer 1

We need to use the formula provided. The formula is:

`s=\sqrt{(1)/(n-1)\sum_{i\mathop{=}1}^n(X_i-\bar{X})^2}`

Where:

`\bar{X}=mean\text{ }data\text{ }set`
`X_i=i\text{-}th\text{ }value\text{ }of\text{ }the\text{ }data\text{ }set`

Then, first, we need to find the mean:

`\bar{X}=\frac{\sum_{i\mathop{=}1}^nX_i}{n}`

The data set is: 2, 3, 4, 4,7. Since there are 5 values, n = 5.

Now calculate:

`\bar{X}=(2+3+4+4+7)/(5)=(20)/(5)=4`

Now, we need to find the square of the difference between each value of the data set and the mean:

`\begin{gathered} (X_i-\bar{X})^2: \\ (2-4)^2=(-2)^2=4 \\ (3-4)^2=(-1)^2=1 \\ (4-4)^2=0^2=0 \\ (4-4)^(2)=0^(2)=0 \\ (7-4)^2=3^2=9 \end{gathered}`

And now we need to find the sum of those numbers:

`\sum_{i\mathop{=}1}^n(X_i-\bar{X})^2=4+1+0+0+9=14`

Now, we can calculate the standard deviation:

`s=\sqrt{(1)/(n-1)\sum_{i\mathop{=}1}^n(X_i-\bar{X})^2}=\sqrt{(1)/(5-1)\cdot14}=\sqrt{(14)/(4)}=\sqrt{(7)/(2)}\approx1.870828`

Thus, the correct answer is option a.) 1.87