Question

Consider a sample with data values of 10, 20, 12, 17, and 16. Compute the -score for each of the five observations (to 2 decimals). Enter negative values as negative numbers. Observed value -score

Answer 1

Hence, the score for each of the five observations are
`-1.25,1.25,-0.75,0.50,0.25`

Given :

Sample with data values of
`x_i`
`10,20,12,17` and
`16`

Sample size
`n=5`

To find:

Compute the score for each of the five observations.

Explanation :

`\because` Sample mean
`\bar{x}=(\sum x_i)/(n)`

`\Rightarrow \bar{x}=(10+20+12+17+16)/(5)=(75)/(5)`

`\Rightarrow \bar{x}=15`

Standard deviation
`\sigma=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}`

`\Rightarrow \sigma=\sqrt{((10-15)^2+(20-15)^2+(12-15)^2+(17-15)^2+(16-15)^2)/(5-1)}`

`\Rightarrow \sigma=\sqrt{((-5)^2+(5)^2+(-3)^2+(2)^2+(1)^2)/(4)}`

`\Rightarrow \sigma=\sqrt{(25+25+9+4+1)/(4)}`

`\Rightarrow \sigma=\sqrt{(64)/(4)} =√(16)`

`\Rightarrow \sigma=4`

`\because` The score of the observations
`Z` is
`\frac{x-\bar{x}}{\sigma}`.

So, when
`(x=10),`
`Z=(10-15)/(4)=-1.25`

when
`(x=20),`
`Z=(20-15)/(4)=1.25`

when
`(x=12),`
`Z=(12-15)/(4)=-0.75`

when
`(x=17},`
`Z=(17-15)/(4)=0.50`

when
`(x=16)`
`Z=(16-15)/(4)=0.25`