Question

Which data set has the highest standard deviation (without doing calculations)?

A) 1,2,3,4

B) 1,1,1,4

C) 1,2,2,4

D) 4,4,4,4

Answer 1

B)1,1,1,4

Explanation:

A.1,2,3,4

Mean=
`\bar x=(1+2+3+4)/(4)=2.5`

x
`x-\bar x`
`(x-\bar x)^2`

1 -1.5 2.25

2 -0.5 0.25

3 0.5 0.25

4 1.5 2.25

`\sum(x-\bar x)^2=2.25+0.25+0.25+2.25=5`

B.

Mean=
`\bar x=(1+1+1+4)/(4)=1.75`

`(x-\bar x)^2`

0.5625

0.5625

0.5625

5.0625

`\sum(x-\bar x)^2=0.5626+0.5625+0.5625+5.0625=6.75`

C.

Mean=
`\bar x=(1+2+2+4)/(4)=2.25`

`(x-\bar x)^2`

1.5625

0.0625

0.0625

3.0625

`\sum (x-\bar x)=1.5625+0.0625+0.0625+3.0625=4.75`

D.
`Mean=\bar x=(4+4+4+4)/(4)=4`

`(x-\bar x)^2`

0

0

0

0

`\sum (x-\bar x)^2=0+0+0+0=0`

We know that

S.D is directly proportional to
`\sum (x-\bar x)^2`.

When
`\sum (x-\bar x)^2` is highest then the S.D is also highest.

We can see that the value of
`\sum (x-\bar x)^2` is highest in option B.

Therefore, S.D of the date set of option B is highest.