Suppose a population $A$ has 100 observations $101,102, \ldots, 200$, and another population $B$ has 100 observations $151,152, \ldots, 250$. If $V_{A}$ and $V_{B}$ represent the variances of the two populations, respectively, then $\frac{V_{A}}{V_{B}}$ is

JEE Main · 2006 · 30 Apr 2006 · Offline · Statistics

Suppose a population

A

has 100 observations

101, 102, \dots, 200

, and another population

B

has 100 observations

151, 152, \dots, 250

. If

V_{A}

and

V_{B}

represent the variances of the two populations, respectively, then

\frac{V _{A}}{V _{B}}

Answer: (A)

Both populations consist of

100

consecutive integers;

A

runs from

101

200

and

B

from

151

250

. Variance measures the spread of the data about its mean, and adding the same constant to every observation (here a shift of

50

) moves the mean by that constant while leaving each deviation

x_{i} - \overset{x}{ˉ}

unchanged. Since the two lists are identical apart from this shift, their deviations, and hence their variances, are equal. Therefore

\frac{V _{A}}{V _{B}} = 1

JEE Main · 2006 · 30 Apr 2006 · Offline · Single correct option