BTGmoderatorDC wrote:If sets A and B have the same number of terms, is the standard deviation of set A greater than the standard deviation of set B?

(1) The range of set A is greater than the range of set B.
(2) Sets A and B are both evenly spaced sets.
Source: Veritas Prep

$$\# A = \# B$$
$${\sigma _A}\mathop > \limits^? {\sigma _B}$$

$$\left( 1 \right)\,\,{R_A} > {R_B}\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{A = }}\left\{ {0,1} \right\}\,,\,\,B = \left\{ {0,0} \right\}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left\{ \matrix{
{\rm{A = }}\left\{ {0,0,0,10} \right\}\,\, \hfill \cr
B = \left\{ {0,0,9,9} \right\}\,\, \hfill \cr} \right.\,\, \Rightarrow \,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,{\rm{finite}}\,\,{\rm{APs}}\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{A = }}\left\{ {0,2,4} \right\}\,,\,\,B = \left\{ {0,1,2} \right\}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,{\rm{A = }}\left\{ {0,1,2} \right\}\,,\,\,B = \left\{ {0,2,4} \right\}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,{\rm{distance}}\,\,{\rm{between}}\,\,{\rm{terms}}\,\,{\rm{and}}\,\,{\rm{mean}}\,\,{\rm{in}}\,\,A\,\,{\rm{is}}\,\,{\rm{larger}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

BTGmoderatorDC wrote:If sets A and B have the same number of terms, is the standard deviation of set A greater than the standard deviation of set B?

(1) The range of set A is greater than the range of set B.
(2) Sets A and B are both evenly spaced sets.

Standard deviation describes how much the values in a set deviate from the mean. A larger standard deviation indicates that the values are deviating more -- getting farther away from -- the mean. So the question can be rephrased:

Do the values in set A deviate more from the mean than the do values in set B?

Let SD = standard deviation.

Statement 1:
We know that the distance between the biggest and smallest values in A is greater than the distance between the biggest and smallest values in B.
But we don't know the mean, and to determine which set has a greater SD, we need to know how all the numbers in each set -- not just the biggest and smallest -- are deviating from the mean.
INSUFFICIENT.

Statement 2:
When values are evenly spaced, the mean = the median, and all the values are symmetrical about the median.
For example, if m = median, and all the values are consecutive even or odd integers, the set will look like this:

...m-6, m-4, m-2, m, m+2, m+4, m+6...

But to determine which set has a greater SD, we need to know in each set the distance between each successive pair of values. For example:

If A = consecutive even integers = {2,4,6} and B = consecutive multiples of 3 = {3,6,9}, then the values in B deviate more from the mean and B has the larger SD.
If A = consecutive multiples of 3 = {3,6,9} and B = consecutive even integers = {2,4,6}, then the values in A deviate more from the mean and A has the larger SD.
INSUFFICIENT.

Statements 1 and 2:
A and B have the same number of values.
A and B are both evenly spaced sets, so the values in each set are symmetrical about the mean.
The range in A is greater.
For A to have a greater range, the distance between each successive pair in A must be greater than the distance between each successive pair in B. In other words, the values in A are more spread out.
Thus, the values in A are deviating more from the mean, and A has a larger SD.
SUFFICIENT.

The correct answer is C.