STANDARD DEVIATIONDefinitionStandard Deviation (SD, or STD or

) - a measure of the dispersion or variation in a distribution, equal to the
square root of variation or the arithmetic mean (average) of squares of deviations from the arithmetic mean.

In
simple terms, it shows how much variation there is from the "average"
(mean). It may be thought of as the average difference from the mean of
distribution, how far data points are away from the mean. A low
standard deviation indicates that data points tend to be very close to
the mean, whereas high standard deviation indicates that the data are
spread out over a large range of values.


Properties
;

only if
all elements in a set is equal;
Let standard deviation of

be

and mean of the set be

:
Standard deviation of

is

Standard deviation of

is

if a new element

is added to

set and standard deviation of a new set

is

, then:
-

if

-

if

-

if

-

is the lowest if
Tips and TricksGMAC
in majority of problems doesn't ask you to calculate standard
deviation. Instead it tests your intuitive understanding of the
concept. In 90% cases it is a faster way to use just average of

instead of true formula for standard deviation, and treat standard deviation as "
average difference between elements and mean".
Therefore, before trying to calculate standard deviation, maybe you can
solve a problem much faster by using just your intuition.
Advance
tip. Not all points contribute equally to standard deviation. Taking
into account that standard deviation uses sum of squares of deviations
from mean, the most remote points will essentially contribute to
standard deviation. For example, we have a set A that has a mean of 5.
The point 10 gives

in sum of squares but point 6 gives only

.
25 times the difference! So, when you need to find what set has the
largest standard deviation, always look for set with the largest range
because remote points have a very significant contribution to standard
deviation.
ExamplesExample #1Q: There is a set

.
If we create a new set that consists of all elements of the initial set
but decreased by 17%, what is the change in standard deviation?
Solution:
We don't need to calculate. Decrease all elements in a set by a
constant value will decrease standard deviation of the set by the same
value. So, the decrease in standard deviation is 17%.
Example #2Q: There is a set of consecutive even integers. What is the standard deviation of the set?
(1) There are 39 elements in the set.
(2) the mean of the set is 382.
Solution:
Before reading Data Sufficiency statements, what can we say about the
question? What should we know to find standard deviation? "consecutive
even integers" means that all elements strictly related to each other.
If we shift the set by adding or subtracting any integer, does it
change standard deviation (average deviation of elements from the
mean)? No. One thing we should know is the number of elements in the
set, because the more elements we have the broader they are distributed
relative to the mean. Now, look at DS statements, all we need it is
just first statement. So, A is sufficient.
Example #3Q: Standard deviation of set

is 18.3. How many elements are 1 standard deviation above the mean?
Solution: Let's find mean:

Now, we need to count all numbers greater than 42+18.3=60.3. It is one number - 76. The answer is 1.
Example #4Q:
There is a set A of 19 integers with mean 4 and standard deviation of
3. Now we form a new set B by adding 2 more elements to the set A. What
two elements will decrease the standard deviation the most?
A) 9 and 3
B) -3 and 3
C) 6 and 1
D) 4 and 5
E) 5 and 5
Solution:
The closer to the mean, the greater decrease in standard deviation. D
has 4 (equal our mean) and 5 (differs from mean only by 1). All other
options have larger deviation from mean.
Normal distributionIt
is a more advance concept that you can rarely see in GMAT but
understanding statistic properties of standard deviation can help you
to be more confident about simple properties stated above.
In
probability theory and statistics, the normal distribution or Gaussian
distribution is a continuous probability distribution that describes
data that cluster around a mean or average. Majority of statistical
data can be characterized by normal distribution.


covers 68% of data

covers 95% of data

covers 99% of data
Official GMAC Books:The Official Guide, 12th Edition: DT #9; DT #31; PS #199; DS #134;
The Official Guide, 11th Edition: DT #31; PS #212;