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Math: Standard Deviation

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Math: Standard Deviation [#permalink] New post 12 Dec 2009, 05:48
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STANDARD DEVIATION

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This post is a part of [GMAT MATH BOOK]

created by: walker
edited by: bb, Bunuel

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Definition

Standard Deviation (SD, or STD or \sigma) - a measure of the dispersion or variation in a distribution, equal to the square root of variance or the arithmetic mean (average) of squares of deviations from the arithmetic mean.

variance = \frac{\sum(x_i - x_{av})^2}{N}

\sigma = \sqrt{variance} = \sqrt{\frac{\sum(x_i - x_{av})^2}{N}}

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.

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Properties

\sigma \ge 0;

\sigma = 0 only if all elements in a set is equal;

Let standard deviation of \{x_i\} be \sigma and mean of the set be \m:

Standard deviation of \{\frac{x_i}{a}\} is \sigma^'=\frac{\sigma}{a}. Decrease/increase in all elements of a set by a constant percentage will decrease/increase standard deviation of the set by the same percentage.

Standard deviation of \{x_i+a\} is \sigma^'=\sigma. Decrease/increase in all elements of a set by a constant value DOES NOT decrease/increase standard deviation of the set.

if a new element y is added to \{x_i\} set and standard deviation of a new set \{\{x_i\},y\} is \sigma^', then:

1) \sigma^'>\sigma if |y-\m|>\sigma

2) \sigma^'=\sigma if |y-\m|=\sigma

3) \sigma^'<\sigma if |y-\m|<\sigma

4) \sigma^' is the lowest if y=\m


Tips and Tricks

GMAC 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 |x_i-x_{av}| 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 (10-5)^2=25 in sum of squares but point 6 gives only (6-5)^2=1. 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.


Examples

Example #1
Q: There is a set \{67,32,76,35,101,45,24,37\}. 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 as we know rule that decrease in all elements of a set by a constant percentage will decrease standard deviation of the set by the same percentage. So, the decrease in standard deviation is 17%.

Example #2
Q: 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 #3
Q: Standard deviation of set \{23,31,76,45,16,55,54,36\} is 18.3. How many elements are 1 standard deviation above the mean?
Solution: Let's find mean: \m=\frac{23+31+76+45+16+55+54+36}{8}=42
Now, we need to count all numbers greater than 42+18.3=60.3. It is one number - 76. The answer is 1.

Example #4
Q: 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 distribution

It is a more advance concept that you will never 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.

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\m-\sigma<x<\m+\sigma covers 68% of data

\m-2\sigma<x<\m+2\sigma covers 95% of data

\m-3\sigma<x<\m+3\sigma 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;

Generated from [GMAT ToolKit]


Resources

Bunuel's post with PS SD-problems: [PS Standard Deviation Problems]
Bunuel's post with DS SD-problems: [DS Standard Deviation Problems]

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Last edited by walker on 17 Dec 2009, 16:27, edited 7 times in total.
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Re: Math: Standard Deviation [#permalink] New post 12 Dec 2009, 21:06
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Well, I'm a little confused in regard to Example #4.
Quote:
Example #4
Q: There is a set A of 19 integers with mean 11 and standard deviation of 4. 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 mean our numbers the larger is decreasing 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.

What on earth is the mean? 11 or 4? I guess it might be 4 by the explanation that follows.
Actually, the SD value is an unrelative piece of information, we can solve this problem by mean value alone.
Correct me if I'm wrong, any comment is welcome, thanks a lot.
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Re: Math: Standard Deviation [#permalink] New post 12 Dec 2009, 21:09
Expert's post
sayysong wrote:
Well, I'm a little confused in regard to Example #4.
What on earth is the mean? 11 or 4? I guess it might be 4 by the explanation that follows.
Actually, the SD value is an unrelative piece of information, we can solve this problem by mean value alone.
Correct me if I'm wrong, any comment is welcome, thanks a lot.


Thanks! You are right, there is a typo. mean should be 4.

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Re: Math: Standard Deviation [#permalink] New post 17 Dec 2009, 13:56
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On the last part, it should be s mu + sigma, mu + 2 sigma, mu + 3 sigma
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Re: Math: Standard Deviation [#permalink] New post 21 Dec 2009, 10:54
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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%.


I don't know if I am making this sound more complicated but I think there are two concepts here... May be a rephrase would help

1) Decrease/increase in all elements of a set by a constant value DOES NOT decrease/increase standard deviation of the set. See here - gmat-club-test-m11-doubt-81138.html
2) Decrease in all elements of a set by a constant percentage will decrease standard deviation of the set by the same percentage.

2) is different from 1) because the same percentage decrease/increase results in different values for the different elements of the set and is not the same as increase/decrease by a constant value.
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Re: Math: Standard Deviation [#permalink] New post 21 Dec 2009, 14:52
Expert's post
kaptain wrote:
...May be a rephrase would help...


Thanks, I'll rephrase it. Definitely, it is not very clear.
+1

... I've added your statements to properties part of the post and corrected the example.

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Re: Math: Standard Deviation [#permalink] New post 27 Dec 2009, 21:54
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VERY minor typo in Example #3.

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Should be dividing by 8, not 5. :)
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Re: Math: Standard Deviation [#permalink] New post 09 Jan 2010, 17:35
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Quote:
Q: 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.


I do not get it , how could we calculate Standard deviation of even consecutive integers without knowing the set ?

or it means that Standard deviation of any 39 consecitive even integers will be same as they are equally distributed with respect to the avarge ?
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Re: Math: Standard Deviation [#permalink] New post 09 Jan 2010, 18:45
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GMATMadeeasy wrote:
...or it means that Standard deviation of any 39 consecitive even integers will be same as they are equally distributed with respect to the avarge ?


Exactly. Let's look at simple example,

{4,6,8} and {1004,1006,1008} they have the same STD because all elements of second set are shifted by constant number 1000.

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Re: Math: Standard Deviation [#permalink] New post 12 Feb 2010, 23:18
hi, thanks a lot for this post. I really needed some theory on Standard Deviation.
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Re: Math: Standard Deviation [#permalink] New post 25 Feb 2010, 04:44
Really helpful. The suggestion people have posted are also valid and give any one going thru the theory an idea of how well or what kind of concentrated efforts he's taking... especially if he does not come up with similar questions as others :)
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Re: Math: Standard Deviation [#permalink] New post 26 Feb 2010, 11:17
thank you for posting this concise explanation of SD

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Re: Math: Standard Deviation [#permalink] New post 03 Mar 2010, 23:05
Thanks. Has anycome ever come across a normal distribution question on actual GMAT?
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Re: Math: Standard Deviation [#permalink] New post 11 Mar 2010, 12:43
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Brief note--there is one Data Sufficiency question in the OG that references Variance, which is the square of the standard deviation. Though it probably won't be tested, we may want to define the term Variance as a precaution.

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Re: Math: Standard Deviation [#permalink] New post 08 Sep 2010, 03:19
Thanks for the gr8 post :)
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Re: Math: Standard Deviation [#permalink] New post 14 Sep 2010, 02:09
Thanks a lot walker. Its a great help on SD basics...
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Re: Math: Standard Deviation [#permalink] New post 19 Oct 2010, 19:59
It's very useful and easy to understand the concept, Thanks Again Walker :-)

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Re: Math: Standard Deviation [#permalink] New post 20 Oct 2010, 07:39
One suggestion : We can include information about what is range in this doc.
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Re: Math: Standard Deviation [#permalink] New post 10 Jan 2011, 17:13
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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.


Not to be a nit pick but the largest range does not imply the largest standard deviation.
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Re: Math: Standard Deviation [#permalink] New post 10 Jan 2011, 22:44
Great post. Thanks.!!
Re: Math: Standard Deviation   [#permalink] 10 Jan 2011, 22:44
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