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STANDARD DEVIATION : Calculation of Standard Deviation (SD)

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amitdgr
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STANDARD DEVIATION : Calculation of Standard Deviation (SD) [#permalink] New post   Updated on: 14 Oct 2008, 19:04
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STANDARD DEVIATION :

    Calculation of Standard Deviation (SD):
      i) Find the mean of the set of numbers
      ii) Find the difference between each of the numbers and the mean
      iii) Square the differences and take the mean of the differences (Thanks Ian)
      iv) Take the positive square root of this value

    It is very unlikely that a GMAT problem will involve calculating the SD of a given series.

      1.SD does not change when the same constant is added or subtracted to all the members of the set.

      2.A set of numbers with range of zero means that all of the numbers are the same, hence the dispersion of the numbers from its mean is zero. In other words, If the range is 0, then the SD must also be 0, because there is no variance.

      3.If Range or SD of a list is 0, then the list will contain all identical elements.

      4.You only need to know the difference between values and total number of values to compute SD.

      5.If we know all the numbers of the list, there is a definite SD, regardless of what it is, we can compute it and get an answer – this is helpful for DS questions.

      6.The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD.

      7.The sum of the deviations of the elements from the mean must be 0

      8.Closer the more values to the MEAN, lower the SD

      9.If you multiply all terms by x then SD =x times old SD and the mean = x times old mean

      10.For comparing the SD for two sets any information about mean ,median,mode and range are insufficient unless you can determine the individual terms from the given data.

      11.If mean = maximum value it means that all values are equal and SD is 0.

      12.Variance is the square of the standard deviation

    RANGE :
      1. Range is the difference between the largest number and smallest number in a set

      2. If Range of a list is 0, then the list will contain all identical elements. Or if a set contains all identical elements, then range = 0

      3. If a set contains only one item, then range = 0

    MODE :

      Mode is the most frequently recurring number/numbers among the given set of numbers. It can be more than one

    MEAN AND MEDIAN:

      1. Arithmetic Mean (Average) = total of quantities / number of quantities

      2. The median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers, and the average of the two middle numbers if there are an even number of numbers

      3. For a set of consecutive integers, the median is the the average of the first and the last integer

      4. For a given set of consecutive even/odd (or evenly spread) numbers mean = median

      5. For a given set of consecutive integers median = mean

https://www.integratedlearning.net/gmat/sample.asp

-----------------------------------------------------------------------------------------------------------------

Thanks Ian for pointing out the mistakes.

Guys, Please add if i missed something.

Originally posted by amitdgr on 18 Aug 2008, 02:34.
Last edited by amitdgr on 14 Oct 2008, 19:04, edited 4 times in total.
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Re: Statistics - Rules and Tips [#permalink] New post   Updated on: 17 Nov 2009, 14:13
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Tip1.
Always keep in mind:
Range >= SD >=0

Case1: when the set of numbers given are +ve and different: 2 3 4 : Range = 2, SD = sq-root (2/3). R > SD
Case2: when the set of numbers given are -ve and different: -4 -3 -2 : Range = 2, SD = sq-root (2/3). R > SD
Case3: Only in case of all numbers are equal, then Range = SD = 0, already mentioned above.

There cud be a problem where you are asked to arrange the Mean, median, mode, range & SD, any of these in ascending order. or framed differently for DS questions.

Tip2:
Edit: Set of consecutive integers has mean 0, means set is symmetric about the origin. And yes # of terms should be odd. Thanks Bunuel

Case1: either all numbers are zero
Case2: or there are odd number of integers
Helpful for DS again.

Kudos if you like the tips. More to come.........

Edit: Thanks Bunuel, edited the post as advised by you

Originally posted by ctrlaltdel on 17 Nov 2009, 12:27.
Last edited by ctrlaltdel on 17 Nov 2009, 14:13, edited 5 times in total.
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Re: Statistics - Rules and Tips [#permalink] New post   24 Aug 2008, 18:40
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amitdgr wrote:

6. Calculation of Standard Deviation (SD):
i) Find the mean of the set of numbers
ii) Find the difference between each of the numbers and the mean
iii) Square the differences and add them together
iv) Take the positive square root of this value

Please add if i missed something.


Step iii) above is not quite right- you want to take the mean of the squares, not the sum.
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Re: Statistics - Rules and Tips [#permalink] New post   24 Aug 2008, 18:48
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amitdgr wrote:
[list]

11. SD ranks the dispersion (deviation) of the numbers in a list. The more alike the numbers are, the less the dispersion, so the less the standard deviation

13. The more uneven members are dispersed around their arithmetic average, the more their SD

21. For data with approximately the same mean, the greater the range, the greater the SD

26. For odd number of consecutive integers median = mean

Please add if i missed something.


In 11: this could be confusing. A set such as {0,0,0,0, 1000000, 1000000, 1000000, 1000000} contains numbers which are very much 'alike' but the set has a huge standard deviation: all of its elements are very far from the mean.

In 13: I'm not exactly sure what you're trying to say here.

In 21: This is not a mathematical rule; indeed it's very often untrue. The set S = {100, 50, 50, 50, 50, 50, 50, 50, 0} has the same mean as the set T = {99, 99, 99, 99, 99, 99, 1, 1, 1, 1, 1, 1}, and has a larger range. It has a much smaller standard deviation, however.

In 26. For any number of consecutive integers, median = mean. You can remove the word 'odd'. Indeed, in any arithmetic progression, the median = mean.
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STANDARD DEVIATION : Calculation of Standard Deviation (SD) [#permalink] New post   17 Nov 2009, 13:08
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ctrlaltdel wrote:
Tip2:
There is a formula: (I don't know if it will be relevant for GMAT)
3*Median = 2*Mode + Mean
Helpful for DS.


I think it's not true. If set is for example: {1,1,1,5,7}
Mean=3
Median=1
Mode=1

And equation doesn't hold true. Check the source.

ctrlaltdel wrote:
Tip3:
If the Mean of set of consecutive numbers is zero, then:
Case1: either all numbers are zero
Case2: or there are odd number of integers
Helpful for DS again.


If we have set of consecutive integers how can all numbers be zero?

Set of consecutive integers has mean 0, means set is symmetric about the origin. And yes # of terms should be odd.
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Re: Statistics - Rules and Tips [#permalink] New post   17 Nov 2009, 14:07
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ctrlaltdel wrote:
https://www.pinkmonkey.com/studyguides/subjects/stats/chap4/s0404601.asp

4.6 Empirical Relation Between Mean, Median And Mode

A distribution in which the values of mean, median and mode coincide (i.e. mean = median = mode) is known as a symmetrical distribution. Conversely, when values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed distribution. In moderately skewed or asymmetrical distribution a very important relationship
exists among these three measures of central tendency. In such distributions the distance between the mean and median is about one-third of the distance between the mean and mode, as will be clear from the diagrams 1 and 2. Karl Pearson expressed this relationship as:

Mode = mean - 3 [mean - median]

Mode = 3 median - 2 mean

Knowing any two values, the third can be computed.


This is about the special kind of distribution which you never see on GMAT.

Again if set is for example: {1,1,1,5,7}
Mean=5
Median=1
Mode=1

Put these values in the formula in red. You'll see the equation won't hold true.

I wouldn't advise to study this source in GMAT preparation.
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Re: Statistics - Rules and Tips [#permalink] New post   24 Aug 2008, 15:57
Thanks amitdgr for the info.
+1 for you
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Re: Statistics - Rules and Tips [#permalink] New post   24 Aug 2008, 18:27
yes statistics class is all coming back to me now. it was only last year tho lol. thanks for the info although I'm pretty sure you don' t need to have advanced knowledge of stats for the GMAT, do you? It does kind of help you understand your score report tho.....
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Re: Statistics - Rules and Tips [#permalink] New post   20 Sep 2008, 06:32
Thanks Ian .. I guess I mistook some of the concepts :)
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Re: Statistics - Rules and Tips [#permalink] New post   24 Jan 2009, 22:35
Thanks man. Into my book marks.
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Re: Statistics - Rules and Tips [#permalink] New post   19 Aug 2009, 19:28
Thanks very much for the effort....however, do we need this sort of stat knowledge for GMAT?
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Re: Statistics - Rules and Tips [#permalink] New post   28 Aug 2009, 03:41
Thanx for the this useful stuff!!!
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Re: Statistics - Rules and Tips [#permalink] New post   07 Oct 2009, 05:11
shortcut**

S.D=Sq.root(mean of the squares-square of the mean)
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Re: Statistics - Rules and Tips [#permalink] New post   16 Oct 2009, 10:39
Thank you so much amitdgr!!
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Re: Statistics - Rules and Tips [#permalink] New post   16 Oct 2009, 10:47
Good compilation .. Thanks dude
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Re: Statistics - Rules and Tips [#permalink] New post   20 Oct 2009, 18:45
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amitdgr wrote:

so basically if I have the following numbers

112, 113, 114, 115, 116, 117
I dont need to do the sum/6
i basically can do 112+117/6
which gives me the mean and the median!!!

thank you soo much that saves lots of time especially with big numbers!!!

h.



Typo.........Should be divided by 2........
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Re: Statistics - Rules and Tips [#permalink] New post   20 Oct 2009, 19:10
Quote:
1.SD does not change when the same constant is added or subtracted to all the members of the set.

Could you explain how this verbatim would affect the below problem?

A certain list of 100 data has an average of 6 and standard deviation of d where d is positive. Which of the followig pairs of data, when added to the list must result in a ist of 102 data with the standard deviation less than d?

Answer choices -

1. 0 and 6
2. 0 and 12
3. 0 and 0
4. -6 and 0
5. 6 and 6

Thanks,
Vannu
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Re: Statistics - Rules and Tips [#permalink] New post   21 Oct 2009, 09:06
It is totally Ok. Welcome.
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Re: Statistics - Rules and Tips [#permalink] New post   13 Nov 2009, 03:28
Great post!

Just out of curiousity... has anyone actually seen a standard deviation problem in the real thing?
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Re: Statistics - Rules and Tips [#permalink] New post   Updated on: 17 Nov 2009, 14:15
Edit: Sorry for the confusion :oops:

Originally posted by ctrlaltdel on 17 Nov 2009, 13:52.
Last edited by ctrlaltdel on 17 Nov 2009, 14:15, edited 1 time in total.
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Re: Statistics - Rules and Tips [#permalink]
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