All About Statistics

7/23/2019 All About Statistics

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CONTENT

1.The Meaning of Arithmetic Mean

2.Some Mean Questions

3.Finding Arithmetic Mean Using Deviations

4.Application of Arithmetic Means

5.Means Questions on Median

6.A Range of Questions

7.Dealing with Standard Deviation

8.Dealing with Standard Deviation II

9.Some Trick Standard Deviation Questions

10.! Important "oncepts for Statistics Questions on the #MAT

11.$ow to Quickl Solve Standard Deviation Questions on the #MAT

12.A %&' (evel #MAT Question on Statistics)

13.A %&'* (evel Question on SD

14.+ther Resources on Statistics
http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563151http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563158http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563159http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563176http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563188http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563196http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563202http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563208http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563213http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563219http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563224http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563225http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563229http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563234http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563158http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563159http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563176http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563188http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563196http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563202http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563208http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563213http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563219http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563224http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563225http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563229http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563234http://gmatclub.com/forum/statistics-made-easy-all-in-one-topic-203966.html#p1563151


2/29

The Meaning of Arithmetic Mean

BY KARISHMA, VERITAS PREP

Lets start today with statistics mean, median, mode, range and standard deviation. The topics are

simple but the fun lies in the questions. Some questions on these topics can be extremely trickyespecially those dealing with median, range and standard deiation. !nyway, we will tackle meantoday.

So what do you mean by the arithmetic mean of some obserations" # guess most of you will reply thatit is the $Sum of %bserations&Total number of obserations. 'ut that is how you calculatemean. (yquestion is $what is mean" Loosely, arithmetic mean is the number that represents all theobserations. Say, if # know that the mean age of a group is )*, # would guess that the age of +obbie,who is a part of that group, is )*. %f course +obbies actual age could be anything but the best guesswould be )*.

Say, # tell you that the aerage age of a group of )* people is ) yrs. -an you tell me the sum of theages of all )* people" # am sure you will say that it is )*) / )*. 0ou can think of it in two ways1

(ean / Sum of all ages&2o of people

So Sum of all ages / (ean 32o of people4 / ))*

%r

Since there are )* people and each persons age is represented by ), the sum of their ages / )*).'asically, the total sum was distributed eenly among the )* people and each person got ) yrs.

2ow, lets say you made a mistake. ! boy whose age you thought was 5* was actually 6*. 7hat is thecorrect mean" !gain, you can think of it in two ways1

2ew sum / )* 8 )* / )9*

2ew aerage / )9*&)* / )9

%r

0ou can say that there is an extra )* that has to be distributed eenly among the )* people, so eachperson gets ) extra. :ence, the aerage becomes ) 8 ) / )9.

!s you might hae guessed, we will work on the second interpretation. Lets look at an example now.
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,-ample ./The average age of a group of n people is 15 yrs. One more person aged 39 joins thegroup and the new average is 17 yrs. What is the value of n?

3!4 ;3'4 )*3-4 ))

3irst tell me, if the age of the additional person were ) yrs, what would hae happened tothe aerage" The aerage would hae remained the same since this new persons age would hae beenthe same as the age that represents the group. 'ut his age is 6; ) / 5? more than the aerage. 7eknow that we need to eenly split the extra among all the people to get the new aerage. 7hen 5? issplit eenly among all the people 3including the new guy4, eeryone gets 5 extra 3since aerage ageincreased from ) to )@4. There must be 5?&5 / )5 people now 3including the new guy4 i.e. n must be)) 3without including the new guy4.

This uestion is dis!ussed$,R,.

Lets look at another similar example though a little trickier. Try soling it on your own first. #f notlogically, try using the formula approach. Then see how elegant the solution becomes once you start$thinking instead of Aust $calculating.

,-ample 0/When a person aged 39 is added to a group of n people" the average age in!reases #y$. When a person aged 15 is added instead" the average age de!reases #y 1. What is the value of n?

3!4 @3'4 B3-4 ;3


4/29

ome Mean !"e#tion#


# hope the theory of arithmetic mean we discussed aboe is clear to you. Lets see the theory in actiontoday. # will pick some mean questions from arious sources 3%fficial Cuide, C(!T prep tests, etc.4 andwe will try to use the concepts we learned last week to sole them.

Lets start with a simple question.

Question ./%or the past n days the average daily produ!tion at a !ompany was &' units. (ftoday)s produ!tion of 1'' units raises the average to &5 units per day" what is the value of n?

3!4 6*3'4 )B3-4 )*3


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!gain, this was a relatiely straight forward question. Lets look at a tricky one now.

Question !/* set of num#ers has an average of 5'. (f the largest element is greater than 3times the smallest element" whi!h of the following values !annot #e in the set?

3!4 B3'4 ;*3-4 )563


6/29

$in%ing Arithmetic Mean ing 'e(iation#


#n this post is again focused on arithmetic mean. Lets start our discussion by considering the case ofarithmetic mean of an arithmetic progression.

7e will start with an example. 7hat is the mean of ?6, ??, ?, ?9, ?@" 3:int1 #f you are thinking aboutadding the numbers, thats not the way # want you to go.4

!s we discussed in our preious posts, arithmetic mean is the number that can represent&replace allthe numbers of the sequence. 2otice in this sequence, ?? is one less than ? and ?9 is one more than?. So essentially, two ?s can replace both ?? and ?9. Similarly, ?6 is 5 less than ? and ?@ is 5 morethan ? so two ?s can replace both these numbers too.

The sequence is essentially ?, ?, ?, ?, ?.

:ence, the arithmetic mean of this sequence must be ?D 3#f you hae doubts, you can calculate andfind out.4

#t makes sense, doesnt it" The middle number in the sequence of consecutie positie integers will bethe mean. The deiations of all numbers to the left of the middle number will balance out thedeiations of all the numbers to the right of the middle number.

3#n this post, we will assume that the gien numbers are in increasing&decreasing order. #f that is notthe case, you can always put them in increasing order and use these concepts.4

%nce again, what is the mean of );5, );6, );?, );, );9, );@, );B"

#t is ); since it is the middle numberD

%k, what about );5, );6, );?, );, );9, );@" 7hat is the mean in this case" There is no middle numberhere since there are 9 numbers. The mean here will be the middle of the two middle numbers which is);?. 3the middle of the third and the fourth number4. #t doesnt matter that );?. is not a part of thislist. #f you think about it, arithmetic mean of some numbers neednt be one of the numbers.

7hat about @), @6, @, @@, @;" 7hat will be the mean in this case" =en though these numbers are notconsecutie integers, the difference between two adAacent numbers in the list is the same 3it is anarithmetic progression4. So the deiations of the numbers on the left of the middle number will cancelout the deiations of the numbers on the right of the middle number 3@) is ? less than @ and @; is ?more than @. @6 is 5 less than @ and @@ is 5 more than @4. :ence, the mean here will be @ 3Aust likeour first example4.

ust to reMinforce1

)*5, )*9, ))* N (ean / )*9

)*5, )*9, ))*, ))? MN (ean / )*B 3(iddle of the second and third numbers4

Lets twist this concept a little now. 7hat is the mean of 69, ?*, ?5, ?6, ??, ?@"


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This is not an arithmetic progression. So do we need to sum and then diide by 9 to get the mean" 2otso fastD Lets try and use the deiations concept we hae Aust learned.

Cien sequence1 69, ?*, ?5, ?6, ??, ?@

#t seems that the mean would be around ?5, right" Some numbers are less than ?5 and others are morethan ?5.

69 is 9 less than ?5.

?* is 5 less than ?5.

%erall, the numbers less than ?5 are 985 / B less than ?5.

?6 is ) more than ?5.

?? is 5 more than ?5.

?@ is more than ?5

%erall, the numbers more than ?5 are )858 / B more than ?5.

The deiations of the numbers less than ?5 get balanced out by deiations of the numbers greater than?5D :ence, the aerage must be ?5.

This method is especially useful in cases inoling big numbers which are close to each other.

,-ample ./What is the average of 5$" 53" &3" &7" 4'" 99" 5'?

7hat would you say the aerage is here" Oerhaps, around ?@*"

Lets see1

?5 is )B less than ?@*.

?6 is )@ less than ?@*.

?96 is @ less than ?@*.

?9@ is 6 less than ?@*.

%erall, the numbers less than ?@* are )B 8 )@ 8 @ 8 6 / ? less.

?B* is )* more than ?@*.

?;; is 5; more than ?@*.

*? is 6? more than ?@*.

%erall, the numbers more than ?@* are )* 8 5; 8 6? / @6 more than ?@*.


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The shortfall is not balanced by the excess. There is an excess of @6 ? / 5B.

So what is the aerage" #f we assume the aerage of these @ numbers to be ?@*, there is an excess of5B. 7e need to distribute the excess eenly among all the numbers and hence the aerage will increaseby 5B&@ / ?. 3Co back to the first post on arithmetic mean if this is not clear.4

:ence, the required mean is ?@* 8 ? / ?@?.

3#f we had assumed the mean to be ?@?, the shortfall would hae balanced the excess.4

Lets go through one more example using this concept1

,-ample 0/What is the mean of 99" 1'3" 1'" 1'9" 1$'" 1$3" 1$4" 13'?

Lets start by guessing a mean for this sequence. Say, around ))"

Lets see if the shortfall is balanced by the excess.

;; is )9 less, )*6 is )5 less, )*? is )) less and )*; is 9 less than )).

%erall shortfall / )9 8 )5 8 )) 8 9 / ?

)5* is more, )56 is B more, )5B is )6 more and )6* is ) more than )).

%erall excess / 8 B 8 )6 8 ) / ?)

7e are close, but not quite there yetD There is a shortfall of ?. Since there are a total of B numbers,the aerage must be ?&B / *. less than )). :ence, the aerage here is ))?.

%nce you get a hang of this method and understand what you are doing, it is much faster than addingall the big numbers and then diiding the sum since you only deal with small numbers in this method.

Lets wrap up this post here. #n the next post, we will see these concepts in actionD

A))*ication of Arithmetic Mean#


9/29


#n the aboe post we discussed arithmetic means of arithmetic progressions in C(!T math problems.2ow, lets see those concepts in action.

Question ./(f is the sum of the even integers from $'' to &'' in!lusive" and y is the num#er ofeven integers from $'' to &'' in!lusive" what is the value of 6 y?

3!4 5**?**3'4 5*)?**3-4 5**?*53


10/29

7e can find the middle number i.e. the aerage by Aust aeraging the first and the last terms.

(nm)+(n+m)2=2n2=n

A(erage=(200+600)2=400

Sum of all terms in the sequence / x / !rithmetic (ean 2umber of terms / ?**5*)

-+,=400+201+201=401+201

*nswer +-This uestion is dis!ussed$,R,.

This question was simple. 0ou could hae found the sum using the formula n2+(2a+(n1)%)that we

saw in the !O post. 'ut this method is more intuitie since if you dont want to, you dont hae to use

any formula here. !nyway, lets go on to our second question for today.

Question 0/The sum of n !onse!utive positive integers is 5. What is the value of n?

Statement #1 n is een

Statement ##1 n Q ;

Solution/>irst # will gie the solution of this question and then discuss the logic used to sole it.

#n how many ways can you write n consecutie integers such that their sum is ?" Lets see whether wecan get such numbers for some alues of n.

n / ) MN 2umbers1 ?n / 5 MN 2umbers1 55 8 56 / ?n / 6 MN 2umbers1 )? 8 ) 8 )9 / ?n / ? MN 2o such numbersn / MN 2umbers1 @ 8 B 8 ; 8 )* 8 )) / ?n / 9 MN 2umbers1 8 9 8 @ 8 B 8 ; 8 )* / ?

Lets stop right here.

Statement #1 n must be een.

n could be 5 or 9. Statement # alone is not sufficient.

Statement ##1 n Q ;n can take many alues less than ; hence statement 5 alone is not sufficient.

'oth statements together1 Since n can take alues 5 or 9 which are een and less than ;, bothstatements together are not sufficient.

*nswer +,-This uestion is dis!ussed$,R,.
http://gmatclub.com/forum/if-x-is-the-sum-of-the-even-integers-from-200-to-600-inclusive-and-y-204307.htmlhttp://gmatclub.com/forum/if-x-is-the-sum-of-the-even-integers-from-200-to-600-inclusive-and-y-204307.htmlhttp://gmatclub.com/forum/the-sum-of-n-consecutive-positive-integers-is-45-what-is-126861.htmlhttp://gmatclub.com/forum/if-x-is-the-sum-of-the-even-integers-from-200-to-600-inclusive-and-y-204307.htmlhttp://gmatclub.com/forum/the-sum-of-n-consecutive-positive-integers-is-45-what-is-126861.html


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2ow, the interesting thing is how do we get these numbers for different alues of n. :ow do we knowthe alues that n can take" #ts pretty easy really. >ollow my thought here.

%f course, n can be ). #n that case we hae only one number i.e. ?.

n can be 5. 7hy" 7hen we diide ? by 5, we get 55.. Since 555. is ?, we hae to find 5

consecutie integers such that their arithmetic mean is 55.. The integers are obiously 55 and 56.

n can be 6. 7hen we diide ? by 6, we get ). So we need 6 consecutie integers such that their meanis ). They are )?, ), )9.

7hen we diide ? by ?, we get )).5.


12/29

Mean# !"e#tion# on Me%ian


!s promised, we discuss medians hereD -onceptually, the median is ery simple. #t is Aust the middlenumber. !rrange all the numbers in increasing&decreasing order and the number you get right in themiddle, is the median. So it is quite straight forward when you hae odd number of numbers since youhae a Rmiddle number. 7hat about the case when you hae een number of numbers" #n that case, itis Aust the aerage of the two middle numbers.

(edian of 5, , )*U is

(edian of 6, @B, )*5, **U is (78+102)2=90

#f its that simple, why are we discussing it" because it isnt Rthat simpleD -onceptually it is, but

when the test writers make questions using median and arithmetic mean together, they make someery mean questionsD # will show you with an example, but first, we will look at a simpler question.

Question ./*" : and 2 have re!eived their ath midterm s!ores today. They find that thearithmeti! mean of the three s!ores is 74. What is the median of the three s!ores?

3)4 ! scored a @6 on her exam.

354 - scored a @B on her exam.

Solution/+ecall from the arithmetic mean postthat the sum of deiations of all scores from themean is *.

i.e. if one score is less than mean, there has to be one score that is more than the mean.e.g. #f mean is @B, one of the following must be true1!ll scores are equal to @B.!t least one score is less than @B and at least one is greater than @B.>or example, if one score is @* i.e. B less than @B, another score has to make up this deficit of B.Therefore, there could be a score that is B9 3B more than @B4 or there could be two scores of B5 eachetc.

Statement )1 ! scored @6 on her exam.

>or the mean to be @B, there must be at least one score higher than @B. 'ut what exactly are the othertwo scores" 7e hae no ideaD Various cases are possible1

@6, @B, B6 or

@6, @?, B@ or

@*, @6, ;) etc.

#n each case, the median will be different. :ence this statement alone is not sufficient.
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Statement 51 - scored @B on her exam.

2ow we know that one score is @B. =ither the other two will also be @B or one will be less than @B andthe other will be greater than @B. #n either case, @B will be the middle number and hence will be themedian. This statement alone is sufficient.

*nswer +:-This uestion is dis!ussed$,R,.

7ere you tempted to say 3-4 is the answer" # hope this question shows you that median can be a littletricky. Lets go on to the tougher question now.

Question 0/%ive logs of wood have an average length of 1'' !m and a median length of 11& !m.What is the maimum possi#le length" in !m" of the shortest pie!e of wood?

3!4 *3'4 @93-4 B?3irst thing that comes to mind median is the 6rd term out of so the lengths arranged in increasingorder must look like this1

EEE EEE ))9 EEE EEE

The mean is gien and we need to maximiGe the smallest number. 'asically, the smallest number shouldbe as close to the mean as possible. This means the greatest number should be as close to the mean aspossible too 3if the shortfall deiation is small, the excess deiation should by equally small4.

#f this doesnt make sense, think of a set with mean 5*1

);, 5*, 5) 3smallest number is ery close to meanW greatest number is ery close to the mean too4), 5*, 6; 3smallest number is far away from the mean, greatest number is far away too4

Psing the same logic, lets make the greater numbers as small as possible 3so the smallest number canbe as large as possible4. The two greatest numbers should both be at least ))9 3since ))9 is themedian4. 2ow the lengths arranged look like this1

EEE EEE ))9 ))9 ))9

Since the mean is )** and each of the 6 large numbers are already )9 more than )** i.e. total )96 /

?B more than the mean 3excess deiation is ?B4, the deiations of the two small numbers should be atotal of ?B less than the mean. To make the smallest number as great as possible, each of the smallnumbers should be ?B&5 / 5? less than the mean i.e. they both should be @9.

*nswer +:-.This uestion is dis!ussed$,R,.

:opefully, it made sense to you.EEEEEEEEEEEEEEEEE
http://gmatclub.com/forum/a-b-and-c-have-received-their-math-midterm-scores-today-they-find-th-203982.htmlhttp://gmatclub.com/forum/five-logs-of-wood-have-an-average-length-of-100-cm-and-a-median-length-203984.htmlhttp://gmatclub.com/forum/a-b-and-c-have-received-their-math-midterm-scores-today-they-find-th-203982.htmlhttp://gmatclub.com/forum/five-logs-of-wood-have-an-average-length-of-100-cm-and-a-median-length-203984.html


14/29

A ange of !"e#tion#


Lets discuss the idea of Rrange today. #t is simply the difference between the smallest and thegreatest number in a set. -onsider the following examples1

+ange of J5, 9, )*, 5, *K is * 5 / ?B

+ange of JM5*, )**, B*, 6*, 9**K is 9** 3M5*4 / 95*

and so onI

Thats all the theory we hae on the concept of rangeD So lets Aump on to some questions now 3thereinlies the challenge4D

Question ./Whi!h of the following !annot #e the range of a set !onsisting of 5 odd multiples of9?

3!4 @53'4 )??3-4 5BB3


15/29

*nswer +,-.This uestion is dis!ussed$,R,.

%n to another one now1

Question 0/(f the arithmeti! mean of n !onse!utive odd integers is $'" what is the greatest ofthe integers?

3)4 The range of the n integers is )B.

354 The least of the n integers is )).

Solution/ 7e hae discussed mean in case of arithmetic progressions in the preious posts. #f mean ofconsecutie odd integers is 5*, what do you think the integers will look like"

);, 5) or)@, );, 5), 56 or), )@, );, 5), 56, 5 or)6, ), )@, );, 5), 56, 5, 5@ or

)), )6, ), )@, );, 5), 56, 5, 5@, 5;etc.


16/29

'ea*ing /ith tan%ar% 'e(iation


#n this post, we will work our way through the concepts of Standard


17/29

going to dele deep into it but will quickly recap so that we can moe ahead. +ecall that if you plot thenumbers on the number line, it gies you a sense of how far the numbers are from the mean. Thefarther the numbers, higher is the S


18/29

Lets discuss each of these four cases now.

-ase )1 S / J6, 6, 6K or T / J*, )*, 5*K

T has higher S


19/29

deviation of d" where d is positive. Whi!h of the following two test s!ores" when added to the list"must result in a list of 3'$ test s!ores with a standard deviation less than d?

3!4 @ and B*3'4 B* and B3-4 @* and @

3" : < =$'"1'" '" 1'" $'> and 2< =3'" 35" '" 5" 5'>-" whi!h of the following represents the !orre!t ordering+largest to smallest- of the sets in terms of the a#solute in!rease in their standard deviation?

3!4 !, -, '3'4 !, ', -3-4 -, !, '3


20/29

*nswer +,-This uestion is dis!ussed$,R,.

Simple enough, right" S< questions are generally straight forward once you understand the basics well.

ome Tric, tan%ar% 'e(iation !"e#tion#BY KARISHMA, VERITAS PREP

#n the aboe post, we promised you a couple of tricky standard deiation 3Sor each tank, ?*H of the olume of water that was in the tank at the beginning of theexperiment was remoed during the experiment.

Statement 51 The aerage olume of water in the tanks at the end of the experiment was B* gallons.

Solution/

7e hae B water tanks. This implies that we hae B elements in the set 3olume of water in each of theB tanks4. S< of the olume of water in the tanks is 5* gallons. 7e need to find the new S< i.e. the Sor each tank, ?*H of the olume of water that was in the tank at the beginning of the

experiment was remoed during the experiment.

#nitial S< is 5*. 7hen ?*H of the water is remoed from each tank, the leftoer water is 9*H of theinitial olume of water i.e. *.9initial olume of water. This means that each element of the initial setwas multiplied by *.9 to obtain the new set. The S< will change. #t will become *.9preious S< i.e.*.95* / )5 3think of the formula of S< we discussed in the first S< post4. This statement alone issufficient.

Statement 51 The aerage olume of water in the tanks at the end of the experiment was B* gallons.

The aerage olume doesnt gie us the S< of the new set. :ence, this statement alone is notsufficient.

*nswer +*-This uestion is dis!ussed$,R,.

2ow that we are done with the easier one, lets go on to the tougher one.

Question 0/ is a !olle!tion of four odd integers. The range of set is . @ow many distin!tvalues !an standard deviation of tae?

3!4 6
http://gmatclub.com/forum/sets-a-b-and-c-are-shown-below-if-number-100-is-included-111455.htmlhttp://www.veritasprep.com/gmat/http://gmatclub.com/forum/during-an-experiment-some-water-was-removed-from-each-of-94166.htmlhttp://gmatclub.com/forum/during-an-experiment-some-water-was-removed-from-each-of-94166.htmlhttp://gmatclub.com/forum/sets-a-b-and-c-are-shown-below-if-number-100-is-included-111455.htmlhttp://www.veritasprep.com/gmat/http://gmatclub.com/forum/during-an-experiment-some-water-was-removed-from-each-of-94166.html


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3'4 ?3-4 3


22/29

3 m)ortant Conce)t# for tati#tic# !"e#tion# on the MATBY KARISHMA, VERITAS PREP

7e hae discussed these three concepts of statistics in detail1

*rithmeti! meanis the number that can represent&replace all the numbers of the sequence. #t liessomewhere in between the smallest and the largest alues.

edianis the middle number 3in case the total number of numbers is odd4 or the aerage of twomiddle numbers 3in case the total number of numbers is een4.

;tandard deviationis a measure of the dispersion of the alues around the mean.

! conceptual question is how these three measures change when all the numbers of the set are ariedis a similar fashion.

>or example, how does the mean of a set change when all the numbers are increased by say, )*" :owdoes the median change" !nd what about the standard deiation" 7hat happens when you multiplyeach element of a set by the same number"

Lets discuss all these cases in detail but before we start, we would like to point out that the discussionwill be conceptual. 7e will not get into formulas though you can arrie at the answer by manipulatingthe respectie formulas.

7hen you talk about mean or median or standard deiation of a list of numbers, imagine the numberslying on the number line. They would be spread on the number line in a certain way. >or example,

XX*XaXXXbXcXXXXXXXdXXXeXXXXXXXXfXgXXXXXXX

-ase #1

7hen you add the same positie number 3say x4 to all the elements, the entire bunch of numbersmoes ahead together on the number line. The new numbers a, b, c, d, e, f and g would look likethis

XX*XXXXXXaXXXbXcXXXXXXXdXXXeXXXXXXXXfXgXXXXXX

The relatie placement of the numbers does not change. They are still at the same distance from eachother. 2ote that the numbers hae moed further to the right of * now to show that they hae moedahead on the number line.

The mean lies somewhere in the middle of the bunch and will moe forward by the added number. Say,


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if the mean was d, the new mean will be %=%+-.

;o when you add the same num#er to ea!h element of a list"

Aew mean < Old mean 6 *dded num#er.

%n similar lines, the median is the middle number 3d in this case4 and will moe ahead by the added

number. The new median will be %=%+-

;o when you add the same num#er to ea!h element of a list"

Aew median < Old median 6 *dded num#er

Standard deiation is a measure of dispersion of the numbers around the mean and this dispersion doesnot change when the whole bunch moes ahead as it is. Standard deiation does not depend on wherethe numbers lie on the number line. #t depends on how far the numbers are from the mean. Sostandard deiation of 6, , @ and ; is the same as the standard deiation of )6, ), )@ and );. The

relatie placement of the numbers in both the cases will be the same. @en!e" if you add the samenum#er to ea!h element of a list" the standard deviation will stay the same.

-ase ##1

Lets now moe on to the discussion of multiplying each element by the same positie number.

The original placing of the numbers on the number line looked like this1

XX*XaXXXbXcXXXXXXXdXXXeXXXXXXXXfXgXXXXXXX

The new placing of the numbers on the number line will look something like this1

XX*XXXaXXXXXXbXXXcXXXXXXXXXXXXdXXXXXXXXXeXM etc

The numbers spread out. To understand this, take an example. Say, the initial numbers were )*, 5* and6*. #f you multiply each number by 5, the new numbers are 5*, ?* and 9*. The difference betweenthem has increased from )* to 5*.

#f you multiply each number by x, the mean also gets multiplied by x. So, if d was the mean initially, d

will be the new mean which is -+%.

Aew mean < Old mean B ultiplied num#er

Similarly, the median will also get multiplied by x.

Aew median < Old median B ultiplied num#er

7hat happens to standard deiation in this case" #t changesD Since the numbers are now further apartfrom the mean, their dispersion increases and hence the standard deiation also increases. The newstandard deiation will be x times the old standard deiation. 0ou can also establish this using the


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standard deiation formula.

Aew standard deviation < Old standard deviation B ultiplied num#er

The same concept is applicable when you increase each number by the same percentage. #t is akin tomultiplying each element by the same number. Say, if you increase each number by 5*H, you are, in

effect, multiplying each number by ).5. So our case ## applies here.

2ow, think about what happens when you subtract&diide each element by the same number.

o/ to !"ic*, o*(e tan%ar% 'e(iation !"e#tion# on theMAT

BY RON AWAD, VERITAS PREP

The quantitatie section of the C(!T is designed to test your understanding and application of

concepts you learned in high school. The exam focuses on core mathematical concepts such as algebra,geometry and statistics. :oweer some concepts are more engrained in the high school curriculum thanothers. =eryones done addition, multiplication, subtraction and diision, but sometimes figuring outfactorials or square roots may be a little more unusual.

Oerhaps no concept perplexes students on the C(!T more than the standard deiation. The standarddeiation 3often represented by C4 is measure of dispersion around the mean. #t indicates how closethe numbers in a set are to the sets aerage.As a simple example, the sets J, )*, )K and JB, )*, )5Kboth hae the same mean 3)*4W howeer they do not hae the same standard deiation.

Ynowing how to calculate the standard deiation is not required on the C(!T, but knowing how itscalculated gies you a tremendous edge in answering questions. #ts a four step process1

.6 Find the average 7mean6 of the set8

06 Find the di9erences :etween each element of the set and that average8

!6 S;uare all the di9erences and take the average of the di9erences8 Thisgives ou the variance8


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represents and the idea behind it are fair game on the test. %ne of the simple takeaways from themath behind the process is that, the farther the number is from the mean of the set, the more thestandard deiation will increase. Specifically, the distance increases with the square of the difference,so looks much farther out than 5.

This kind of concept can be tested on the exam, but if you know what youre looking for, you can

answer standard deiation questions ery quickly. Lets look at an example1

%or the set =$" $" 3" 3" " " 5" 5" >" whi!h of the following values of will most in!rease thestandard deviation?

3!4 )3'4 53-4 63


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A 750 e(e* MAT !"e#tion on tati#tic#


#n this post, we hae a ery interesting statistics question for you. !boe, we hae already discussedstatistics concepts such as mean, median, range.

This question needs you to apply all these concepts but can still be easily done in under two minutes.2ow, without further ado, lets go on to the question there is a lot to discuss there.

Question/*n automated manufa!turing unit employs A eperts. Their average monthly salary is7''' while the median monthly salary is only 5'''. (f the range of their monthly salaries is1'"'''" what is the minimum value of A?

3!4)*3'4)53-4)?3


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* is @*** less than @*** and *** is 5*** less than @*** which means we hae a total of F;*** less than@***. %n the other hand, )*,*** is 6*** more than @***. The deiations on the two sides of mean donot balance out. To balance, we need to add two more people at a salary of F)*,*** so that the totaldeiation on the right of @*** is also F;***. 2ote that since we need the minimum number of experts,we should add new people at )*,*** so that they quickly make up the deficit in the deiation. #f weadd them at B*** or ;*** etc, we will need to add more people to make up the deficit at the right.

2ow we hae

* I *** I )****, )****, )****

2ow the mean is @*** but note that the median has gone awry. #t is )*,*** now instead of the ***that is required. So we will need to add more people at *** to bring the median back to ***. 'utthat will disturb our mean againD So when we add some people at ***, we will need to add some at)*,*** too to keep the mean at @***.

*** is 5*** less than @*** and )*,*** is 6*** more than @***. 7e dont want to disturb the totaldeiation from @***. So eery time we add 6 people at *** 3which will be a total deiation of 9***

less than @***4, we will need to add 5 people at )*,*** 3which will be a total deiation of 9*** morethan @***4, to keep the mean at @*** this is the most important step. =nsure that you haeunderstood this before moing ahead.

7hen we add 6 people at *** and 5 at )*,***, we are in effect adding an extra person at *** andhence it moes our median a bit to the left.

Lets try one such set of addition1

* I ***, ***, ***, *** I )****, )****, )****, )****, )****

The median is not F*** yet. Lets try one more set of addition.

* I ***, ***, ***, ***, ***, ***, *** I )****, )****, )****, )****, )****, )****, )****

The median now is F*** and we hae maintained the mean at F@***.

This gies us a total of ) people.

*nswer +-This uestion is dis!ussed$,R,.

Cranted, the question is tough but note that it uses ery basic concepts and that is the hallmark of agood C(!T questionD

Try to come up with some other methods of soling this.

EEEEEEEEEEEEEEEEE
http://gmatclub.com/forum/an-automated-manufacturing-unit-employs-n-experts-such-that-the-range-195844.htmlhttp://gmatclub.com/forum/an-automated-manufacturing-unit-employs-n-experts-such-that-the-range-195844.htmlhttp://gmatclub.com/forum/an-automated-manufacturing-unit-employs-n-experts-such-that-the-range-195844.html


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A 750 e(e* !"e#tion on 'BY KARISHMA, VERITAS PREP

!boe, we looked at a @*8 leel question on mean, median and range concepts of Statistics. :ere wehae a @*8 leel question on standard deiation concept of Statistics. 7e do hope you enAoy checkingit out.

'efore you begin, you might want to reiew the post that discusses standard deiation1


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M #f x and y were different, we could select the alues of x and y in 6-5 ways1 x could be ) and y couldbe 6W x could be ) and y could be W x could be 6 and y could be .

>or clarification, lets enumerate the different ways in which we can write set S1

J), ), ), K, J), 6, 6, K, J), , , K, J), ), 6, K, J), ), , K, J), 6, , K

These are the 9 ways in which we can choose the numbers in our example.

7ill all of them hae unique standard deiations" or the set J), ), ), K, mean is 5. Three of the numbers are distance ) away from mean and onenumber is distance 6 away from mean.

>or the set J), , , K, mean is ?. Three of the numbers are distance ) away from mean and onenumber is distance 6 away from mean.

The deiations in both cases are the same MN ), ), ) and 6. So when we square the deiations, addthem up, diide by ? and then find the square root, the figure we will get will be the same.

Similarly, J), ), 6, K and J), 6, , K will hae the same S

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