How to Solve: Statistics

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I have learned a lot from gmatclub and am done with my gmat too. So, i have decided to contribute back.

As part of this i have decided to share the knowledge i have regarding various topics related to gmat quant.

Hope it will be useful. This post is about how to solve "Statistics"

TheoryMean (Arithmetic Mean) is the average of the all the numbers in the set.

Mean = (Sum of all the numbers in the set) / total number of numbers in the set

Suppose the set is {1,2,3,4,5}

Then, Mean = (1+2+3+4+5)/5 = 15 / 5 = 3

1. If all the numbers in the set are increased/decreased by the same number(k) then the mean also gets increased/decreased by the same number(k)

Suppose the set is {a,b,c,d,e}

then the Mean = (a+b+c+d+e)/5

Now, lets increase all the numbers by k. So, the new set is {a+k,b+k,c+k,d+k,e+k)

New Mean = (a+k +b+k +c+k +d+k + e+k)/5

= (a+b+c+d+e + 5k)/5 = ((a+b+c+d+e)/5 ) + k = Old Mean + k

2. If all the numbers in the set are multiplied/divided by the same number(k) then the mean also gets multiplied/divided by the same number(k)

Proof same as above. In this case if we multiple all the numbers by k then

New Mean = k* (Old Mean)

SUGGESTION: Don't try remembering the points 1 and 2 above. It does not take much time to calculate them!

Median is the middle value of the set.

In case of even number of numbers in the set: Median is the mean of the two middle numbers (after the numbers are arranged in the increasing / decreasing order)

Example: If the set is {5,1,4,6,3,2} then we will arrange the set as {1,2,3,4,5,6} and median will be mean of middle two terms. Middle two terms in this case are 3 and 4 so Median = (3+4)/2 = 3.5

In case of odd number of numbers in the set: Median is the middle number (after the numbers are arranged in increasing/ decreasing order )

Example: If the set is {4,5,3,1,2} then we will arrange the set as {1,2,3,4,5} and the median will be the middle number which is 3

1. If all the numbers in the set are increased/decreased by the same number(k) then the median also gets increased/decreased by the same number(k)

Proof same as for mean.

2. If all the numbers in the set are multiplied/divided by the same number(k) then the median also gets multiplied/divided by the same number(k)

Proof same as for mean.

3. In Case of evenly spaced set

Mean = Median = Middle term (if the number of terms is odd)

= Mean of middle terms (if the number of terms is even)

4. In case of consecutive integers: IF the number of integers is even then then the Mean = Median != Integer

Suppose the set is {1,2,3,4,5,6}

then Mean = Median = 3.5

SUGGESTION: Don't try remembering the points 1 and 2 above. It does not take much time to calculate them!

Range of a set is the difference between the highest and lowest value of the set.

Example: Suppose the set is {-1,2,3,6,8} then the range will be

8 -(-1) = 9

1. If all the numbers in the set are increased/decreased by the same number(k) then the range DOES NOT CHANGE!

Suppose the set is {a,b,c} (in increasing order)

Range = c-a

Now, lets increase all the numbers by k then the set will become {a+k, b+k, c+k}

New range = c+k -(a+k) = c-a = Old range

2. If all the numbers in the set are multiplied/divided by the same number(k) then the range also gets multiplied/divided by the same number(k)

Proof similar to that for mean.

Mode is the number which has occurred the maximum number of times in the set.

Suppose the set is {1,1,2,2,3,3,3,3,4,5}

then the mode is 3, as 3 has occurred the maximum number of times in the set.

Standard Deviation is an indicator of how spread the numbers are. Standard Deviation is the Root Mean Square (RMS) of the distance of the values from the mean.

Variance = (sum of (squares of difference of each number from mean )) / total number of numbers

Standard deviation = \sqrt{Variance}

Example:

Suppose the set is (1,2,3,4,5)

then Mean = 3

Variance = { (3-1)^2 + (3-2)^2 + (3-3)^2 + (3-4)^2 + (3-5)^2)} / 5

= (4+1+0+1+4)/5

= 2

Standard Deviation = \sqrt{Variance} = \sqrt{2}

Arithmetic Sequence is the sequence in which each number differs from its previous by a constant value (d)

Arithmetic sequence is generally denoted as

a , a+d , a+2d+,..., a+(n-1)d

where, a is the first term of the sequence.

d is the common difference

n is the number of terms in the sequence.

Tn is the nth term of the sequence.

Tn = a + (n-1)d

A.M. = Arithmetic mean of the sequence = Mean of First term and last term = (a + a+(n-1)d)/2

= (2a+ (n-1)d)/2

Sum of all the terms of the sequence = A.M. * n = (n/2) * (2a + (n-1)d)

Problems:1. If the mean of numbers 28, x, 42, 78 and 104 is 62, then what is the mean of 128, 255, 511, 1023 and x?

A. 395

B. 275

C. 355

D. 415

E. 365

Solution:

the mean of numbers 28, x, 42, 78 and 104 is 62

=> (28+x+42+78+104)/5 = 62

=> x = 58

the mean of 128, 255, 511, 1023 and x = mean of 128, 255, 511, 1023 and 58

= (128+255+511+1023+58)/5 = 1975/5 = 395

So, Answer will be A

link to the problem:http://gmatclub.com/forum/if-the-mean-of-numbers-28-x-42-78-and-104-is-62-then-135288.html2. Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

(A) Zero

(B) One

(C) Two

(D) Three

(E) More than three

Solution:

we need to first decide where will we position x

suppose we keep x at the center then the set in ascending order will be

{4,8,x,12,16}

now the median = x so the mean also has to be x

=> (4+8+x+12+16)/5 = x

=> x= 10

which is possible. So, x=10 is one such value

Now lets put x at the left of 8 so the median will be 8 now. So even the mean has to be 8

=> (4+8+x+12+16)/5 = 8

=> x= 0

Which is possible. So, x=0 is one such value

Now lets put x at the right of 12 so the median will be 12 now. So even the mean has to be 12

=> (4+8+x+12+16)/5 = 12

=> x= 20

Which is possible. So, x=20 is one such value

So, there are three such values for which mean = median.

So, answer will be D

link to the problem:http://gmatclub.com/forum/set-s-consists-of-5-values-not-necessarily-in-ascending-144018.html3.

http://gmatclub.com/forum/set-a-3-3-3-4-5-5-5-has-a-standard-deviation-of-138266.htmlLooking for a Quant Tutor?

Check out my post for the same

starting-gmat-quant-classes-tutoring-bangalore-online-135537.htmlHope it helps!

Good Luck!

_________________

Ankit

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