Statistics Test-2

Given:

• Let the integers be a, b, c, d, e ….
• Set P = {Respective Distance of a, b, c, , d, e …. from -1}
• Set Q = {Respective Distance of a, b, c, d, e …from 1}

To Find : the statement, which is true about the standard deviation of sets P and Q

Approach:

1. To find the relation between the standard deviation of sets P and Q, we would need to find the relation between the terms that are present in sets P and Q
2. Terms in set P
   • We know that distance of a number x from -1 on the number line can be written as |x –(-1)| = |x +1|
   • We will use the above understanding to write down the terms of set P
3. Terms in set Q
   • We know that distance of a number x from 1 on the number line can be written as |x -1|
   • We will use the above understanding to write down the terms of set Q
4. Once we write down the terms of sets P and Q, we will find the relation between the terms of these 2 sets. Then, we 'll use the standard properties of standard deviation to compare the standard deviation of sets P and Q.

Working out:

1. Terms in set P
   • Set P = {|a+1|, |b+1|, |c+1|…….}
2. Terms in set Q
   • Set Q = {|a-1|, |b-1|, |c-1|…….} = {|(a + 1) -2|, |(b+1) -2|, |(c+1) -2|……..}
   • We can observe that if we subtract 2 from each of the terms of set P, we will get the terms of set Q.
3. We know from the property of standard deviation that reducing all the terms of a set by the same constant does not change the standard deviation of the set.
4. So, we can write Standard Deviation(P) = Standard Deviation(Q)

Answer : A  

Step 1 &2: Understand Question and Draw Inference

Given:

• List A: 4, 8, 20, x (don ’t know the value of x, other numbers arranged in ascending order)
• List B: -4, 3, 6, 8, 12 (numbers arranged in ascending order)

To find: x = ?

Step 3 : Analyze Statement 1 independent

(1) The range of the numbers in List A is equal to the range of the numbers in List B

• Range of List B = 12 –(-4 ) = 12 + 4 = 16
• So, Range of List A = 16
• (Greatest number of List A) –(Smallest number of List A) = 16
  • In List A, difference between 20 and 4 is 16
  • This means, Greatest number of List A = 20
  • And, Smallest number of List A = 4
• This means, 4 ≤x ≤20

Not sufficient to find a unique value of x.

Step 4 : Analyze Statement 2 independent

(2) The median of the numbers in List A is equal to the median of the numbers
in List B.

• Median of List B = 6
• So, Median of List A = 6
• In List A, possible arrangements of the 4 elements in ascending order and the corresponding medians:

• Thus, from Statement 2, we infer that x ≤4

Not sufficient to find a unique value of x

Step 5: Analyze Both Statements Together (if needed)

• From Statement 1: 4 ≤x ≤20
• From Statement 2: x ≤4
• combing the 2 statements: x = 4Combin

Thus, the two statements together are sufficient to find a unique value of x
Answer: Option C

Given:

• The list of first n non-negative integers: {0, 1, 2, 3, . . . , n –1}
• Positive integer y is added to each integer in this list: {0 + y, 1 + y, 2 + y, . . . n –1 + y}
  • = {y, y + 1, y + 2, . . . , y + n –1}

To Find:  Which of the 3 statements is/are true?

Approach:

1. Since these 3 statements deal with:
   • Mean of the first n positive integers
   • Mean of the resulting numbers
   • And, Median of the resulting numbers,

We will first find the expressions for these 3 quantities.

2. Then, we ’ll evaluate the 3 statements one by one to determine which is/are true for all values of y and n

Working out:

• Finding the expressions for the 3 quantities featured in Statements I –III
   
  • Finding Mean of the first n positive integers
    • Sum of first n positive integers = 
    • So, the mean of the first n positive integers =  
      • (n+2/2)
• Finding Mean of the Resulting Numbers
  • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n –1}
    • These numbers form an increasing arithmetic sequence of n terms.
      • First term of the sequence = y
      • Last term of the sequence = y + n -1
      • So, the sum of these numbers =

• Finding Median of the Resulting Numbers
  • The resulting numbers are: {y, y + 1, y + 2, . . . , y + n –1}
  • The total number of elements in this set is (y + n –1) –y + 1 = n
    • These numbers form an increasing arithmetic sequence of n terms.
    • Now, in an ordered list that has:
      • An even number of elements (say 4 elements), the median of the list is equal to the average of the middle 2 elements of the list
      • An odd number of elements (Say 5 elements), the median of the list is equal to the middle element in the list

• Case 1: If n is odd,
• Then, Median = the middle element in the list of resulting numbers
• The first term in the list is y + 0 and the last term is y +(n –1)
• So, the Median = 
• (Note: If the above expression for the Median is not intuitive to you, you can arrive at it by taking a few easy values of n. For example:
  • If n = 3, the list is {y, y + 1, y + 2}. So, the median = y + 1
  • If n = 5, the list is {y, y + 1. y + 2, y + 3, y + 4}. So, the median = y + 2
  • Similarly, if n = 7, the list goes from y to y + 6 and the median = y + 3
  • From these examples, the pattern for how the value of Median changes with n becomes easy to see)
• Case 2: If n is even,
  • This means, the median of the list is equal to the ​
  • 
  • 

• Evaluating Statement I
  • If the median of the resulting numbers is then n is odd
  • In our calculation of the Median of the Resulting Numbers, observe that the median is always equal to  , whether n is even or odd.
  • Therefore, Statement I is not correct
• Evaluating Statement II
  • The arithmetic mean of the resulting numbers is equal to the median of the resulting numbers
    • From our calculations of the Mean and Median of the Resulting Numbers, we see that:
      • Mean of the Resulting numbers =
      • Median =
  • So, Statement II is indeed true.
• Evaluating Statement III
  • The arithmetic mean of the resulting numbers is y units greater than the arithmetic mean of the first n positive integers.
  • From our calculations above, we see that:
    • Mean of the Resulting numbers = 
    • Mean of the first n positive integers  
  • Note that  is not equal to  . Therefore, it is wrong to say that Mean of the Resulting Numbers is y units greater than the Mean of the first n positive integers.
  • So, Statement III is not true.
• Getting to the answer
  • Of the 3 statements, we see that only Statement II is true.

Looking at the answer choices, we see that the correct answer isOption B

Step 1 &2: Understand Question and Draw Inference

Given : Integers a, b, c >0
To find : (a+c) /2

Step 3 : Analyze Statement 1 independent

Statement 1 says that ‘The average of a + b and 4 is 6 ’

• This equation doesn ’t indicate to us the value of a + c
• So, Statement 1 is not sufficient to answer the question

Step 4 : Analyze Statement 2 independent

Statement 2 says that ‘The average of a + c and b is 18 ’

• We need to find the value of a + c. However, the above equation gives us the value of a + b + c
  • Multiple values of a + c will satisfy this equation
    • For example, a + c = 35 and b = 1 or a + c = 34 and b = 2 etc.
• Therefore, Statement 2 alone is not sufficient to answer the question

Step 5: Analyze Both Statements Together (if needed)

• From Statement 1: a + b = 8
• From Statement 2: a + b + c = 36
• Combining the 2 statements: c = 36 –8 = 28
• However, we do not yet know the value of a
• So, we are still unable to find the value of a + c
  So, even the 2 statements together are not sufficient to answer the question

Answer: Option E

Given:

• 70 students scored more than Ricky
• 428 students scored less than Ricky
• None scored equal to Ricky
  • Number of students = 70 + 428 + 1 = 499
  • When arranged in ascending order, Ricky ’s score would be at 429 place.
• Mean score = m
• Standard deviation of the scores = d

To Find: : The range in which Ricky ’s score should lie

Approach:

1. As the options are given in terms of m and d, we first need to understand the distribution curve. 
2. The above distribution is symmetric about the mean m
   • 68% of the distribution lies within 1 standard deviation, the distribution would have  68% / 2 = 34  of data sets on either sides of the mean.
   • Similarly, 95% of the distribution lies within 2 standard deviation, the distribution would have 95 % / 2 = 47.5 % of data sets on either sides of the mean.
3. Since we know the position of Ricky ’s score and we know the number of students who appeared in the Olympiad, we can calculate the number of students between consecutive standard deviations of the mean score.

Working out:

1. Since, 499 students appeared in the Olympiad, the mean score would lie at position 250
   • So, number of students whose score lies within 1 standard deviation = 68% of 500 = 340
   • So, number of students whose score is between m and m+d = 340 / 2 = 170  i.e. position of scores of students who lie between m and m + d will be between 250 and (250+170) = 420
   • We need not bother with scores of students whose position is less than 250, as we are concerned about the position of student who is at 429^th place.
2. As Ricky ’s score is at 429 place, his score does not lie between m and m+ d. Let ’s see if he lies between m + d and m + 2d.
3. Number of students whose score lie within 2 standard deviations = 95% of 500 = 475
   • Number of students whose score lie between m and m + 2d = 475/2 = 237
4. So, number of students whose score lie between m + d and m + 2d = 237 –170 = 67
5. So, the position of students whose score lie between m + d and m + 2d will be between 420 and 420 + 67 = 487
   • As Ricky ’s score lies at a position of 429, his score would lie between m + d and m + 2d

Answer : D  

Given:

2 sets of Prime Numbers –let ’s call them Sets A and B:

• Set A = {2, 3, 5, 7}
• Set B = {11, 13, 17, 19}

To find : Difference between Average(Set B) and Average (Set A)

• That is, Average(Set B) –Average (Set A)

Approach:

1. In order to find the answer, we need to know the values of Average(Set A), Average(Set B)
2. Since we know all the elements in each of Set A and B, we will be easily able to find the required averages.

Working out:

Correct Answer –Option B

Step 1 &2: Understand Question and Draw Inference

• Let the number of students be x
• Number of rounds in the quiz = 3

To Find : Median number of rounds cleared by the students

• Median = {0, 1, 2 or 3}

Step 3 : Analyze Statement 1 independent

1. 20  percent of the students could not clear round 1 of the quiz.

• 0.2x could not clear round 1
• 0.8x cleared round 1
  • Since more than half cleared the first round, we can be sure that Median is not equal to zero
• But we do not know how many of them cleared round 2 and round 3.So,
  • Median may be 1, 2 or 3

Insufficient to find an answer.

Step 4 : Analyze Statement 2 independent

2. 40 percent of the students could clear round 2 but could not clear round 3.

• 0.4x could clear round 2 but not clear round 3
• All the students who cleared Round 2 must have cleared Round 1 first
• This means, we can be sure that at least 0.4x students definitely cleared Round 1

However, we do not know the number of students who cleared round 3 and the number of students who could not clear round 1
Insufficient to answer.

Step 5: Analyze Both Statements Together (if needed)

1. From Statement 1, we inferred that 0.8x cleared round 1 and 0.2x didn ’t
2. From Statement 2, we know that 0.4x could clear round 2 but not clear round 3

Let there be a% of students who cleared round 1 and b% of students who cleared round 2

• a% ≥20%
• b% ≤100%
• From st-2 we know that b% - a% = 40%. Since we do not know the division of the rest 40% of the students, we will not have a unique value for the median. For example:
  • If a% = 20% and b% = 60%, median = 2
  • If a% = 60% and y% = 100%, median = 1

Insufficient to answer.
Answer: E

Given:

• {50, 60, 70, 80}
• {35, 40, 45, 50}
• {90, 110, 130, 150}

To Find:  Arrange the sets in increasing order of the (Mean / Standard deviation) ratio.

Approach:

1. Let ’s assume the mean of set-I be m and standard deviation be d.
2. We would try to express the terms of the other two sets in terms of set-I for establishing a relation between the standard deviations and the means of the set
3. Also, we will use the following properties of standard deviation to calculate the standard deviation of the other sets in terms of d:
   • If all the numbers of a set are multiplied or divided by the same constant x, the standard deviation is also multiplied or divided by |x| respectively.
   • If all the numbers of a set are increased or decreased by the same constant x, the standard deviation of the set does not change.

Working out:

1. Set-I: {50, 60, 70, 80}
   • Assuming the mean of set {50, 60, 70, 80} as m and standard deviation be d
   • Ratio = m/d ………(1)
2. Set-II: {35, 40, 45, 50}
   • {50, 60, 70, 80} →Dividing all the terms of the set by 2, results in the standard deviation of d/2 and mean of m/2 . The set becomes = {25, 30, 35, 40}
   • Adding 10 to all the numbers of the set {25, 30, 35, 40} does not change its standard deviation but increases its mean by 10. The set becomes = {35, 40, 45, 50}
   • Hence, the set {35, 40, 45, 50} has a standard deviation of d/2  and mean of  
   • 
3. Set-III: {90, 110, 130, 150}
   • ​​ {50, 60, 70, 80}→Multiplying all the terms of the set by 2, results in standard deviation of 2d and mean of 2m. The set becomes = { 100, 120, 140, 160}
   • Subtracting 10 from all the terms of the set {100, 120, 140, 160} does not change its standard deviation but decreases the mean by 10. The set becomes = {90, 110, 130, 150}
   • Hence, the set {90, 110, 130, 150} has a standard deviation of 2d and mean of 2m –10
   • 
4. Comparing (1), (2) and (3), we can arrange the sets in the ratio of mean to standard deviation as
   a. III

Answer : D

Step 1 &2: Understand Question and Draw Inference

• Set P →n integers

To Find: Standard Deviation (P)

Step 3 : Analyze Statement 1 independent

(1) The range of Set P is equal to zero

• Range = largest integer –smallest integer
  • 0 = largest integer –smallest integer
  • largest integer = smallest integer
• This tells us that:
  • all the integers of set P are equal OR
  • set P has only 1 element

In both the cases, the standard deviation is 0.
Sufficient to answer.

Step 4 : Analyze Statement 2 independent

(2) The mean of Set P is equal to the median of Set P

• As the mean of the set is equal to the median of the set, we can say that the set is symmetric about the mean and the median
• However, it does not tell us anything about the other terms of the set. So, we cannot say anything about the standard deviation of the set.

Insufficient to answer.

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step- 3, this step is not required.

Answer : A

Step 1 &2: Understand Question and Draw Inference

Given: A triangle ABC

• Let the angles of the triangle be x, y, z in increasing order of magnitude.
• We know x + y + z = 180^o . . . (1)

To find: Is z –x >90^o ?

Step 3 : Analyze Statement 1 independent

(1) The median angle of triangle ABC is 70^o .

But we do not know if x <10^o . So, Statement 1 alone is not sufficient.

Step 4 : Analyze Statement 2 independent

(2) The difference between the two larger angles of triangle ABC is 10^o

But we do not know if    So, Statement 2 alone is not sufficient.

Step 5: Analyze Both Statements Together (if needed)

• From Statement 1: y = 70
• From Statement 2: z –y = 10
• By combining these two equations, we ’ll get a unique value of z
• From (1), we ’ll get a unique value of x

Since we now know the values of x, y and z, we ’ll be able to answer the question on the range of the angles.
The two statements together are sufficient to answer the question.