MCQ s' Measures of Central Tendency and Dispersion - Quantitative Aptitude for CA

Q1: A Professor has given assignment to students in a Statistics class. A student Jagan computes the arithmetic mean and standard deviation for a set of 100 observations as 50 and 5 respectively. Later on, Sonali points out to Jagan that he has made a mistake in taking one observation as 100 instead of 50. What would be the correct mean if the wrong observation is corrected?
(a) 50.5
(b) 49.9
(c) 49.5
(d) 50.1
Ans: (c)

Sol:
According to question, we have

n = 100, x̄ = 50 and δ = 5

⇒ Σx_incorrect = 100 × 50 = 5000

⇒ Σx_correct = 5000 – 100 + 50 = 4950

⇒ Correct mean = 4950/100

⇒ Correct mean = 49.5

Q2: Find the Harmonic mean of 2, 4, and 6.
(a) 3.00
(b) 3.30
(c) 3.75
(d) 4.00
Ans: (b)

Sol:
Harmonic mean of 2, 4 and 6 is given by
H.M. =
⇒ H.M. = 3.27 = 3.30 (approx)

Q3: The Geometric Mean of 3, 7, 11, 15, 24, 28, 30, 0 is
(a) 6
(b) 0
(c) 9
(d) 12
Ans: (b)

Sol:
We know,

Geometric Mean is given by

(a₁ × a₂ × ... × an)^1/n

= (3 × 7 × 11 × 15 × 24 × 28 × 30 × 0)^1/8

= (0)^1/8

= 0

Q4: If the AM and GM for two numbers are 6.50 and 6 respectively, then the two numbers are
(a) 6 and 7
(b) 9 and 4
(c) 10 and 3
(d) 8 and 5
Ans: (b)

Sol:
If a and b are two positive observations such that
(a+b)/2 = 6.5  ⇒ AM = 6.5 …(i)
and √(ab) = 6  ⇒ GM = 6 …(ii)
a = 36
∴ (a – b)² = (a + b)² – 4ab
(a – b)² = (13)² – 4 × 36
(a – b)² = 169 – 144
(a – b)² = 25
a – b = 5 …(iii)
Adding (i) and (iii) we get
2a = 18
a = 9
From (i), we get
b = 13 – 9 = 4
Therefore, the two numbers are 9 and 4.

Q5: Which of the following measure of central tendency depends on the position of the observation?
(a) Mean
(b) Median
(c) Mode
(d) Harmonic Mean
Ans: (b)

Sol:
 We know,
Median depends on the position of the observation whereas mean is the average of all the observations and Mode is the value that occur maximum number of times.

Q6: Which one of the following measures of central tendency is based on only fifty percent (50%) of the central values?
(a) Geometric Mean
(b) Harmonic Mean
(c) Median
(d) Mode
Ans: (b)

Sol:
We have,
Median is based on only fifty percent (50%) of the central values.

Q7: For open-end classification, which of the following is the best measure of central tendency?
(a) AM
(b) GM
(c) Median
(d) Mode
Ans: (c)

Sol:
We know,
For open-end classification, Median is the best measure of central tendency.

Q8: If two variable x and y are related as 2x – y = 3. If the median of x is 10, what is the median of y?
(a) 4
(b) 17
(c) 5
(d) 6
Ans: (b)

Sol:
Given, Two variable x and y are related as
2x – y = 3
⇒ 2x_med – y_med  = 3
⇒ 2(10) – y_med  = 3
⇒ y_med  = 17

Q9: Find out the mode from the following data:
100, 110, 125, 225, 325, 125, 90, 80, 455, 375, 125
(a) 325
(b) 110
(c) 455
(d) 125
Ans: (d)

Sol:
Given data: 100, 110, 125, 225, 325, 125, 90, 80, 455, 375, 125

Here, 125 appears thrice.

Therefore, the mode of the data is 125.

Q10: If the mode of the following data is 13, then the value of x in the dataset is
13, 8, 6, 3, 8, 13, 2x + 8, 8, 13, 3, 5, 7
(a) 5
(b) 6
(c) 7
(d) 8
Ans: (a)

Sol:

 Given dataset:

13, 8, 6, 3, 8, 13, 2x + 3, 13, 3, 5, 7

Since, 8 and 13 are repeating thrice and 13 is the given mode.
Therefore, the value of 2x + 3 should be equal to 13 to the mode be 13.
i.e., 2x + 3 = 13 or x = 5.

Q11: The Arithmetic Mean (A.M.) and Mode of the data are 32 and 26 respectively, then find the median of the data.
(a) 30
(b) 12
(c) 6
(d) 29
Ans: (a)

Sol:
 Given,  
Mean = 32 and Mode = 26  
We know,  
Mean – Mode = 3 (Mean – Median)  
⇒ 32 – 26 = 3 (32 – Median)  
⇒ 6/3 = 32 – Median  
⇒ Median = 32 – 2 = 30  

Q12: If the mode of a data is 18 and mean is 24, then median is
(a) 18
(b) 24
(c) 22
(d) 21
Ans: (c)

Sol:
We know,  
Mean – Mode = 3 (Mean – Median)  
24 – 18 = 3 (24 – Median)  
Median = 22

Q13:  For a moderately skewed distribution of marks in Statistics for a group of 200 students, the mean marks and median marks were found to be 55.60 and 52.40 respectively. What are the modal marks?
(a) 55.43
(b) 48
(c) 53.36
(d) 46
Ans: (d)

Sol:
Given,
Mean = 55.60 and Median = 52.40
We know,
Mode = 3 Median – 2 Mean
⇒ Mode = 3 (52.40) – 2 (55.60)
⇒ Mode = 46

Q14: If the difference between Mean and Mode is 69, then the difference between Mean and Median will be
(a) 63
(b) 31.5
(c) 23
(d) None
Ans: (c)

Sol: 
We know that,
Mean – Mode = 3 (Mean – Median)
⇒ 69 = 3 (Mean – Median)
⇒ 69/3 = Mean – Median
⇒ Mean – Median = 23

Q15: If x and y are related as 4x + 2y + 12 = 0 and mean deviation of x is 4.5, then the mean deviation of y is
(a) 9
(b) -9
(c) 11
(d) 4.5
Ans: (a)

Sol:
Given, x and y are related as 4x + 2y + 12 = 0
⇒ 2y = –4x – 12
⇒ y = –2x – 6
We know,
Mean deviation remains unchanged due to change in origin but changes in same ratio due to a change in   scale. 
Thus, mean deviation of y is given by
M . D_y = |–2| × M . D_x
⇒ M . D_y = 2 × 4.5 
⇒ M . D_y = 9

Q16: The quartiles of a variable are 45, 52 and 65 respectively. Its quartile deviation is
(a) 10
(b) 20
(c) 25
(d) 8.30
Ans: (a)

Sol:
We know,
Quartile deviation = (Q₃ − Q₁)/2

Q17: If the quartile deviation is 12 and the first quartile is 25, then the value of the third quartile is|
(a) 37
(b) 49
(c) 61
(d) 60
Ans: (b)

Sol:
Given: Quartile deviation is 12 and the first quartile is 25
We know,
Q.D = Q₃ − Q₁/2
⇒ 12 = Q₃ − 25/2
⇒ Q₃ − 25 = 24
⇒ Q₃ = 49

Q18: If the quartile deviation of x is 8 and 3x + 6y = 20, then the quartile deviation of y is
(a) 4
(b) 3
(c) 5
(d) None of these
Ans: (a)

Sol:
Given,
Relation between x and y is 3x + 6y = 20
⇒ 6y = –3x + 20

Therefore, quartile deviation of y is given by

Q D_y = 1/2 x8
Q D_y = 4

Q19: The coefficient of the range of the data: 7, 8, 4, 1, 9, 12, 18, 16, 94, 3, 5, –6 is
(a) 133.6 
(b) 163.3
(c) 166.3 
(d) 113.6
Ans: (d)

Sol:

Given data: 7, 8, 4, 1, 9, 12, 18, 16, 94, 3, 5, –6

Here, Largest observation = 94

Smallest observation = –6

Therefore, 
Coefficient of range =

Q20: In a dataset, 25 per cent of values are smaller than 30 and one-fourth of values are larger than 70, then the coefficient of quartile deviation is ______ %.
(a) 40
(b) 50
(c) 60
(d) 70
Ans: (a)

Sol:
Given, 25 percent of values are smaller than 30 and one-fourth of values are larger than 70
i.e., Q₁ = 30 and Q₃ = 70
Thus, coefficient of quartile deviation