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Test: Measures of Central Tendency - 2 - Commerce MCQ


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15 Questions MCQ Test - Test: Measures of Central Tendency - 2

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Test: Measures of Central Tendency - 2 - Question 1

Let x be the mean of squares of first n natural numbers and y be the square of mean of first n natural numbers. If x/y = 55/42, then what is the value of n ?

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 1

Concept:

  • Mean of x1, x2, ....,xn is 
  • 1 + 2 + 3 + ....+ n = 
  • 12 + 22 + 32 + ....+ n2

Calculation:
Given: x is the mean of squares of first n natural numbers and y is the square of mean of first n natural numbers


⇒ 2 × 42(2n + 1) = 3 × 55(n + 1)
⇒ 168n + 84 = 165n + 165
⇒ 3n = 81
⇒ n = 27
∴ The correct option is (3).

Test: Measures of Central Tendency - 2 - Question 2

If a variable takes discrete values a + 4, a - 3.5, a - 2.5, a - 3, a - 2, a + 0.5, a + 5 and a - 0.5 where a > 0, then the median of the data set is

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 2

Given:
The given values =  a + 4, a – 3.5, a – 2.5, a – 3, a – 2, a + 0.5, a + 5 and a – 0.5

Concept used:
If n is odd
Median = [(n + 1)/2]th observations
If n is even
Median = [(n/2)th + (n/2 + 1)th observations]/2

Calculation:
a + 4, a – 3.5, a – 2.5, a – 3, a – 2, a + 0.5, a + 5 and a – 0.5
Arrange the data in ascending order
⇒ a – 3.5, a – 3, a – 2.5, a – 2, a – 0.5, a + 0.5, a + 4, a + 5
Here, the n is 8, which is even
Median =  [(n/2)th + (n/2 + 1)th observations]/2
⇒ [(8/2) + (8/2 + 1)/2] term
⇒ 4th + 5th term
⇒ [(a – 2 + a – 0.5)/2]
⇒ [(2a – 2.5)/2]
⇒ a – 1.25
∴ The median of the data set is a – 1.25

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Test: Measures of Central Tendency - 2 - Question 3

Find the mean of given data:

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 3

Formula used:
The mean of grouped data is given by,

Xi = mean of ith class
fi = frequency corresponding to ith class

Given:

Calculation:
Now, to calculate the mean of data will have to find ∑fiXi and ∑fi as below,

Then,
We know that, mean of grouped data is given by

= 1535/43
= 35.7
Hence, the mean of the grouped data is 35.7

Test: Measures of Central Tendency - 2 - Question 4

Find the median of the given set of numbers 2, 6, 6, 8, 4, 2, 7, 9

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 4

Concept:
Median:
The median is the middle number in a sorted- ascending or descending list of numbers.
Case 1: If the number of observations (n) is even

Case 2: If the number of observations (n) is odd

Calculation:
Given values 2, 6, 6, 8, 4, 2, 7, 9
Arrange the observations in ascending order:
2, 2, 4, 6, 6, 7, 8, 9
Here, n = 8 = even
As we know, If n is even then,


Hence Median = 6

Test: Measures of Central Tendency - 2 - Question 5

If the mode of the following data is 7, then the value of k in the data set 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k + 5, 9, 7, and 13 is:

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 5

Concept:
Mode is the value that occurs most often in the data set of values.

Calculation:
Given data values are 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k + 5, 9, 7, and 13
In the above data set, values 6, and 7 have occurred more times i.e., 3 times
But given that mode is 7.
So, 7 should occur more times than 6.
Hence the variable 2k + 5 must be 7
⇒ 2k + 5 = 7
⇒ 2k = 2
∴ k = 1

Test: Measures of Central Tendency - 2 - Question 6

Let the average of three numbers be 16. If two of the numbers are 8 and 10, what is the remaining number?

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 6

Concept:

Calculation:
Here n = 3. Let's say that the third number is x.

Test: Measures of Central Tendency - 2 - Question 7

What is the mean of first 99 natural numbers ?

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 7

Concept:
Suppose there are ‘n’ observations {x1,x2,x3,…,xn}

Sum of the first n natural numbers

Calculation:
To find:  Mean of the first 99 natural numbers
As we know, Sum of first n natural numbers 
Now, Mean

Test: Measures of Central Tendency - 2 - Question 8

The mean of six numbers is 47. If one number is excluded, their mean becomes 41. The excluded number is

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 8

Concept:
Mean = (Sum of observations) / (Total number of observations)

Calculation:
Let the six numbers be a, b, c, d, e, f
So, Mean 
⇒ a + b + c + d + e + f = 282          ....(1)
Let, the excluded number be a, 
So mean of remaining five numbers 
⇒ b + c + d + e + f = 205                ....(2)
∴ a + 205 = 282              (from (1) and (2))
⇒ a = 77
Hence, option (1) is correct.

Test: Measures of Central Tendency - 2 - Question 9

Find the median of the series of all the even terms from 4 to 296.

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 9

Concept:

  • Arithmetic progression is a sequence where any two consecutive terms differ by same difference.
  • Median is the middlemost data of set (example: 3, 4, 5, 6, 7 here median is 5.)

Important tip:

  • If the given sequence is arithmetic sequence, then median = (first term + last term)/2 = Mean.
     

Calculation:
The sequence is 4, 6, 8, 10 …. 296
Here common difference = 8 – 6 = 6 – 4 = 2     (which is constant)
Given sequence is an AP
∴ Median = (first term + last term)/2 = (4 + 296)/2 = 150.
Hence, option (3) is correct.

Test: Measures of Central Tendency - 2 - Question 10

Find the value of ‘n’ if the mean of the set of the numbers 8, 5, n, 10, 15, 21 is given as 11.

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 10

Concept:
The mean (or average) of a number of observations is the sum of the values of all the observations divided by the total number of observations. It is denoted by the symbol , read as ‘x bar’.

Calculation:
Given set of number is 8, 5, n, 10, 15, 21.

Test: Measures of Central Tendency - 2 - Question 11

Find the median of the data set: 6, 3, 8, 2, 9, 1?

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 11

Concept:
Median
Case 1: If number of observation (n) is even

Case 2: If number of observation (n) is odd

Calculation:
Arrange the observations in the ascending order are
1, 2, 3, 6, 8, 9
Here, n = 6 = even.
So, 3rd and 4th observation are 3 and 6

Test: Measures of Central Tendency - 2 - Question 12

Find the value ‘p + q’, if mean of set of numbers 3, 6, 7, 14, p, 34, 26, q, 12 is given as 22.

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 12

Concept:
The mean (or average) of a number of observations is the sum of the values of all the observations divided by the total number of observations. It is denoted by the symbol, read as ‘x bar’.

Calculation:
Given data 3, 6, 7, 14, p, 34, 26, q and 12.
Mean 


∴ p + q = 96

Test: Measures of Central Tendency - 2 - Question 13

The mean of 25 observations is 36. If the mean of the first 13 observations is 32 and that of the last 13 observations is 39 , the 13th observation is: 

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 13

Given:
The mean of 25 observations is 36
The mean of the first 13 observations is 32 and that of the last 13 observations is 39 

Concept used:
Mean = sum of all observation/total number of observation

Calculation:
The sum of all 25 observation = 25 × 36 = 900
Sum of first 13 observations = 13 × 32 = 416
Sum of last 13 observations = 13 × 39 = 507
∴ 13th term = (416 + 507) - 900 = 923 - 900 = 23

Test: Measures of Central Tendency - 2 - Question 14

What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 14

Given:
The given data is 5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Concept used:
The mode is the value that appears most frequently in a data set
At the time of finding Median
First, arrange the given data in the ascending order and then find the term

Formula used:
Mean = Sum of all the terms/Total number of terms
Median = {(n + 1)/2}th term when n is odd 
Median = 1/2[(n/2)th term + {(n/2) + 1}th] term when n is even
Range = Maximum value – Minimum value 

Calculation:
Arranging the given data in ascending order 
2, 3, 3, 4, 4, 4, 5, 6, 8, 9, 9, 10, 11, 15, 19
Here, Most frequent data is 4 so 
Mode = 4
Total terms in the given data, (n) = 15 (It is odd)
Median = {(n + 1)/2}th term when n is odd 
⇒ {(15 + 1)/2}th term 
⇒ (8)th term
⇒ 6 
Now, Range = Maximum value – Minimum value 
⇒ 19 – 2 = 17
Mean of Range, Mode and median = (Range + Mode + Median)/3
⇒ (17 + 4 + 6)/3 
⇒ 27/3 = 9
∴ The mean of the Range, Mode and Median is 9

Test: Measures of Central Tendency - 2 - Question 15

If mean and mode of some data are 4 & 10 respectively, its median will be:

Detailed Solution for Test: Measures of Central Tendency - 2 - Question 15

Concept:
Mean: The mean or average of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.
Mode: The mode is the value that appears most frequently in a data set.
Median: The median is a numeric value that separates the higher half of a set from the lower half. 

Relation b/w mean, mode and median:
Mode = 3(Median) - 2(Mean)

Calculation:
Given that,
mean of data = 4 and mode of  data = 10
We know that
Mode = 3(Median) - 2(Mean)
⇒ 10 = 3(median) - 2(4)
⇒ 3(median) = 18
⇒ median = 6
Hence, the median of data will be 6.

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