Test: Measures of Central Tendency - 2 - Mechanical Engineering MCQ

Concept:

• Mean of x₁, x₂, ....,x_n 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).

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
⇒ 4^th + 5^th term
⇒ [(a – 2 + a – 0.5)/2]
⇒ [(2a – 2.5)/2]
⇒ a – 1.25
∴ The median of the data set is a – 1.25

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 ∑f_iX_i and ∑f_i 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

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

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

Concept:

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

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

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

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.

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.

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.

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

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

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

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

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.