Measures of Central Tendency MCQ Quiz - Objective Question with Answer for Measures of Central Tendency - Download Free PDF

Calculation:

Given,

The sequence is 

Number of terms,

The sum of the first squares is 

Sum up to :

Sum up to :

Sum of the required terms:

The arithmetic mean is,

∴ The arithmetic mean is .

Hence, the correct answer is option 3.

Calculation:

Given,

Number of observations, n = 100

Original arithmetic mean,

Transformation applied to each observation:

New value,

The new arithmetic mean is,

∴ The new arithmetic mean is 2.25.

Hence, the correct answer is Option 1.

Concept:

• The problem involves the calculation of Mean, Mean Deviation about Mean (α), Mean Deviation about Median (β), and Variance (σ²) from a given frequency distribution.
• The Mean is the average of all values weighted by their frequencies.
• Mean deviation about mean (α) is the average of absolute deviations from the mean, weighted by frequencies.
• Mean deviation about median (β) is the average of absolute deviations from the median, weighted by frequencies.
• Variance (σ²) is the average of the squares of deviations from the mean, weighted by frequencies.

Calculation:

Given,

Total frequency sum = 19

Median of the distribution = 6

Let the unknown frequencies be f₁ and f₂.

Total frequency, N = 12 + f₁ + f₂

Sum of frequencies = 19 ⇒ 12 + f₁ + f₂ = 19 ⇒ f₁ + f₂ = 7

Median class is 6. Hence, the cumulative frequency before 6 is 5 + f₁ (must be less than 10)

Since 5 + f₁ 1 + f₂ > 9, find f₁ and f₂ accordingly:

⇒ 6 + f₁ = 10 ⇒ f₁ = 4, f₂ = 3

Now compute Σf_ix_i:

Σf_ix_i = 4×5 + 5×4 + 6×1 + 8×3 + 9×3 + 11×1 + 12×2 = 133

Mean, x̄ = Σf_ix_i / Σf_i = 133 / 19 ≈ 7

Mean deviation about mean (α):

α = Σf_i|x_i - x̄| / Σf_i = 48 / 19

⇒ 19α = 48     (Q ⇒ 3)

Mean deviation about median (β):

β = Σf_i|x_i - Median| / Σf_i = 49 / 19

⇒ 19β = 49     (R ⇒ 2)

Variance σ²:

σ² = Σf_ix_i² / Σf_i - (mean)² = 146 / 19

⇒ 19σ² = 146     (S ⇒ 1)

Now compute:

7f₁ + 9f₂ = 7×4 + 9×3 = 28 + 27 = 55     (P ⇒ 5)

∴ The correct matching is P → 5, Q → 3, R → 2, S → 1.

Hence, Option 3 is the correct answer.

Explanation:

Given:

⇒ 

⇒ Mean when x_n is replaced by k

⇒ New mean = 

= 

∴ Option (d) is correct

Explanation:

Mean  = 

⇒ 130 = 

⇒ 780 = 2n² + 3n + 1

⇒ 2n² + 3n – 779 =0

⇒ 2n² + 41n – 38n – 779 = 0

⇒ n(2n + 41) – 19(2n + 41) = 0 

⇒ (2n + 41) (n – 19) = 0

⇒ x = 19 ..... ( n = is not possible)

∴ Option (b) is correct.

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

Formula used:

The mean of grouped data is given by,

Where, 

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

= 

= 35.7

Hence, the mean of the grouped data is 35.7

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.

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:

Calculation:

We know that, 

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:

To find number of observations between 250 and 300.

first we have to draw a frequency distribution table from this data.

∴ The Number of observation in between 250-300 = 38 - 15 = 23.

Formula used:

If the total number of observations given is odd, then the formula to calculate the median is:

Median = {(n+1)/2}th term

If the total number of observations is even, then the median formula is:

Median  = [(n/2)th term + {(n/2)+1}th term]/2

where n is the number of observations.

Mode

The mode is the value that appears most frequently in a data set.

Given:

Calculation:

Since the frequency of x = 14 is 9 which is the maximum.

So, mode = 14

for frequency distribution,

So, the total number of observation = (1 + 4 + 7 + 5 + 9 + 3) = 29

So, 29 is ODD number, For odd number, the Median formula is, 

⇒ Median = 15^th term

⇒ Frequency of the 15^th term

According to the table, the value of 15^th is at x = 13

so the median = 13

Concept:

Suppose there are ‘n’ observations {}

Mean  

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:

.

Calculation:

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

∴ 

⇒ x + 18 = 48

⇒ x = 30.