Variance and Standard Deviation MCQ Quiz - Objective Question with Answer for Variance and Standard Deviation - Download Free PDF

Calculation:

Given,

Number of observations, n = 100

Original standard deviation,

Transformation applied to each observation:

New value,

The additive constant (5) does not affect the standard deviation, and scaling by 20 divides the standard deviation by 20, so the new standard deviation is,

∴ The new standard deviation is 0.5.

Hence, the correct answer is Option 2.

Calculation:

Explanation:

The Coefficient of Variation (C.V.) formula is given as:

Substituting values:

The Coefficient of Variation for Y is:

∴ C.V.(X)

Hence, the correct answer is Option 2.

Calculation:

Given,

The sum and sum of squares of the observations corresponding to length X in cm and weight Y in gm of 50 tropical tubers are given as:
= 200, = 250, = 900, = 1400 

The formula for variance is:

Where N = 50 is the number of observations.

Variance of X:

Substituting the given values:

Variance of Y:

Substituting the given values:

Conclusion:

Hence, the correct answer is Option 2.

Explanation:

Given:

Variance σ = 

= 

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)⇒ σ + M² = 

∴ Option (b) is correct.

Calculation

Let the two unknown items be x and y. Then,

Mean = 4 ⇒ 

⇒ x + y = 11...(i)

and variance = 5.2

⇒  - (mean)2 = 5.2

⇒ 41 + x2 + y2 = 5(5.2 + 16)

⇒ 41 + x2 + y2 = 106

⇒ x2 + y2 = 65... (ii)

On solving Eqs. (i) and (ii), we get

x = 4, y = 7 or x = 7, y = 4

sum = 28

Formula Used∶

• σ² = ∑(x_i – x)²/n
• Standard deviation is same when each element is increased by the same constant

Calculation:

Since each data increases by 10,

There will be no change in standard deviation because (x_i – x) remains same.

∴ The standard deviation of 10, 11, 12, 13 _____ 19 will be will be K.

Alternate Method 

Calculation:

Let the numbers be x_1, x₂, x₃, x₄.

The mean of four numbers x_1, x₂, x₃, x₄ = 37

The sum of four numbers x_1, x₂, x₃, x₄ = 37 × 4 = 148.

The mean of the smallest three numbers x_1, x₂, x₃ = 34

The sum of the smallest three numbers x_1, x₂, x₃ = 34 × 3 = 102.

∴ The value of the largest number x₄ = 148 – 102 = 46.

The Range (Difference between largest and smallest value) x₄ – x₁ = 15.

∴ Smallest number x₁ = 46 – 15 = 31.

Now,

The sum of x₂, x₃ = Total sum – (sum of smallest and largest number).

⇒ 148 – (46 + 31)

⇒ 148 – 77

⇒ 71

Now,

The mean of the Largest three numbers x₂, x₃, x₄ = (71 + 46)/3 = 117/3 = 39

⇒ n = total frequency

Mid value of 10 – 12 = (10 + 12)/2 = 11

Mid value of 12 – 14 = (12 + 14 )/2 = 13

Mid value of 14 – 16 = (14 + 16 )/2 = 15

Mid value of 16 – 18 = (16 + 18 )/2 = 17

Mid value of 18 – 20 = (18 + 20 )/2 = 19

⇒ Mean = 14.67

∴The mean marks of the given data are 14.67

Concept:

Standard deviation:

The standard deviation of the observation set  is given as follows:

Where  and  .

Calculations:

First, we will calculate the mean of the given observations.

Therefore, the numerator inside the square root term of the standard deviation formula will simply be equal to  .

Now we observe that  .

Therefore, the standard deviation is given as follows:

Therefore, the standard deviation of the given observations is 2.

Concept:

If every observation is multiplied by a number, then the mean is also multiplied by the same number 

If every observation is multiplied by a number, new variance = (number)² × old variance

Calculation:

Here, mean (x̅) = 4 and variance (σ²) = 2

Number of observation (n) = 10

New mean = 2 × (mean)

= 2 × 4

⇒ 8

New var = (number)2 × old variance

⇒ 2² × 2

⇒ 8

Hence, option (3) is correct.

Concept:

Calculation:

Given:

Variance = 81 and coefficient variation = 30%

We know that, 

As we know, 

⇒ Mean = 30

Calculation:

Given values 6.5, 3.4, 8.6, 2.9

By arranging the given in the ascending order, we get

2.9, 3.4, 6.5, 8.6

⇒ Median = (3.4 + 6.5)/2 = 9.9/2 = 4.95

Explanation:

The terms used in calculating standard deviation are deviations from the mean of the observations.
As each number/observation is increased by 2, the deviations from the mean remain the same.
Hence standard deviation remains the same.
On the other hand, median gives the middle term or the average of the two middle terms accordingly when the total number of terms is odd or even.
Thus, it has to increase by 2. 

Hence new median = 20 + 2 = 22 and SD = 4

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

n → odd term

⇒ Median = [(5 + 1)/2]^th term

⇒ Median = 3^th term

∴ Median of 1, 2, 3, 4 and 5 is 3.

Mean,