Variance and Standard Deviation MCQ Quiz - Objective Question with Answer for Variance and Standard Deviation - Download Free PDF
Last updated on Apr 11, 2025
Latest Variance and Standard Deviation MCQ Objective Questions
Variance and Standard Deviation Question 1:
Let \(\rm \Sigma_{i=1}^9x_i^2=885\) If M is the mean and σ is the standard deviation of x1, x2, x3.....x9 then what is the value of M2 + σ2?
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 1 Detailed Solution
Explanation:
Given:
\(\rm \Sigma_{i=1}^9x_i^2=885\)
Variance σ = \(\rm \Sigma_{i=1}^9x_i^2 - (mean)^2\)
= \(\frac{855}{9} -(M)^2\)
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)⇒ σ + M2 = \(\frac{855}{9} = 95\)
∴ Option (b) is correct.
Variance and Standard Deviation Question 2:
The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6 , then the product of other two is
Answer (Detailed Solution Below) 28
Variance and Standard Deviation Question 2 Detailed Solution
Calculation
Let the two unknown items be x and y. Then,
Mean = 4 ⇒ \(\frac{1+2+6+x+y}{5}=4\)
⇒ x + y = 11...(i)
and variance = 5.2
⇒ \(\frac{1^{2}+2^{2}+6^{2}+x^{2}+y^{2}}{5}\) - (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
Variance and Standard Deviation Question 3:
The mean of five observations is 4 and their variance is 5.2 . If three of these observations are 1, 2 and 6, then the other two observations are
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 3 Detailed Solution
Concept Used:
Mean: \(\bar{x} = \frac{\sum x_i}{n}\)
Variance: \(\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2\)
Calculation:
Given:
Mean of 5 observations = 4
Variance of 5 observations = 5.2
Three observations: 1, 2, 6
Let the other two observations be x and y.
Mean: \(\frac{1 + 2 + 6 + x + y}{5} = 4\)
⇒ \(9 + x + y = 20\)
⇒ \(x + y = 11\) ...(1)
Variance: \(\frac{1^2 + 2^2 + 6^2 + x^2 + y^2}{5} - 4^2 = 5.2\)
⇒ \(\frac{1 + 4 + 36 + x^2 + y^2}{5} - 16 = 5.2\)
⇒ \(\frac{41 + x^2 + y^2}{5} = 21.2\)
⇒ \(41 + x^2 + y^2 = 106\)
⇒ \(x^2 + y^2 = 65\) ...(2)
From (1), \(y = 11 - x\)
Substitute in (2):
⇒ \(x^2 + (11 - x)^2 = 65\)
⇒ \(x^2 + 121 - 22x + x^2 = 65\)
⇒ \(2x^2 - 22x + 56 = 0\)
⇒ \(x^2 - 11x + 28 = 0\)
⇒ \((x - 4)(x - 7) = 0\)
⇒ \(x = 4\) or \(x = 7\)
If \(x = 4\), \(y = 11 - 4 = 7\)
If \(x = 7\), \(y = 11 - 7 = 4\)
The other two observations are 4 and 7.
Hence option 1 is correct
Variance and Standard Deviation Question 4:
The coefficient of skewnes of a distribution is 0.32. If its standard deviation is 6.5 and mean is 29.6, then the mode of the distribution is given by
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 4 Detailed Solution
Concept Used:
Coefficient of Skewness = \(\frac{Mean - Mode}{Standard Deviation}\)
Calculation
Given:
Coefficient of Skewness = 0.32
Standard Deviation (SD) = 6.5
Mean = 29.6
Substitute the given values into the formula:
⇒ \( 0.32 = \frac{29.6 - Mode}{6.5} \)
⇒ \( 0.32 \times 6.5 = 29.6 - Mode \)
⇒ \( 2.08 = 29.6 - Mode \)
⇒ \( Mode = 29.6 - 2.08 \)
⇒ \( Mode = 27.52 \)
∴ The mode of the distribution is 27.52.
Hence option 2 is correct
Variance and Standard Deviation Question 5:
Let A = [aij] be a 2 × 2 matrix such that aij ∈ {0, 1} for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is:
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 5 Detailed Solution
\(|A|=\left|\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right|\)
= a11 a22 – a21
a12 = {–1, 0, 1}
\(\begin{array}{c|c|c|c} \mathrm{x} & \mathrm{P}_{\mathrm{i}} & \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}} & \mathrm{P}_{1} \mathrm{X}_{\mathrm{i}}{ }^{2} \\ -1 & \frac{3}{16} & -\frac{3}{16} & \frac{3}{16} \\ 0 & \frac{10}{16} & 0 & 0 \\ 1 & \frac{3}{16} & \frac{3}{16} & \frac{3}{16} \\ \hline & & \sum \mathrm{P}_{\mathrm{i}} X_{\mathrm{i}}=0 & \sum \mathrm{P}_{\mathrm{i}} X_{\mathrm{i}}{ }^{2}=\frac{3}{8} \end{array}\)
\(\therefore \operatorname{var}(\mathrm{x})=\sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}^{2}-\left(\sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}\right)^{2} \)
\(=\frac{3}{8}-0=\frac{3}{8} \)
Top Variance and Standard Deviation MCQ Objective Questions
If the standard deviation of 0, 1, 2, 3 ______ 9 is K, then the standard deviation of 10, 11, 12, 13 _____ 19 will be:
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 6 Detailed Solution
Download Solution PDFFormula Used∶
- σ2 = ∑(xi – 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 (xi – x) remains same.
∴ The standard deviation of 10, 11, 12, 13 _____ 19 will be will be K.
Alternate Method
The mean of four numbers is 37. The mean of the smallest three of them is 34. If the range of the data is 15, what is the mean of the largest three?
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 7 Detailed Solution
Download Solution PDFCalculation:
Let the numbers be x1, x2, x3, x4.
The mean of four numbers x1, x2, x3, x4 = 37
The sum of four numbers x1, x2, x3, x4 = 37 × 4 = 148.
The mean of the smallest three numbers x1, x2, x3 = 34
The sum of the smallest three numbers x1, 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 = 39What is the standard deviation of the observations
\(-\sqrt{6}, -\sqrt{5},- \sqrt{4}, -1, 1, \sqrt{4}, \sqrt{5}, \sqrt{6} \ ?\)
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 8 Detailed Solution
Download Solution PDFConcept:
Standard deviation:
The standard deviation of the observation set \(\rm \{x_i,i=1,2,3,\cdots\}\) is given as follows:
\(\rm \sigma=\sqrt{\dfrac{\sum\left(x_i-\mu\right)^2}{N}}\)
Where \(\rm N=\mbox{size of the observation set}\) and \(\rm \mu=\mbox{mean of the observations}\) .
Calculations:
First, we will calculate the mean of the given observations.
\(\begin{align*} \mu &= \dfrac{-\sqrt6-\sqrt5-\sqrt4-1+1+\sqrt4+\sqrt5+\sqrt6}{8}= 0 \end{align*}\)
Therefore, the numerator inside the square root term of the standard deviation formula will simply be equal to \(\rm (x_i-\mu)^2=x_i^2\) .
Now we observe that \(\rm N=8\) .
Therefore, the standard deviation is given as follows:
\(\begin{align*} \sigma &= \sqrt{\dfrac{\left(-\sqrt6\right)^2+\left(-\sqrt5\right)^2+\left(-\sqrt4\right)^2+\left(-1\right)^2+\left(1\right)^2+\left(\sqrt4\right)^2+\left(\sqrt5\right)^2+\left(\sqrt6\right)^2}{8}}\\ &= \sqrt{\dfrac{32}{8}}\\ &= \sqrt4\\ &= 2 \end{align*}\)
Therefore, the standard deviation of the given observations is 2.
The data given below shows the marks obtained by various students.
Marks |
Number of students |
10 – 12 |
6 |
12 – 14 |
8 |
14 – 16 |
5 |
16 – 18 |
7 |
18 - 20 |
4 |
What is the mean marks (Correct up to two decimal places) of given data?
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 9 Detailed Solution
Download Solution PDF\(\bar x\left( {mean} \right)\; = \;\frac{{\sum fx}}{n}\)
⇒ n = total frequency
\(\sum fx = Sum\;of\;the\;product\;of\;mid - interval\;values\;and\;their\;corresponding\;frequency\;\;\)
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
\(\Rightarrow \;Mean\; = \;\frac{{11\; \times \;6\; + \;13\; \times \;8\; + \;15\; \times \;5\; + \;17\; \times \;7\; + \;19\; \times \;4}}{{6\; + \;8\; + \;5\; + \;7\; + \;4}} = \;\frac{{440}}{{30}}\)
⇒ Mean = 14.67
∴The mean marks of the given data are 14.67
The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively.
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 10 Detailed Solution
Download Solution PDFConcept:
If every observation is multiplied by a number, then the mean is also multiplied by the same number
\(\text { Variance }=\sigma^{2}=\frac{\sum\left(x_{i}-\bar{x}\right)^{2}}{\text {} n}\)
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.
If the variance of a distribution is 81 and the coefficient variation is 30%, find mean.
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 11 Detailed Solution
Download Solution PDFConcept:
\({\rm{Coefficient\;of\;variation}} = {\rm{}}\frac{{{\rm{standard\;deviaiton}}}}{{{\rm{Mean}}}} \times 100\)
\(\rm S.D = \sqrt {Variance}\)
Calculation:
Given:
Variance = 81 and coefficient variation = 30%
We know that, \(\rm S.D = \sqrt {Variance}\)
\(\rm S.D = \sqrt {Variance} = \sqrt {81} = 9\)
As we know, \({\rm{Coefficient\;of\;variation}} = {\rm{}}\frac{{{\rm{standard\;deviaiton}}}}{{{\rm{Mean}}}} \times 100\)
\(\Rightarrow {\rm{Coefficient\;of\;variation}} = {\rm{}}\frac{9}{\rm Mean} \times 100 \)
\(\Rightarrow \rm 30 = \frac{9}{{Mean}} \times 100\)
⇒ Mean = 30
Find the median of the data set: 6.5, 3.4, 8.6, 2.9?
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 12 Detailed Solution
Download Solution PDFCalculation:
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 = 4.95The median and SD of a distributed are 20 and 4 respectively. If each item is increased by 2, the new median and SD are
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 13 Detailed Solution
Download Solution PDFExplanation:
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
The variance of the data 2, 4, 6, 8, 10 is
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 14 Detailed Solution
Download Solution PDFMean, \(\bar x = \frac{{2 + 4 + 6 + 8 + 10}}{5} = \frac{{30}}{5} = 6\)
\(\therefore Variance = \frac{1}{n}\sum {\left( {{x_i} - \bar x} \right)^2}\;\)
\(= \frac{1}{5}\left\{ {{{\left( {2 - 6} \right)}^2} + {{\left( {4 - 6} \right)}^2} + {{\left( {6 - 6} \right)}^2} + {{\left( {8 - 6} \right)}^2} + {{\left( {10 - 6} \right)}^2}} \right\}\)
\(= \frac{1}{5}\left\{ {16 + 4 + 0 + 4 + 16} \right\} = \frac{1}{5} \times 40 = 8\)
Consider the following grouped frequency distribution:
X |
F |
0-10 |
8 |
10-20 |
12 |
20-30 |
10 |
30-40 |
P |
40-50 |
9 |
If the mean of the above data is 25.2, then what is the value of p?
Answer (Detailed Solution Below)
Variance and Standard Deviation Question 15 Detailed Solution
Download Solution PDF
X |
F |
Z |
F × Z |
0-10 |
8 |
5 |
40 |
10-20 |
12 |
15 |
180 |
20-30 |
10 |
25 |
250 |
30-40 |
P |
35 |
35P |
40-50 |
9 |
45 |
405 |
Z is the midpoint of range of x for every data
sum of all F × Z = 875 + 35P
mean of above data is 25.2 & sum of all F = 8 + 12 + 10 + P + 9 = 39 + P
mean \(= \frac{{875 + 35P}}{{39 + P}} = 25.2\)
⇒ 9.8 P = 107.8
⇒ P = 11