Mathematics · Quantitative Aptitude

Statistics and Dispersion

515 Questions

Statistics and dispersion involve the calculation of mean, standard deviation, variance, and coefficient of variation for data sets. These questions also cover probability distributions and cumulative frequency analysis. Such quantitative aptitude topics are heavily featured in banking and SSC examinations.

Standard deviationNormal distributionMean calculationCumulative frequencyCoefficient of variation

Statistics and Dispersion Questions

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

If the covariance between x and y is $30$, variance of x is $25$ and variance of y is $144$, then what is the correlation coefficient?

  1. $0.4$
  2. $0.5$
  3. $0.6$
  4. $0.7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given:-
$Cov(x,y)=30$
$V(x)=25$
$V(y)=144$
As we know formula of Corelation coefficient is$:-$
Let $r$ be Corelation coefficient of $x,y$ 
Then,
$r=\dfrac{Covariance(x,y)}{\sqrt{V(x)\times V(y)}}$
on solving$:-$
$\Rightarrow r=\dfrac{30}{\sqrt{25\times 144}}$
$\Rightarrow r=0.5$
Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

Calculate the coefficient of correlation between $x$ and $y$ for the data

x 1 2 3 4 5 6 7 8 9 10
y 3 10 5 1 2 9 4 8 7 6
  1. $0.12$
  2. $0.19$
  3. $0.22$
  4. $0.62$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$x\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { x=55 } $         $y\\ 3\\ 10\\ 5\\ 1\\ 2\\ 9\\ 4\\ 8\\ 7\\ 6\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { y=55 } $                 $X=x-\overline { x } \\ -4.5\\ -3.5\\ -2.5\\ -1.5\\ -0.5\\ \quad 0.5\\ \quad 1.5\\ \quad 2.5\\ \quad 3.5\\ \quad 4.5\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 0$                          $XY\\ \quad 11.25\\ -15.75\\ \quad 1.25\\ \quad 6.75\\ \quad 1.75\\ \quad 1.75\\ -1.25\\ \quad 6.25\\ \quad 1.75\\ \quad 2.25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 16\\ $                   ${ \quad X }^{ 2 }\\ 20.25\\ 12.25\\ 06.25\\ 02.25\\ 00.25\\ 00.25\\ 02.25\\ 06.25\\ 12.25\\ 20.25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ 82.50$

${ \quad Y }^{ 2 }\\ 06.25\\ 20.25\\ 00.25\\ 12.25\\ 12.25\\ 02.25\\ 06.25\\ 02.25\\ 00.25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 82.50$

Therefore, $\overline { x } =\cfrac { 55 }{ 10 } \\ \quad =5.5$
$\cfrac { \sum { y }  }{ 10 } =5.5$

Therefore, $r=\cfrac { \sum { XY }  }{ \sqrt { \sum { { X }^{ 2 }\sum { { Y }^{ 2 } }  }  }  } \\ =\cfrac { 16 }{ 82.5 } \\ =0.19$
Multiple choice economics measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

If $\sum\limits _{i = 1}^9 {\left( {{x _i} - 5} \right) = 9}$ and $\sum\limits _{i = 1}^9 {{{\left( {{x _i} - 5} \right)}^2}}  = 45$, then the standard deviation of the $9$ items ${x _1},{x _2},.....,{x _9}$ is

  1. $2$
  2. $3$
  3. $9$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
S.D of $xi-5$ is

$\sigma =\sqrt{\dfrac{\sum _{i=1}^{9}(xi-5)^2}{9}-\left [ \dfrac{\sum _{i=1}^{9}(xi-5)^2}{9} \right ]^2}$

$\sigma =\sqrt{5-1}=2$
Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

If the coefficient of variation and standard deviation of a distribution are 50% and 20 respectively, then its mean is

  1. 40

  2. 30

  3. 20

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $\sigma = 20$, coefficient of variation $=50$ %
We know coefficient of variation $=\cfrac{\sigma }{\bar{x}}\times 100=50$
$\Rightarrow \bar{x} = 2\times \sigma = 40$

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

The sum of squares of deviations for $10$ observations taken from mean $50$ is $250 $. Then Co-efficient of variation is

  1. $10\%$
  2. $40\%$
  3. $50\%$

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\sum(x-\overline{x})^2=250$, $\overline{x}=50$
$\Rightarrow$  Standard deviation $(\sigma)=\sqrt{\dfrac{250}{10}}=\sqrt{25}=5$
$\Rightarrow$  Coefficient of variation $=\sqrt{\dfrac{\sum(x-\overline{x})^2}{n}}$
                                             $=\dfrac{\sigma}{Mean}\times 100$

                                             $=\dfrac{5}{50}\times 100$

                                             $=10\%$
Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

The Coefficient of Variation is given by:

  1. $\dfrac{Mean}{\ Standard \ \ deviation } \times 100$
  2. $\dfrac{\ Standard \ \ deviation }{Mean}$
  3. $\dfrac{Standard \ \ deviation }{Mean }\times 100$
  4. $\dfrac{Mean}{Standard \ Deviation}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The coefficient of variation (CV) is a standardized measure of dispersion 

. It is defined as the ratio of the standard deviation to the mean.
$CV\quad =\quad \cfrac { \sigma  }{ Mean }\times100 $

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

If mean of a series is 40 and variance 1486, then coefficient of variation is 

  1. $0.9021$
  2. $0.9637$
  3. $0.8864$
  4. $0.9853$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If mean of the given dist. be $\bar{x}$ and S.D be $\sigma $
then given $\bar{x} = 40, \sigma^2 = 1486$
$\therefore$ Coefficient of variation $=\cfrac{\sigma}{\bar{x}}=\cfrac{\sqrt{1486}}{40}=.9637$

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

If the coefficient of variation and standard deviation of a distribution are 50% and 20 respectively, the its mean is

  1. 40

  2. 30

  3. 20

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know if a distribution having mean $\bar{x}$ and standard deviation $\sigma$
then coefficient of variation $=\cfrac{\sigma}{\bar{x}}\times 100$
$\therefore \cfrac{20}{\bar{x}}\times 100=50\Rightarrow \bar{x} = 40$
Hence required mean is $=40$

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

The sum of the squares of deviation of 10 observations from their mean 50 is 250, then coefficient of varition is

  1. 10%

  2. 40%

  3. 50%

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $\displaystyle \Sigma \left ( x _{i}-\overline{x} \right )^{2}=250$,$n=10,\overline{x}=50$

Now, $\sigma=\sqrt{\dfrac{1}{n}\Sigma \left ( x _{i}-\overline{x} \right )^{2}}$

$= \sqrt{\dfrac{1}{10}\times 250}=5$ 
Hence coefficient of variation $\displaystyle =\dfrac{\sigma }{\overline{x}}\times 100=\dfrac{5}{50}\times 100=10$%

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

The sum of the squares of deviation of 10 observations from their mean 50 is 250, then coefficient of variation is

  1. 10%

  2. 40%

  3. 50%

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given,   $\sum (x-\bar{x})^2 = 250, n = 10, \bar{x} =50$
Thus standard deviation $ = \sqrt{\cfrac{\sum (x-\bar{x})^2}{n}}=\sqrt{25}=5$
$\therefore$ Coefficient of variation $=\cfrac{\sigma}{\bar{x}}\times 100 =\cfrac{5}{50}\times 100$ % $= 10$%

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

The mean of a distribution is 4. If its coefficient of variation is 58%. Then the S.D. of the distribution is

  1. 2.23

  2. 3.23

  3. 2.32

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given,  mean $\bar{x} = 4,$ and coefficient of variation $=58$ %
If S.D of the given distribution is $\sigma$ then we know that,
Coefficient of variation $=\cfrac{\sigma}{\bar{x}}\times 100$ %
$\Rightarrow 58 = \cfrac{\sigma}{4}\times 100\Rightarrow \sigma = \cfrac{58\times 4}{100}=2.32$

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

For the given data, SD $= 10$, AM $= 20$ the coefficient of variation is ...........

  1. $47$
  2. $24$
  3. $44$
  4. $50$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Coefficient of variation is the ratio of standard deviation to the mean.


Given that $SD=10$ and $AM=20$

Therefore of coefficient of variation is $\dfrac{SD}{AM}\times100=\dfrac{10}{20}\times100=50\%$

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

The mean of a distribution is $14$ and standard deviation is $5$. What is the value of the coefficient of variation?

  1. $57.7\%$
  2. $45.7\%$
  3. $35.7\%$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Coefficient of variation is given by $CV = \dfrac{SD}{Mean}\times 100 $
$\Rightarrow \dfrac{5}{14}\times 100 = 35.7\%$

Multiple choice maths measures of dispersion coefficient of variance variance and standard deviation statistics and probability range and mean deviation

If the standard deviation of a set of scores is $1.2$ and their mean is $10$, then the coefficient of variation of the scores is

  1. $12$
  2. $0.12$
  3. $20$
  4. $120$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given : standard deviation$(\sigma)=1.2,$ mean$(\overline {X})=10$.

Coefficient of variation(C.V.) $=\dfrac{\sigma}{\overline {X}}\times 100=\dfrac{1.2}{10}\times 100=12$
$\therefore$ C.V. $=12$
Hence, option $A$ is correct.