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 mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

Harmonic mean is a part of _______________.

  1. Positional average

  2. Mathematical average

  3. Both a & b

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Mathematical average refers to all such average where a figure is taken out through mathematical methods from the a given series that represents the whole series. Harmonic mean is a mathematical tool which is used to calculate average of a certain series. Therefore, it is a part of mathematical average. 

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

The value of Spearman's rank coefficient lies between 

  1. $2$ and $3$
  2. $1$ and $2$
  3. $0$ and $1$
  4. $-1$ and $1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Spearman's rank cofficient : $R=1-\dfrac { 6\sum { { d } _{ i }^{ 2 } }  }{ n({ n }^{ 2 }-1) } $

Its values lies between $-1$ and $1$
So option $D$ is correct.

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

FInd the rank correlation from the following data:

S. No. 1 2 3 4 5 6 7 8 9 10
Rank Differences -2 -4 -1 3 2 0 -2 3 3 -2

  1. 0.64

  2. 0.50

  3. 0.45

  4. 0.34
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Rank Difference $(d)$ | $d^2$ | | --- | --- | --- | | 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. | -2 -4 -1 3 2 0 -2 3 3 -2 | 4 16 1 9 4 0 4 9 9 4 |

 $\sum d^2=60,\quad n=10$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}$

$r=1-\cfrac{6(60)}{10(10^2-1)}$

$r=1-\cfrac{360}{990}$

$r=0.6363....\approx 0.64$

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

Find the spearman's rank coefficient of correlation from the following data:

X 48 33 40 9 16 16 65 25 16 57
Y 13 13 24 6 15 4 20 9 6 19
  1. $0.76$
  2. $0.52$
  3. $0.61$
  4. $0.85$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Rank | $Y$ | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 48 33 40 9 16 16 65 25 16 57 | 3 5 4 10 7 7 1 6 7 2 | 13 13 24 6 15 4 20 9 6 19 | 5 5 1 8 4 10 2 7 8 3 | 2 0 3 2 3 3 1 1 1 1 | 4 0 9 4 9 9 1 1 1 1 |

$n=10,\quad \sum d^2=39$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 39}{10(10^2-1)}=1-\cfrac{234}{990}=0.76$

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

Find the rank correlation coefficient between the heights of fathers and sons from the following data:

Heights of fathers in inches  65 66 67 67 68 69 70 72
Height of sons in inches 67 68 65 68 72 72 69 71
  1. $0.67$
  2. $0.58$
  3. $0.42$
  4. $0.92$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Rank | Height(Son) | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 65 66 67 67 68 69 70 72 | 8 7 5 5 4 3 2 1 | 67 68 65 68 72 72 69 71 | 7 5 8 5 1 1 4 3 | 1 2 3 0 3 2 2 2   | 1 4 9 0 9 4 4 4 |

$n=08,\quad \sum d^2=35$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 35}{8(8^2-1)}=1-\cfrac{210}{504}=0.58$

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

Based on the following data, find coefficient of rank correlation.

x 43 96 74 38 35 43 22 56 35 80
y 30 94 84 13 30 18 30 41 48 95
  1. $0.3456$
  2. $0.5621$
  3. $0.6303$
  4. $0.7326$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the ranks of students obtained in Physics be $x$ and the ranks of students obtained in Chemistry be $y$.

 $X$  $Y$  Rank $X$       $(x)$  Rank $Y$      $(y)$  $d=x-y$  $d^2$
 $43$ $30$   $5.5$  $7$ $-1.5$   $2.25$
 $96$  $94$  $1$ $2$   $-1$  $1$
 $74$  $84$  $3$  $3$  $0$  $0$
 $38$  $13$  $7$  $10$  $-3$  $9$
 $35$  $30$  $8.5$  $7$  $1.5$  $2.25$
 $43$  $18$  $5.5$  $9$ $-3.5$   $12.25$
 $22$  $30$  $10$ $7$   $3$  $9$
 $56$  $41$  $4$  $5$  $-1$  $1$
 $35$  $48$  $8.5$  $4$ $4.5$   $20.25$
 $80$  $95$  $2$  $1$  $1$  $1$
       $\sum$  $0$  $58$


In the $X$ series $43$ has repeated twice and given ranks $5.5$ instead of $5$ and $6$. 

For this the correction factor is $\dfrac{2(4-1)}{12}=\dfrac{1}{2}$.

Also $35$ has repeated twice and given ranks $8.5$ instead of $8$ and $9$. For this the correction factor is $\dfrac{2(4-1)}{12}=\dfrac{1}{2}$.

In the $Y$ series $30$ has repeated thrice and given ranks $7$ instead of $6,7,8$. 

For this the correction factor is $\dfrac{3(9-1)}{12}=2$.

So, the total correction factors $C.F=\dfrac{1}{2}+\dfrac{1}{2}+2=3$

The rank correlation coefficient is given by,

$r=1-\dfrac{6(\sum d^2-C.F)}{n(n^2-1)}$
$=1-\dfrac{6(58+3)}{10(100-1)}$
$=1-\dfrac{276}{10 \times 99}$
$=1-\dfrac{366}{990}$
$=1-0.3696$
$=0.6303$
Therefore the rank correlation coefficient is $0.6303$.

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

If the correlation coefficient between $x$ and $y$ is $0.6$, covariance is $27$ and variance of $y$ is $25$, then what is the variance of $x$?

  1. $9/5$
  2. $81/25$
  3. $9$
  4. $81$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$Correlation \:coefficient = \dfrac{cov (x,y)}{std\: deviation (x) \times std\: deviation (y)}$

Let std deviation of $x$ be $x$.

We have $correlation \:coefficient=0.6, cov(x,y)=27, std\:deviation(y)=\sqrt{25}=5$

Substituting respective values

$0.6=\dfrac{27}{5\times x}$

$\Rightarrow x=9$

So variance of $x$ is $9^2=81$

Multiple choice introduction to ratio and percentages comparing quantities maths

There are four numbers whose product is $9261000$ and each of these four numbers is formed by $3$ distinct prime numbers. The average of all the four numbers is:

  1. $61.75$
  2. $67.25$
  3. $82.33$
  4. $Data\ insufficient$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product of the four numbers is 9261000. If we assume the numbers are equal or near each other, the cube root of 9261000 is 210. However, the question states each number is formed by 3 distinct prime numbers. This implies a specific set of numbers. Given the options, 61.75 is the only plausible arithmetic mean.

Multiple choice business mathematics and statistics random variable and mathematical expectation discrete and continuous data random variable random variable and its types

The mean of discrete obervations $y _1, y _2$ , ................ , $y _n$ is given by

  1. $\displaystyle \frac{\sum _{i=1}^{n} y _i}{n}$
  2. $\displaystyle \frac{\sum _{i=1}^{n} y _i}{\sum _{i=1}^{n}i}$
  3. $\displaystyle \frac{\sum _{i=1}^{n} y _i f _i}{n}$
  4. $\displaystyle \frac{\sum _{i=1}^{n} y _i f _i}{\sum _{i=1}^{n}y _if _i}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mean of terms = $\dfrac{Sum}{number}$


Therefore, Mean = $\displaystyle \dfrac{\sum _{i=1}^{n} y _i}{n}$

Multiple choice maths measure of central tendency assumed mean method assumed mean method of finding mean mean and median

Sum of squares of deviation of $10$ observations measured from $5$ is $17$ and sum of squares of observations is $170$ then mean of observation is

  1. $40.3$
  2. $4.5$
  3. $4$
  4. $4.03$
  5. $4.3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$x _1,x _2,x _3,x _4...x _10\x _1^2+x _2^2+x _3^2+x _4^2+....x _10^2=17 (Given) \rightarrow (i)$

$(x _1-5)^2+(x _2-5)^2+(x _3-5)^2+....+(x _10-5)^2=17(Given)\rightarrow (i)$
$Mean (M)=\cfrac{x _1+x-2+x _3+....x _10}{10}\ \Rightarrow x _1+x _2+x _3+....+x _10=10M\rightarrow(iii)$
From equation $(ii)$ we get
$x _1^2+25-10x _1+x _2^2+25-10x _2+x _3^2+25-10x _3+...+x _10^2+25-10x _10=17\ \Rightarrow (x _1^2+x _2^2+....x _10^2)-10(x _1+x _2+x _3+....+x _10)+25\times10=17\ \Rightarrow 170-10(10M)+250=17\ \Rightarrow420-100M=17\ \Rightarrow100M=403\ \Rightarrow M=\cfrac{403}{100}=4.03$

Multiple choice maths measure of central tendency assumed mean method assumed mean method of finding mean mean and median

The sum of the deviations of a set of values $x 1, x _2$, ...... $x _n$ measured from $50$ is $-10$ and the sum of deviations of the values from $46$ is $70$. The mean is __________.

  1. $49$
  2. $49.5$
  3. $49.75$
  4. $50$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sum of deviations from $50$ is $-10$


$\Rightarrow \sum(xi - 50) = -10$
     $\sum x _i - 50\sum1 = -10$
     $\sum x _i - 50n = -10$

$\therefore y-50n=-10.....(1)$

Sum of deviations from $46$ is $70$

$\Rightarrow \sum(x _i - 46) = 70$
     $\sum x _i - 46\sum1 = 70$
     $\sum x _i - 46n = 70$

$\therefore y-46n=70.....(2)$


Solving $(1)$ and $(2)$, we get
$4n = 80$ i.e. $n=20$

Putting value of $n$ in $(1)$, we get
$y=990$

Mean $= \dfrac{\sum x _i}{n} = \dfrac{y}{n} = \dfrac{990}{20} = 49.5$

Multiple choice computer and ms office mathematical methods for economics economics

Find the co-efficient of mean deviation from Median from the following data.Median deviation =4.2,Median =21

  1. 0.3

  2. 0.20

  3. 0.25

  4. 0.22
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The coefficient of mean deviation from the median is calculated as (Mean Deviation) / (Median). Here, 4.2 / 21 = 0.20.