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 business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

__________ is the positive square root of the mean of squared deviations from mean. 

  1. Mean Deviation

  2. Standard Deviation

  3. Quartile Deviation

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

Standard deviation is defined mathematically as the square root of the variance, which is the mean of the squared deviations from the arithmetic mean.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

If the A.M of a set of observations is $9$ and its G.M is $6$. Then the H.M of the set of observations is ____.

  1. $4$
  2. $6$
  3. $3$

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

For any set of positive numbers, the relationship between Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) is GM^2 = AM * HM. Given AM=9 and GM=6, then 36 = 9 * HM, so HM = 4.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

To find the median, it is necessary to arrange the data in _______.

  1. ascending order

  2. descending order

  3. ascending or descending order

  4. any of them
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To find the median, the data must be ordered by magnitude, which can be done in either ascending or descending order.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

Calculate standard deviation of the following data.

X $0-10$ $10-20$ $20-30$ $30-40$ $40-50$
f $10$ $8$ $15$ $8$ $4$
  1. $-12$
  2. $12$
  3. $12.36$
  4. $152.77$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Marks Mid-values(X) f dxi$=X-25/10$ $fdx _if$ $dx _i^2$
$0-10$ $5$ $10$ $-2$ $-20$ $40$
$10-20$ $15$ $8$ $-1$ $-8$ $8$
$20-30$ $25$ $15$ $0$ $0$ $0$
$30-40$ $35$ $8$ $1$ $8$ $8$
$40-50$ $45$ $4$ $2$ $8$ $16$
Total $45$ $-12$ $72$

$\sqrt{\displaystyle\frac{72}{45}-\frac{144}{45\times 45}}\times 10$
$=12.36$

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The standard deviation of $10, 16, 10, 16, 10, 10, 16, 16$ is?

  1. $4$
  2. $6$
  3. $3$
  4. $0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Assume $A=13;$ hence $d=A-13$

x d $d^2$
$10$ $-3$ $9$
$16$ $3$ $9$
$10$ $-3$ $9$
$16$ $3$ $9$
$10$ $-3$ $9$
$10$ $-3$ $9$
$16$ $-3$ $9$
$16$ $3$ $9$
$\displaystyle\sum d=0$ $\displaystyle\sum d^2=72$

Use following formula:
$\sqrt{\displaystyle\frac{\displaystyle\sum d^2}{n}-\left[\displaystyle\frac{\displaystyle\sum d}{n}\right]^2}$
$\sqrt{\displaystyle\frac{72}{8}-\left[\displaystyle\frac{0}{8}\right]^2}$
$\sqrt{9}$
$=3$.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

Calculate standard deviation of the following data.

X $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$
f $2$ $7$ $10$ $12$ $15$ $11$ $10$ $6$ $3$
  1. $3.95$
  2. $1.99$
  3. $14$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
X f fx X-Mean $f(X-x)^2$
$10$ $2$ $20$ $-4$ $32$
$11$ $7$ $77$ $-3$ $63$
$12$ $10$ $120$ $-2$ $40$
$13$ $12$ $156$ $-11$ $2$
$14$ $15$ $210$ $0$ $0$
$15$ $11$ $165$ $1$ $11$
$16$ $10$ $160$ $2$ $40$
$17$ $6$ $102$ $3$ $54$
$18$ $3$ $54$ $4$ $48$
Total $76$ $fX=1064$ $f(X-X)^2=300$

$^-{X}=\displaystyle\frac{1064}{76}=14$
$\sigma x=\sqrt{\displaystyle\frac{300}{76}}$
$=\sqrt{3.95}$
$=1.99$.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

Find the mean and standard deviation of the following observations: X $=2, 5, 7, 8, 13$.

  1. $7$ & $3.63$
  2. $3.63$ & $7$
  3. $7.63$ & $3$
  4. $3$ & $7.63$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle\frac{2+5+7+8+13}{5}=7$
$\sqrt{\displaystyle\frac{4+25+49+64+169}{5}}-49=3.63$.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

If the standard deviation of the numbers $2,3,a$ and $11$ is $3.5,$ then which of the following is true?

  1. $3a^2-32a+84=0$
  2. $3a^2-34a+91=0$
  3. $3a^2-23a+44=0$
  4. $3a^2-26a+55=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Numbers are $2,3,a$ and $11$
$N=4$
Standard deviation $\sigma=3.5$
Mean of numbers $(\overline{x})=\dfrac{2+3+11+a}{4}=\dfrac{16+a}{4}$

$\Rightarrow$  $\sigma^2=\dfrac{1}{N}\sum(x _i-\overline{x})^2$
$\Rightarrow$  $3.5\times 3.5\times 4\times  16=(8+a)^2+(4+a)^2+(16-3a)^2+(a-28)^2$
$\Rightarrow$  $784=64+a^2+16a+16+a^2+8a+256+9a^2-96a+a^2+784-56a$
$\Rightarrow$  $784=12a^2-128a+1120$
$\Rightarrow$  $196=3a^2-32a+280$

$\Rightarrow$  $3a^2-32a+84=0$

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The two observations $A$ & $B$ are given by $100, 101, ........149$ and $200, 201, .......,249$ with $V _{A}$ and $V _{B}$ are variances of $A$ and $B$ than $V _{A}$ is equal to :

  1. $V _{B}$
  2. $100\ V _{B}$
  3. $50\ V _{B}$
  4. $200\ V _{B}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sigma {x^2} = \frac{{\sum {{d^2}} }}{h}${here deviation are taken  from mean}

Since, A and B have consecutive integers therefore both have same standard deviation and hence variation 
${ V _{ A } }={ V _{ B } }\sum { { d^{ 2 } }\, \, is\, \, same } $

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

If the standard deviation of the values $2,4,6,8$ is $2.33$, then the standard deviation of the values $4,6,8,10$ is

  1. $0$
  2. $2.58$
  3. $4.66$

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

Given data values are $4,6,8,10$


Mean of the data is $\dfrac{4+6+8+10}{4}=\dfrac{28}{4}=7$

Therefore standard deviation is $\sqrt{\dfrac{(4-7)^2+(6-7)^2+(8-7)^2+(10-7)^2}{4-1}}=\sqrt{\dfrac{20}{3}}=2.58$

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

Suppose for $40$ observations, the variance is $50$. If all the observations are increased by $20$, the variance of these increased observation will be

  1. $20$
  2. $50$
  3. $30$

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

The variance for $40$ observations is $50$
Variance is independent of the number of observations. Therefore, variance for $60$ observations is $50$

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The variance of $5$ numbers is $10$. If each number is divided by $2$, then the variance of new numbers is

  1. $5.5$
  2. $2.5$
  3. $5$

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

Given that variance of $5$ number is $10$


Therefore, $Var(X)=10$

Given that each number is divided by $2$

Therefore, variance of new numbers is $Var(\dfrac X2)$

$Var(\dfrac X2)=\dfrac 14Var(X)=\dfrac 1 4(10)=2.5$

Hence, variance of new numbers is $2.5$