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

The mean deviation from the median is _________ that measured from any other value.

  1. equal to

  2. less than

  3. greater than

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

The value of the mean deviation is minimum if the deviations are taken from the median. So, it is less than that measured from any other value.

A series drawback of the mean deviation is that it cannot be used in statistical inference.

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

The difference between the maximum and the minimum obervations in data is called the ____________.

  1. mean of the data

  2. range of the data

  3. mode of the data

  4. median of the data
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In arithmetic, the range of a set of data is the difference between the largest and smallest values.

So, difference between minimum and maximum values is called range.

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

Coefficient of range $5, 2, 3, 4, 6, 8, 10$ is?

  1. $\dfrac{2}{3}$
  2. $\dfrac{1}{3}$
  3. $\dfrac{3}{5}$
  4. $\dfrac{1}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
${ x _{ m } }=10{ x _{ 0 } }=2$
coefficient of range 
$\begin{array}{l} =\frac { { { x _{ m } }-{ x _{ 0 } } } }{ { { x _{ m } }t{ x _{ 0 } } } }  \\ =\frac { { 10-2 } }{ { 10+2 } } =\frac { 8 }{ { 12 } } =\frac { 2 }{ 3 }  \end{array}$
Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

The highest score of a certain data exceeds in lowest score by $16$ and coefficient of range is $\cfrac{1}{3}$. The sum of the highest score and the lowest score is

  1. $36$
  2. $48$
  3. $24$
  4. $18$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the highest score be $x _{m}$ and 

the lowest score be $x _{0}$
Given that highest score exceeds lowest score by $16$
$\implies x _m=x _0+16\implies x _m-x _0=16$ ————(1)

Coefficient of range is given by $\dfrac{x _m-x _0}{x _m+x _0}$

Given that coefficient of range is $\dfrac 13$

$\implies \dfrac{x _m-x _0}{x _m+x _0}=\dfrac 13$ ———(2)

Substitute (1) in (2) we get

$\dfrac{16}{x _m+x _0}=\dfrac 13$

$\implies x _m+x _0=16\times3=48$

Therefore sum of the highest score and lowest score is $48$

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

For a frequency distribution $8^{th}$ decile is computed by the formula

  1. $ \displaystyle D _{8}= l _{i}+\frac{\frac{N}{8}-C}{f}\times \left ( l _{2}-l _{1} \right )$
  2. $ \displaystyle l _{1}+\frac{\frac{8N}{10}-C}{f}\times \left ( l _{2}-l _{1} \right )$
  3. $ \displaystyle D _{8}= l _{1}+\frac{\frac{N}{10}-C}{f }\times \left ( l _{2}-l _{1} \right )$
  4. $ \displaystyle l _{1}+\frac{\frac{10N}{8}-C}{f }\left ( l _{2}-l _{1} \right )$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A decile is any of the nine values that divide the sorted data into ten equal parts, so that each part represents 1/10 of the sample or population.
For a continuous distribution, the formula for $r^{th}$ decile is given by $D _r = l _1 + \frac{\frac{rN}{10} - C}{f} \times (l _2 - l _1)$
Substituting r = 8, we have $D _8 = l _1 + \frac{\frac{8N}{10} - C}{f} \times (l _2 - l _1)$ 

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

If $n> 1, x> -1, x\neq 0$, then the statement $\left ( 1+x \right )^{n}> 1+nx$ is true for

  1. $ \;n\;\epsilon \;N$
  2. $\forall \;n\;> 1$
  3. $x> -1 \;and\; x\neq 0$

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

$P(1)$ is not true 


For $n=2,P\left( 2 \right) :{ \left( 1+x \right)  }^{ 2 }>1+2x$ is true if $x\neq 0$

Let $P(k):{ \left( 1+x \right)  }^{ k }>1+kx$ be two 

$\therefore{ \left( 1+x \right)  }^{ k+1 }=\left( 1+x \right) { \left( 1+x \right)  }^{ k}>\left( 1+x \right) \left( 1+kx \right)> 1+\left( k+1 \right) x+k{ x }^{ 2}>1+\left(k+1\right) x$

$\left( \because k{ x }^{ 2 }>0 \right) $
$\therefore$ By PMI
Given statement is true for every $n\in N$.

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

The coefficient of mean deviation from median of observations  $40, 62, 54, 90, 68, 76$  is

  1. $2.16$
  2. $0.2$
  3. $5$

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

Arrange the given observations in ascending order
$40,54,62,68,76,90$
Here, number of terms $n=6 (even) $
$\displaystyle \therefore $ Median (M) $\displaystyle =\frac{\left ( \frac{n}{2} \right )th:term+\left ( \frac{n}{2}+1 \right )th:term}{2}=\frac{62+68}{2}=65$

$\Sigma \left | x _{i}-M \right |=25+11+3+3+11+25=78$
Mean deviation from median $\displaystyle =\frac{\Sigma \left | x _{i}-M \right |}{n}=\frac{78}{6}=13 $
$\therefore $ Coefficient of M.D.=$\displaystyle =\frac{M.D.}{median}=\frac{13}{65}=0.2$