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Range and mean deviation - class-X

Description: range and mean deviation
Number of Questions: 24
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Tags: maths statistics and probability
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Coefficient of deviation is calculated by the formula:

  1. $\cfrac { \bar { X } }{ \sigma } \times 100$

  2. $\cfrac { \bar { X } }{ \sigma }$

  3. $\cfrac { \sigma } {\bar { X }} \times 100$

  4. $\cfrac{ \sigma } { \bar { X }}$


Correct Option: C
Explanation:

It is a fundamental concept.
coefficient of deviation $=\cfrac{\sigma}{\bar{x}}\times 100$
where $\sigma$ and $\bar{x}$ are standard deviation and mean respectively.

For a symmetrical distribution lower quartitl is 20 and upper quartile is 40.The value of 50th percentile is

  1. 20

  2. 40

  3. 30

  4. none of these


Correct Option: C
Explanation:

First quartile also called the lower quartile or the 25th percentile(splits off the lowest 25% of data from the highest 75%)
Second quartile also called the median or the 50th percentile (cuts data set in half)
Third quartile  also called the upper quartile or the 75th percentile (splits off the highest 25% of data from the lowest 75%)
Since its a symmetrical distribution therefore the median will be 30

The range of the data 
25,18,20,22,16,6,17,12,30,32,10,19,8,11,20 is

  1. $20$

  2. $16$

  3. $18$

  4. $26$


Correct Option: D
Explanation:

The range of the data=Highest vale-lowest value

Highest value= 36
Lowest value=6
$\therefore$Range of the data=$32-6=24$

The difference between the maximum and the minimum observation in the data is

  1. class interval

  2. frequency

  3. cumulative frequency

  4. range


Correct Option: D
Explanation:

Range =maximum value-minimum value

Hence range is the difference between the maximum and the minimum  observation.

The formula for the coefficient of range is $\dfrac{\text{Range}}{a+b}$. Here, $a$ and $b$ denote:

  1. the mean and median of the data set

  2. the maximum and the minimum value of the data set

  3. the mean and mode value of the data set

  4. the minimum and mean value of the data set


Correct Option: B
Explanation:

Range is the difference between the maximum value and the minimum value of the data set.


Let $a$ be the maximum value of the data set and
$b$ be the minimum value of the data set

Therefore, $range = a-b$

Coefficient of range is the relative measure of the dispersion.

It is given by $\text{coefficient of range}=\dfrac{a-b}{a+b}=\dfrac{range}{a+b}$

The largest of $50$ measurements is $3.84$kg. If the range is $0.46$kg, find the smallest measurement.

  1. $3.38$kg.

  2. $2.38$kg.

  3. $6.38$kg.

  4. None of these


Correct Option: A
Explanation:

$\Rightarrow$  Here, $L=3.84$ and $R=0.46$

$\Rightarrow$  $R=L-S$
  $0.46=3.84-S$
  $S=3.84-0.46$
$\therefore$  $S=3.38\,kg$
$\therefore$   Smallest measurement is $3.38\,kg$

The ________ is the difference between the greatest and the least value of the variate.

  1. Range

  2. Data

  3. Average

  4. Variance


Correct Option: A
Explanation:

$Range$ $as$ $the$ $name$ $indicates$ $gives$ $us$ $all$ $the$ $area$ $available$ $under$ $light$ $and$ $hence$ $statement$ $is$ $true.$

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


Correct Option: B
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.

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


Correct Option: B
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.

For the measure of centre tendency, which the following is not true.

  1. $Z=3M-2\bar{x}$

  2. $2\bar{x}+Z=3M$

  3. $2\bar{x}-3M=-Z$

  4. $2\bar{x}=Z-3M$


Correct Option: D
Explanation:

We have, $Z=3M-2\bar{x}$
$\therefore 2\bar{x}+Z=3M$
or $2\bar{x}-3M=-Z$

$2\bar{x}=-Z+3M$
$\therefore 2\bar{x}=Z-3M $ is not true

Range of data $7, 8, 2, 1, 3, 13, 18$ is?

  1. $10$

  2. $15$

  3. $17$

  4. None of the above


Correct Option: C
Explanation:

$\begin{matrix} 7,8,2,1,3,13,18, \ range\, \, of\, \, data=\left( { \max  imum-\min  imum } \right)  \ =\left( { 18-1 } \right) =17\, \, \, \, \, Ans. \  \end{matrix}$

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

  1. $\dfrac{2}{3}$

  2. $\dfrac{1}{3}$

  3. $\dfrac{3}{5}$

  4. $\dfrac{1}{2}$


Correct Option: A
Explanation:
${ x _{ m } }=10{ x _{ 0 } }=2$
coefficient of range 
$\begin{array}{l} =\frac { { { x _{ m } }-{ x _{ 0 } } } }{ { { x _{ m } }t{ x _{ 0 } } } }  \\ =\frac { { 10-2 } }{ { 10+2 } } =\frac { 8 }{ { 12 } } =\frac { 2 }{ 3 }  \end{array}$

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. $18$


Correct Option: B
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 $48$

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 )$


Correct Option: B
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 
$D _8 = l _1 + \frac{\frac{8N}{10} - C}{f} \times (l _2 - l _1)$ 

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


Correct Option: A
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 $n\in N$.

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


Correct Option: B
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.=$\displaystyle =\frac{M.D.}{median}=\frac{13}{65}=0.2$

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

  1. 2.16

  2. 1.2

  3. 5

  4. none of these


Correct Option: B
Explanation:

Arranging the given data in ascending order
40,54,62,68,76,90
Here, $n=6 (even)$
$M= \dfrac{\text{value of }3^{rd}\text{observation}+\text{value of }4^{th}\text{observation}}{2}$
Median $M=\dfrac{62+68}{2}=65$

Mean deviation about median $M.D=\dfrac{|40-65|+|54-65|+|62-65|+|68-65|+|76-65|+|90-65|}{65}$

$=\dfrac{25+11+3+3+11+25}{65}=1.2$

The difference between the maximum and the minimum observations in the data is

  1. class interval

  2. frequency

  3. cumulative frequency

  4. range


Correct Option: D
Explanation:

The difference between maximum and the minimum observation in the data is range.

For example, suppose an experiment involves finding out the weight of lab rats and the values in grams are 320, 367, 423, 471 and 480. In this case, the range is simply computed as 480-320 = 160 grams.

The coefficient of range of a set of data is given to be $\dfrac18$. Then the ratio of the maximum value in the data to the minimum value is:

  1. $\dfrac81$

  2. $\dfrac98$

  3. $\dfrac97$

  4. $\dfrac87$


Correct Option: C
Explanation:

Coefficient of range of a set of data is given by $\dfrac{max-min}{max+min}$
$\dfrac{max-min}{max+min}=\dfrac{1}{8}$
$8max-8min=max+min$
$7max=9min$
$\dfrac{max}{min}=\dfrac{9}{7}$

The following are the wages of 8 workers in a factory. Find the range and coefficient of range. Wages are in dollars: 1400, 1450, 1520, 1380, 1485, 1495, 1575, 1440.

  1. $0.0231$

  2. $0.03112$

  3. $0.66$

  4. $0.02314$


Correct Option: C
Explanation:

The largest value of data is $x _m=1575$

The smallest value of data is $x _0=1380$
Range$=x _m-x _0=1575-1380=195$

Coefficient of data$=\dfrac{1575-1380}{1575+1380}=\dfrac{195}{2955}=0.0659\approx 0.66$

If the coefficient of range is $0.18$ and the largest value is $7.44$,then the smallest value is?

  1. $3.23$

  2. $4.15$

  3. $5.17$

  4. $5.14$


Correct Option: C
Explanation:

Coefficient of range$=\dfrac{x _m-x _0}{x _m+x _0}=\dfrac{7.44-x _0}{7.44+x _0}$

$0.18(7.44+x _0)=7.44-x _0$
$1.18x _0=7.44-7.44\times 0.18$
$1.18x _0=6.1008$
$x _0=5.17016\approx 5.17$

Find the coefficient of range for the data $43,24,38,56,22,39,45$

  1. $0124$

  2. $0.212$

  3. $0.236$

  4. $0.436$


Correct Option: D
Explanation:
Given data is $43, 24, 38, 56, 2, 39,45$.
The largest value of data is $x _m=56$
The smallest value of data is $x _0=22$
Coefficient of data $=\dfrac{56-22}{56+22}=\dfrac{34}{78}=0.4359\approx 0.436$
Hence, option D is correct.

The weight in Kg of 13 students in a class are $42.5,47.5,48.6,50.5,49,46.2,49.8,45.8,43.2,48,44.7,46.9,42.4$.Find the coefficient of range.

  1. $0.077$

  2. $0.213$

  3. $0.0803$

  4. $0.093$


Correct Option: C
Explanation:

The largest value of data is $x _m=49.8$

The smallest value of data is $x _0=42.4$
Coefficient of data$=\dfrac{49.8-42.4}{49.8+42.4}=\dfrac{7.4}{92.2}=0.08026\approx 0.0803$

Find the coefficient of range for the given data
$59,46,30,23,27,40,52,35,29$

  1. $0.46$

  2. $0.44$

  3. $0.56$

  4. $0.124$


Correct Option: B
Explanation:
Given data is $59, 46, 30, 23, 27, 40, 52, 35, 29$.
The largest value of data is $x _m=59$
The smallest value of data is $x _0=23$
Coefficient of data $=\dfrac{59-23}{59+23}=\dfrac{36}{82}=0.49\approx 0.44$
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