Tag: histograms and frequency distribution diagrams

Questions Related to histograms and frequency distribution diagrams

The third quartile is also called ________.

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. lower quartile

  2. median

  3. mode

  4. upper quartile

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Correct Option: D

The number of partition values in case of quartiles is ______.

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. $4$

  2. $3$

  3. $2$

  4. $1$

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Correct Option: B

From which average sum of deviations is zero?

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. Median

  2. Mean

  3. Mode

  4. None of them

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Correct Option: B

Mean of a set of values is based on _______.

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. all values

  2. $50$ percent values

  3. first and last value

  4. maximum & minimum value

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Correct Option: A

For the following data:

Weekly Income (in Rs.): 58 59 60 61 62 63 64 65 66
No. of Workers: 2 3 6 15 10 5 4 3 1


The value of $\displaystyle\frac{Q _1+Q _3}{2}$ is equal to

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. $Q _1$

  2. $Q _2$

  3. $Q _3$

  4. None of these

Show answer
Correct Option: D
Explanation:

The cumulative frequency of the table is a given below:

Weekly income ( in Rs.) No. of workers Cumulative frequency
58 2 2
59 3 5
60 6 11
61 15 26
62 10 36
63 5 41
64 4 45
65 3 48
66 1 49
N = 49


Lower Quartile: 
We have $\displaystyle\frac{N}{4} = \displaystyle\frac{49}{4} = 12.25$
The cumulative frequency just greater than $\displaystyle\frac{N}{4}$ is $26$ and the corresponding value of the variable is $61$
$\therefore\quad Q _1 = Rs.\space 61$

Middle Quartile (median): 
We have $\displaystyle\frac{N}{2} = \displaystyle\frac{49}{2} = 24.5$
The cumulative frequency just greater than $\displaystyle\frac{N}{2}$ is $26$ and the corresponding value of the variable is $61$
$\therefore\quad Q _2 = Rs.\space 61$

Upper Quartile:
We have $\displaystyle\frac{3N}{4} = \displaystyle\frac{3\times49}{4} = 36.75$
The cumulative frequency just greater than $\displaystyle\frac{3N}{4}$ is $41$ and the corresponding value of the variable is $63$
$\therefore\quad Q _3 = Rs.\space 63$

$\therefore\quad \displaystyle\frac{Q _1+Q _3}{2} = \displaystyle\frac{61+63}{2} = 62$

If 25% of the items are less than 10 and 25%  are more than 40, then the coefficient of quartile deviation is

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. 30

  2. 50

  3. 1.2

  4. 0.6

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Correct Option: A

The quartile deviation of the income of a certain person given in rupees for 12 months in a year:139, 150, 151,151,157,158,160,161,162,162,173,175

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. 4.5

  2. 5.5

  3. 6.2

  4. none of these

Show answer
Correct Option: B
Explanation:
S.NO. Income(Rs)
1 139
2 150
3 151
4 151
5 157
6 158
7 160
8 161
9 162
10 162
11 173
12 175

$N=12$
${ Q } _{ 1 }=$ Size of $\displaystyle\frac { N+1 }{ 4 } th=3.25th$ item
$=$ size of $3rd$ item $+0.25$(size of $4th$ item)$-$(size of $3rd$ item)
$=151+0.25(151-151)=$Rs.$151$
${ Q } _{ 3 }=$ Size of $\displaystyle\frac { 3(N+1) }{ 4 } th=9.75th$ item
$=$ size of $9th$ item $+0.75$(size of $10th$ item)$-$(size of $9th$ item)
$=162+0.75(162-162)=$Rs.$162$
$\therefore$ Quartile Deviation $\displaystyle=\frac { 1 }{ 2 } \left( { Q } _{ 3 }-{ Q } _{ 1 } \right) =\frac { 11 }{ 2 } =$Rs.$5.5$

The quartile deviation of daily wages (in Rs.) of 7 persons given below:
$12,7,15,10,17,17,25$ is

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. $4.5$

  2. $9$

  3. $5$

  4. $14.5$

Show answer
Correct Option: A
Explanation:

Writing the series in ascending order $7,10,12,15,17,19,25$
$\displaystyle{ Q } _{ 1 }=\frac { 15+7 }{ 2 } =11,{ Q } _{ 3 }=\frac { 25+15 }{ 2 } =20$
$\therefore$ Quartile Deviation
$\displaystyle=\frac { { Q } _{ 3 }-{ Q } _{ 1 } }{ 2 } =\frac { 20-11 }{ 2 } =4.5$

For the following data, the value of $Q _1 + Q _3 - Q _2$ is :

Age in years: 20 30 40 50 60 70 80
No. of members: 3 61 132 153 140 51 3
business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. $Q _1$

  2. $Q _2$

  3. $Q _3$

  4. $2Q _2$

Show answer
Correct Option: B
Explanation:

The cumulative frequency table is as given below

Age in years No. of students Cumulative frequency
20                   3 3
30 61 64
40 132 196
50 153 349
60 140 489
70 51 540
81 3 543
N $= $543

Lower Quartile: We have, $\displaystyle\frac{N}{4} = \displaystyle\frac{543}{4} = 135.75$
Cumulative frequency just greater than $N/4$ is $196$ and the corresponding value of the variable is $40$.
$\therefore\quad Q _1 = $ Lower Quartile = $40$ years
Middle Quartile (Medium):
We have, $\displaystyle\frac{N}{2} = \displaystyle\frac{543}{2} = 271.5$
Cumulative frequency just greater than $N/2$ is $349$ and the corresponding value of the variable is $50$.
$\therefore\quad Q _2 = $ Middle quartile = $50$ years
Upper Quartile:
We have $\displaystyle\frac{3N}{4} = \displaystyle\frac{3\times543}{4} = 407.25$
Cumulative frequency just greater than $407.25$ is $489$ and the corresponding value of the variable is $60$.
$\therefore\quad Q _3= $ Upper quartile  $=60$ years
$\therefore\quad Q _1 + Q _3 - Q _2 = 40+60-50 = 50 = Q _2$

If the lower and upper quartiles of a data are $4$ and $12$, then the quartile deviation is ____

business economics and quantitative methods measures of dispersion and skewness quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

  1. $2$

  2. $4$

  3. $6$

  4. $8$

Show answer
Correct Option: B
Explanation:

Quartile deviation is the half of the inter quartile deviation.

Inter quartile deviation is the difference between upper quartile and lower quartile.

Given that, lower quartile is $4$ and upper quartile is $12$

Therefore, Quartile deivation is $\dfrac{\text{upper quartile - lower quartile}}{2}$

$\implies \text{quartile deviation} =\dfrac{12-4}2=\dfrac 82 =4$