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 quartile deviation or semi-interquartile range interquartile range histograms and frequency distribution diagrams

The first quartile divides a frequency distribution in the ratio _____.

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

The first quartile (Q1) represents the 25th percentile, meaning 25% of the data is below it and 75% is above it. The ratio of 25:75 simplifies to 1:3.

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

In a frequency distribution with open ends, one cannot find out ______.

  1. mean

  2. median

  3. mode

  4. all of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The arithmetic mean requires the exact values of all observations. In open-ended distributions, the exact values of the first or last classes are unknown, making the mean impossible to calculate.

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

Mean of a set of values is based on _______.

  1. all values
  2. $50$ percent values

  3. first and last value

  4. maximum & minimum value
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The arithmetic mean is calculated by summing all values in a dataset and dividing by the count, making it dependent on every single observation.

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

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

  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $\dfrac{\text{upper quartile - lower quartile}}{2}$

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

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

Find the quartile deviation of $4, 6, 9, 12, 18, 20, 23, 27, 34, 48$ and $53$

  1. $12.5$
  2. $13$
  3. $12$
  4. $11.5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given series of numbers is already arranged in ascending order.
Thus median is the centre value $ = 20 $
First Quartile $ {Q} _{1} $ will be the mid value of the observations to the left of $ 20 $ which is $ 9 $. 
Similarly, Third Quartile $ {Q} _{3} $ will be the mid value of the observations to the right of $ 20 $ which is $ 34 $.
And Quartile deviation will be $ \dfrac {{Q} _{3} - {Q} _{1}}{2} = \dfrac {34-9}{2} = \dfrac {25}{2} = 12.5 $.

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

For a series the value of mean deviation is 15.The most likely value of its quartile deviation is 

  1. 12.5

  2. 11.6

  3. 13

  4. 9.7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

use:  M.D.$=\displaystyle \frac{4}{4}\sigma ,Q.D=\frac{2}{3}\sigma $ $\displaystyle \Rightarrow \frac{M.D.}{Q.D}=\frac{6}{5}\Rightarrow Q.D.=\frac{5}{6}\left ( M.D. \right )$


$MD=\frac { 4 }{ 5 } \times \sigma \quad ,\quad QD=\frac { 2 }{ 3 } \times \sigma \quad ,\quad \ \frac { MD }{ QD } =\frac { 6 }{ 5 } ,\quad QD\quad =\quad \frac { 5\times 15 }{ 6 } =12.5$

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

If the lower and upper quartiles of a data are 4 and
12 then the quartile deviation is_____

  1. 2

  2. 4

  3. 6

  4. 8
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Lower Quartile $ {Q} _{1} = 4 $

Upper Quartile $ {Q} _{3} = 12 $

And Quartile deviation will be $ \frac {{Q} _{3} - {Q} _{1}}{2} = \frac {12-4}{2} = \frac {8}{2} = 4 $

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

Value of each sample observation is decreased by $25$, the range of the sample observation will ____________.

  1. increase by $25$

  2. remain same
  3. decrease by $25$
  4. increase by $2$ times
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Range is the difference between the maximum and minimum values. If every value is decreased by 25, both the maximum and minimum decrease by 25, so the difference (max - min) remains unchanged.

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

The range from the following observations is 
$10, 18, 20, 28, 15, 17, 22, 25, 29, 32, 34$

  1. $24$
  2. $20$
  3. $30$
  4. $42$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Range is defined as the difference between the highest(or largest ) and lowest(or smallest) observed value in a series. It is the most simple and commonly understandable measures of dispersion.

Range = H - L 
In the given series, H= 34 and L+ 10 
Range = 34-10 = 24.

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

From the following detail find out coefficient of Quartile Deviation.
$3=64,Q1=50$

  1. $12.11$%
  2. $12.28$%
  3. $13.50$%
  4. $12.08$%
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The coefficient of quartile deviation is calculated as (Q3 - Q1) / (Q3 + Q1). Given Q3 = 64 and Q1 = 50, the calculation is (64 - 50) / (64 + 50) = 14 / 114, which is approximately 0.1228 or 12.28%.

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

The $Q _3$ for the following distribution is

X 2 3 4 5 6 7 8 9 10 11
Y 3 6 9 18 20 14 10 10 7 2


  1. $6$
  2. $5$
  3. $8$
  4. $9$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find Q3, calculate the cumulative frequency: 3, 9, 18, 36, 56, 70, 80, 90, 97, 99. The total N is 99, so the 3rd quartile position is 0.75 * (99 + 1) = 75th value, which corresponds to X = 8.

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

The $8th$ decile of the following observations is
$29, 18, 15, 30, 42, 35, 34, 28, 45, 34$

  1. $44.6$
  2. $40.6$
  3. $34.15$
  4. $38.15$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

First, sort the data: 15, 18, 28, 29, 30, 34, 34, 35, 42, 45. The 8th decile (D8) is the 0.8 * (10 + 1) = 8.8th position. Interpolating between the 8th (35) and 9th (42) values: 35 + 0.8 * (42 - 35) = 35 + 5.6 = 40.6.

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

Which of the following relations among the location parameters does not hold?

  1. $Q _2 = Median$
  2. $P _50 = Median$
  3. $D _5 = Median$
  4. $D _4 = Median$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Median is the middle value, which corresponds to the 50th percentile (P50), the 5th decile (D5), and the 2nd quartile (Q2). D4 represents the 40th percentile, not the median.

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

Semi inter-quartile deviation is given by the formula _______.

  1. $Q _3 - Q _1$
  2. $\frac{Q _3 - Q _1}{8}$
  3. $\frac{(Q _3 - Q _1)}{2}$
  4. $\frac{(Q _3 - Q _1)}{4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Semi inter-quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q1 and the third quartile is denoted as Q3 .


Semi inter-quartile deviation = (Q3-Q1) /2

                             = (29-17)/2

                             = 12/2

                             = 6