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 co-efficient of range from the following observations is 
$10, 18, 20, 28, 15, 17, 22, 25, 29, 32, 34$

  1. $0.6$
  2. $0.5$
  3. $0.55$

  4. None of them
Reveal answer Fill a bubble to check yourself
C 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.
Coefficient of range is found by dividing the range with the sum of highest and lowest value of the series. 
Coefficient of range (CR) = R/ (H+L) 
                                          =24 / (34+10) 
                                          = 24/ 44 = 0.55 

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 co-efficient of quartile deviation from the following observations is 
$10, 18, 20, 28, 15, 17, 22, 25, 29, 32, 34$

  1. $0.26$
  2. $0.16$
  3. $0.06$
  4. $0.36$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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 .

The coefficient of quartile deviation is found by dividing the difference between the first quartile and the third quartile by the sum between the first quartile and the third quartile.

In the given series, 10, 15,17,18,20,22,25,28,29,32,34. 

Q1= 17 and Q3=29 

Coefficient of quartile deviation =(Q3-Q1)/ (Q3+Q1)

                                            = (29-17)/ (29+17)

                                            = 12/46 = 0.26 



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 variance of $1, 2, 4, 5$ and x is $2$, then x is _____.

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

Variance = sum((x_i - mean)^2) / n = 2. For data 1, 2, 4, 5, x, the mean is (12 + x) / 5. Solving for x where variance is 2 leads to x = 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

Sum of squares of the deviations is minimum when deviations are taken from ________.

  1. mean

  2. median

  3. mode

  4. zero
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Mean deviation is the arithmetic average of the deviations of the observed values from an  average of the observed values. 
Mean deviation from the mean calculates the arithmetic average of the deviations of the observed values from mean of the observed values. Sum of the squares of the deviations is minimum when deviations are taken from mean as mean is the arithmetic average of the series and disperse the whole series into individual minimum value therefore the sum of their squares is minimum. 
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 quartile deviation from the following observations is 
$10, 18, 20, 28, 15, 17, 22, 25, 29, 32, 34$

  1. $8$
  2. $6$
  3. $10$
  4. $5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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 .

In the given series, 10, 15,17,18,20,22,25,28,29,32,34 

Q1= 17 and Q3=29

Therefore, Quartile deviation = (Q3-Q1) /2

                                                 = (29-17)/2

                                                  = 12/2

                                                  = 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

Quartile deviation is equal to _______.

  1. semi - interquartile range

  2. double the interquartile range

  3. interquartile range

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

 

Quartile deviation also known as 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 .

Quartile deviation or semi inter-quartile deviation = (Q3-Q1) /2


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 mean and standard deviation of a normal distribution are $33$ and $5$ respectively, what is the Inter-quartile range?

  1. $(6.67)$
  2. $(8)$
  3. $(7.74)$
  4. $(6.8)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a normal distribution, the inter-quartile range (IQR) is approximately 1.3489 times the standard deviation. 1.3489 * 5 = 6.7445, which is closest to 6.67.

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 following data show the number of hours worked by 200 statistics students:

Number of Hours Frequency
0-9 40
10-19 50
20-29 70
30-39 40

The class width for this distribution is.........

  1. 9

  2. 10

  3. 11

  4. Varies from class to class
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Class width is the difference between the upper boundary of a class and the lower boundary. For 0-9, 10-19, the width is 10 - 0 = 10.

Multiple choice statistics vital statistics and official statistics guiding rules for tabulation vital statistics textual and tabular presentation of data

Let m be the mid-point be the upper class limit of a continuous frequency distribution The lower class limit of the class is

  1. $2m+L$
  2. $2m-L$
  3. $m-L$
  4. $m+2L$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Mid point=$\dfrac{upper  limit+ lower  limit}{2}$

$\Rightarrow m=\dfrac{L+lower  limit}{2}$
$\Rightarrow Lower   limit=2m-L$

Multiple choice statistics vital statistics and official statistics guiding rules for tabulation vital statistics textual and tabular presentation of data

The class marks in a frequency table are given to be $5,10,15,20,25,30,35,40,45,50$. The class limits of the first classes are 

  1. $3-7,7-17,13-17,17-23,23-27$
  2. $2.5-7.5,7.5-12.5,12.5-17.5,17.5-22.5,22.5-27.5$
  3. $1.5-8.5,8.5-11.5,11.5-18.5,18.5-21.5,21.5-28.5$
  4. $2-8,8-12,12-18,18-22,22-28$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$2.5 - 7.5, 7.5 - 12.5, 12.5 - 17.5, 17.5 - 22.5, 22.5 - 27.5$


Given,

$5,10,15,20,25,30,35,40,45,50$

the class limit is given by,

$\dfrac{5}{2}=2.5$

$\dfrac{5+10}{2}=7.5$

$\dfrac{10+15}{2}=12.5$

$\dfrac{15+20}{2}=17.5$

$\dfrac{20+25}{2}=22.5$

$\dfrac{25+30}{2}=27.5$

Multiple choice statistics vital statistics and official statistics guiding rules for tabulation vital statistics textual and tabular presentation of data

The class marks of a frequency distribution are given as follow 15,20,25,...
The class corresponding to the class mark 20 is 

  1. $12.5-17.5$
  2. $17.5-22.5$
  3. $18.5-21.5$
  4. $19.5-20.5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Width=5

Class marks=20
$\therefore  Lower\  limit=20-\dfrac{5}{2}$
$\Rightarrow \dfrac{40-5}{2}=\dfrac{35}{2}=17.5$
$Upper\  limit=20+\dfrac{5}{2}$
$\Rightarrow \dfrac{40+5}{2}=\dfrac{45}{2}=22.5$
Hence the class corresponding to the class mark 20 is $17.5-22.5.$

Multiple choice business economics and quantitative methods linear regression aspects of correlation scatter graphs and correlation correlation

Co-variance between two variables is _____________.

  1. the average of the product of deviations taken from their averages

  2. a is further divided by the product of their standard deviations

  3. a is further divided by the product of their arithmetic averages

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Covariance is defined as the average of the products of the deviations of two variables from their respective means: Cov(X,Y) = E[(X - E[X])(Y - E[Y])].

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

What is the standard deviation of $7,9,11,13,15$?

  1. $2.4$
  2. $2.5$
  3. $2.7$
  4. $2.8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given numbers are $ 7,9,11,13,15$
Mean of given numbers $=\dfrac { 7+9+11+13+15 }{ 5 } =11$
Standard deviation$=\dfrac { |7-11|+|9-11|+|11-11|+|13-11|+|15-11| }{ 5 } =2.4$
Option A is true