Mathematics · Quantitative Aptitude

Statistics and Dispersion

559 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 $Q _3$ 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

The $P _82$ 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
D Correct answer
Explanation

Total frequency N = 99. The 82nd percentile position is 0.82 * (99 + 1) = 82nd value. Looking at the cumulative frequencies (3, 9, 18, 36, 56, 70, 80, 90, 97, 99), the 82nd value falls within the category X = 9.

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

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

If the minimum value in a set is $15$ and its range is $55$, the maximum value of the set is _______.

  1. $40$
  2. $60$
  3. $80$
  4. $70$
Reveal answer Fill a bubble to check yourself
D 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 = 55

In the given series, H= x and L= 15

Range => x-15 = 55

            => x = 55+15 = 70 

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

Range of a set of values is $60$ and maximum value in the series is $80$. The minimum value of the series is _______.

  1. $140$
  2. $20$
  3. $70$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B 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 = 60 

In the given series, H= 80 and L= x 

Range => 80-x = 60

            => x = 80-60 = 20 

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 $Q 3 + Q _1 = 108, Q _3 - Q _1 = 74$; then Q.D is ______.

  1. $33$
  2. $3.7$
  3. $37$
  4. $0.68$
Reveal answer Fill a bubble to check yourself
C 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 .


Quartile deviation = (Q3-Q1) /2

                             = 74/2

                             = 37

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.