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 range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $\text{coefficient of range}=\dfrac{a-b}{a+b}=\dfrac{range}{a+b}$

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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 )$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $D _8 = l _1 + \frac{\frac{8N}{10} - C}{f} \times (l _2 - l _1)$ 

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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
Reveal answer Fill a bubble to check yourself
B Correct answer
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.=$\displaystyle =\frac{M.D.}{median}=\frac{13}{65}=0.2$

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $M.D=\dfrac{|40-65|+|54-65|+|62-65|+|68-65|+|76-65|+|90-65|}{65}$

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

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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. $\dfrac87$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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

  1. $0124$
  2. $0.212$
  3. $0.236$
  4. $0.436$
Reveal answer Fill a bubble to check yourself
D Correct answer
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.
Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

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. $0.124$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $=\dfrac{59-23}{59+23}=\dfrac{36}{82}=0.49\approx 0.44$
Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

Geometric mean of two observations can be calculated only if ________.

  1. both the observations are positive

  2. one of the two observations is zero

  3. one of them is negative

  4. both of them are zero
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The geometric mean involves taking the nth root of the product of n numbers. If any number is negative, the root may be imaginary; if zero, the product is zero. Thus, it is typically defined for positive observations.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

Extreme value have no effect on _______.

  1. average

  2. median

  3. geometric mean

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

The median is a positional average and is not affected by extreme values (outliers) in the dataset, unlike the arithmetic mean.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

For further algebraic treatment harmonic mean is ______.

  1. suitable

  2. not suitable

  3. sometime suitable

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

Like the geometric mean, the harmonic mean is not suitable for further algebraic treatment because it is based on the reciprocal of the values.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

Harmonic mean gives more weight age to _______.

  1. small values

  2. large values

  3. positive values

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

Harmonic mean gives more weight to smaller values because of the reciprocal in its formula. A small number becomes large when reciprocated, dominating the calculation.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

Mean deviation of $7,10,10,15,10,8,8,7,3,2,10$ through mean is

  1. $3.14$
  2. $8$
  3. $\dfrac {4}{5}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

mean $=\dfrac{\sum {x _1}}{n}=\dfrac{90}{11}\approx 8.18$

derivation through mean = $|x _i-\bar x|$
& mean of it is $=\dfrac{\sum|{x _i-\bar x}|}{n}$
$=\dfrac{(1.18+1.82+1.82+6.82+8.18+0.18+0.18+1.18+5.18+6.18+1.82)}{11}$
$=\dfrac{34.54}{11}=3.14$