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 mathematics and statistics moving averages introduction to time series introduction to time series and forecasting time series and forecasting uses of average in day-to-day life

The fire in a factory is an example of

  1. Secular trend

  2. Seasonal movements

  3. Cyclical variations

  4. Irregular variations
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\Rightarrow$  The fire in factory is an example of $Irregular\, variations.$

$\Rightarrow$  Irregular variations which occur on account of random external events.
$\Rightarrow$ These variations either go very deep downward or too high upward to attain peaks abruptly.
$\Rightarrow$  They do not have any definite pattern and it cannot be predicted in advance. These variations are due to floods, wars, famines, earthquakes, strikes, lockouts, epidemics etc.

Multiple choice business mathematics and statistics moving averages introduction to time series introduction to time series and forecasting time series and forecasting uses of average in day-to-day life

Wheat crops badly damaged on account of rains is

  1. Cyclical movement

  2. Random movement

  3. Secular trend

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

$\Rightarrow$  Wheat crops badly damaged on account of rains is $Random\, movement$

$\Rightarrow$  Random movement which occur on account of irregular external events.
$\Rightarrow$  These variations either go very deep downward or too high upward to attain peaks abruptly. These fluctuations are a result of unforeseen and unpredictably forces which operate in absolutely random or erratic manner.
$\Rightarrow$  They do not have any definite pattern and it cannot be predicted in advance. These variations are due to floods, wars, famines, earthquakes, strikes, lockouts, epidemics etc.

Multiple choice business mathematics and statistics moving averages introduction to time series introduction to time series and forecasting time series and forecasting uses of average in day-to-day life

Indicate which of the following an example of seasonal variations is:

  1. Death rate decreased due to advance in science

  2. The sale of air condition increases during summer

  3. Recovery in business

  4. Sudden causes by wars
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$  Example of seasonal variation is : $The\,sale\,of\,air\,condition\,increases\,during\,summer.$

$\Rightarrow$  Over a span of one year, seasonal variation takes place due to the rhythmic forces which operate in a regular and periodic manner. These forces have the same or almost similar pattern year after year.
$\Rightarrow$  The seasonal variations may be due to various seasons or weather conditions.
$\Rightarrow$  These variations may be also due to man-made conventions & due to habits, customs or traditions.

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

For a symmetrical distribution lower quartitl is 20 and upper quartile is 40.The value of 50th percentile is

  1. 20

  2. 40

  3. 30

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

First quartile also called the lower quartile or the 25th percentile(splits off the lowest 25% of data from the highest 75%)
Second quartile also called the median or the 50th percentile (cuts data set in half)
Third quartile  also called the upper quartile or the 75th percentile (splits off the highest 25% of data from the lowest 75%)
Since its a symmetrical distribution therefore the median will be 30

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 $=\dfrac{59-23}{59+23}=\dfrac{36}{82}=0.49\approx 0.44$