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 maths average arithmetic mean of ap introduction to averages means

Coefficient of variance of a distribution is 60% and the standard deviation is 25. The arithmetic mean of the distribution is

  1. $\cfrac {25}{3}$

  2. 35
  3. $\cfrac {125}{3}$
  4. $\cfrac {25}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The coefficient of variation (CV) is given by (Standard Deviation / Mean) * 100. Rearranging the formula for the mean gives Mean = (Standard Deviation / CV) * 100 = (25 / 60) * 100 = 125 / 3.

Multiple choice maths average arithmetic mean of ap introduction to averages means

Arithmetic Mean is ______ affected by extreme values.

  1. Not

  2. Highly

  3. Less

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

The arithmetic mean takes every value into account equally in its calculation, making it sensitive to outliers and extreme values compared to median or mode.

Multiple choice maths average arithmetic mean of ap introduction to averages means

Suppose a population $A$ has $100$ observations $101,102...200$ and other population $B$ has $100$ observations $151,152...250$. 

Find the difference in their means

  1. $49$
  2. $50$
  3. $51$
  4. $52$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The observations of $A$ are $101,102,......,200$

The sum is $\dfrac{100}{2}[2(101)+(100-1)1]\50(202+99)\50(301)=15050$
Mean is $\dfrac{15050}{100}\150.5$
The observations of $B$ are $151,152,......,250$
The sum is $\dfrac{100}{2}[2(151)+(100-1)1]\50(302+99)\50(401)=20050$
Mean is $\dfrac{20050}{100}=200.50$
The difference is $200.5-150.5=50$

Multiple choice maths average arithmetic mean of ap introduction to averages means

The mean of $3$ observations is $12$ and mean of $5$ observations is $4$ the combined mean is 

  1. 7

  2. 8

  3. 9

  4. 10
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The no .of  observations is $3+5=8$

The sum of observations is $3(12)+5(4)=36+20=56$
The mean is $\dfrac{56}{8}=7$

Multiple choice maths average arithmetic mean of ap introduction to averages means

The sum of $9$ numbers is $246$. If the average of three of them is $24$, what is the average of the remaining numbers?

  1. $30$
  2. $29$
  3. $31$
  4. $25$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The total sum of all 9 numbers is 246. The sum of the first 3 numbers is 3 * 24 = 72. The sum of the remaining 6 numbers is 246 - 72 = 174. The average of the remaining numbers is 174 / 6 = 29.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The arithmetic mean of first ten natural numbers is

  1. $5.5$
  2. $6$
  3. $7.5$
  4. $10$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The arithmetic mean of first ten natural numbers is the average of these natural numbers.


$\therefore$ arithmetic mean

   $=\dfrac{1+2+3+4+5+6+7+8+9+10}{10}$

   $=\dfrac{n(\dfrac{n+1}{2})}{10}\,\,\,\,\,\,\,\rightarrow Here \,\boxed{n=10}$

   $=\dfrac{10\times \dfrac{11}{2}}{10}$ 

   $=\dfrac{11}{2}=\boxed{5.5}\rightarrow $ option -A

Multiple choice maths average arithmetic mean of ap introduction to averages means

The A.M. of 'n' observations is M. If the sum of $(n - 4)$ observation is 'a', what is the mean of remaining $4$ observations?

  1. $nM + a$
  2. $\dfrac {nM - a}{2}$
  3. $\dfrac {nM + a}{2}$
  4. $\dfrac {nM - a}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given $\cfrac { { a } _{ 1 }+{ a } _{ 2 }+{ a } _{ 3 }+{ a } _{ 4 }+....+{ a } _{ n } }{ n } =M\quad \quad \quad (1)\\ \left( { a } _{ 1 }+{ a } _{ 2 }+{ a } _{ 3 }+{ a } _{ 4 }+....+{ a } _{ n-4 } \right) =a\quad \quad \quad (2)$

From $(1)$ and $(2)$
$a+{ a } _{ n-3 }+{ a } _{ n-2 }+{ a } _{ n-1 }+{ a } _{ n }=nM\\ { a } _{ n-3 }+{ a } _{ n-2 }+{ a } _{ n-1 }+{ a } _{ n }=(nM-a)\\ Mean=\cfrac { { a } _{ n-3 }+{ a } _{ n-2 }+{ a } _{ n-1 }+{ a } _{ n } }{ 4 } =\cfrac { nM-a }{ 4 } \\ Mean=\cfrac { (nM-a) }{ 4 } $
Multiple choice maths average arithmetic mean of ap introduction to averages means

The arithmetic mean (average) of the first $n$ positive integers is

  1. $\dfrac {n}{2}$
  2. $\dfrac {n^{2}}{2}$
  3. $n$
  4. $\dfrac {n - 1}{2}$
  5. $\dfrac {n + 1}{2}$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

The sum of the first $n$ positive integers is $s = \dfrac {n(n + 1)}{2}$. Since $A.M. = \dfrac {s}{n}$, the correct choice is (e).

Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

The mean of a data set consisting of $20$ observations is $40$. If one observation $53$ was wrongly recorded as $33$, then the correct mean will be:

  1. $41$
  2. $49$
  3. $40.5$
  4. $42.5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$Mean=\dfrac{S}{n}$


where S=sum of all observations
            n=number of observations

hence, $40=\dfrac{S}{20}\Rightarrow S=800$

but Since, 53 is recorded as 33 so we need to add $(53-33=20)$ to get the correct mean which is 
$Mean _{correct}=\dfrac{800+20}{20}=41$

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

For a set of positive numbers, consider the following statements:
1. If each number is reduced by $2$, then the geometric mean of the set may not always exists.
2. If each number is increased by $2$, then the geometric mean of the set is increased by $2$.
Which of the above statements is/are correct?

  1. $1$ only
  2. $2$ only
  3. Both $1$ and $2$
  4. Neither $1$ nor $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

1. Consider the two numbers $1$ and $4$, geometric mean of $1$ and $4$ is $\sqrt {1 \times 4} = 2$.
When each number is reduced by $2$, the numbers become $-1$ and $2$ whose geometric mean does not exist.
2. Now consider two numbers $2$ and $7$. Their geometric mean is $\sqrt {14}$. The new numbers are $4$ and $9$ whose geometric mean is $\sqrt {4\times 9} = 6$ which is not equal to $2\sqrt {14}$.
Thus only statement $1$ is true.

Multiple choice

Which of the following is NOT a property of the variance?

  1. It is always positive

  2. It is a constant

  3. It is always negative

  4. It is equal to the square of the standard deviation
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The variance is not always negative. It can be positive or zero, depending on the distribution of the random variable.