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 maths concepts of seven and eight digit numbers comparison of numbers comparing numbers operations on rational numbers indian place value chart largest and smallest numbers writing and expanding numbers

A data has highest value $120$ and the lowest value $71.A$ frequency distribution in descending order with seven classes is to be constructed. The limits of the second class interval shall be 

  1. $77$ and $78$
  2. $78$ and $85$
  3. $85$ and $113$
  4. $113$ and $120$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Range of Frequency distribution=Highest Value-Lowest value
 $=120-71=49$

Dividing this into Seven $(7)$ equal classes.

$\Rightarrow \dfrac{49}{7}=7$

Thus the class width should be 7

Now  arranging  in descending order

Class interval $1 \rightarrow (120-7) to\space 120 \rightarrow 113-120$

Class interval $2 \rightarrow (113-7) to \space 113 \rightarrow 106-113$

Hence class interval $1$ and $2$ is $113$ and $120$ 
Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

What is the value of mean for the following data:

Marks No. of Student
$5-14$ $10$
$15-24$ $18$
$25-34$ $32$
$35-44$ $26$
$45-54$ $14$
$55-64$ $10$
  1. $30$
  2. $29$
  3. $33.68$
  4. $34.21$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the mean, use the formula (sum of f*x) / N. The midpoints (x) are 9.5, 19.5, 29.5, 39.5, 49.5, 59.5. Summing f*x gives 3368, and N is 100, resulting in 33.68.

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

If there are two groups containing $30$ and $20$ observations and having $50$ and $60$ as arithmetic means, then the combined arithmetic mean is ________.

  1. $55$
  2. $56$
  3. $54$
  4. $52$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Combined mean = (n1*x1 + n2*x2) / (n1 + n2). Calculation: (30*50 + 20*60) / (30 + 20) = (1500 + 1200) / 50 = 2700 / 50 = 54.

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

What is the value of mean for the following data.

Class interval Frequency
$350-369$ $15$
$370-389$ $27$
$390-409$ $31$
$410-429$ $19$
$430-449$ $13$
$450-469$ $6$
  1. $400$
  2. $400.58$
  3. $394$
  4. $394.50$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Class Intervals Mid-values(x) Frequency(f) fx
$350-369$ $359.5$ $15$ $5,392.5$
$370.389$ $379.5$ $27$ $10,246.5$
$390-409$ $399.5$ $31$ $12,384.5$
$410-429$ $419.5$ $19$ $7,970.5$
$430-449$ $439.5$ $13$ $5,713.5$
$450-469$ $459.5$ $6$ $2,757$
$\displaystyle\sum f=111$ $\displaystyle\sum fx=44,464.5$

Arithmetic mean $=44,464.5/111=400.58$.

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

Consider the following frequency distribution.

Class Intervals $0-10$ $10-20$ $20-30$ $30-40$
Frequency $8$ $10$ $12$ $15$

Arithmetic mean $=$?

  1. $39.65$
  2. $22.55$
  3. $32.55$
  4. $23.56$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Class Intervals Mid-values(x) Frequency(f) fx
$01-0$ $5$ $8$ $40$
$10-20$ $15$ $10$ $150$
$20-30$ $25$ $12$ $300$
$30-40$ $35$ $15$ $525$
$\displaystyle\sum f=45$ $\displaystyle\sum fx =1,015$

Arithmetic mean $=1,015/45=22.55$.

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

Arithmetic mean for grouped data can be calculated by _________.

  1. direct method

  2. assumed mean method

  3. step deviation method

  4. all of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Arithmetic mean refers to the average amount in a given group of data. There are many ways to calculate arithmetic mean like direct method where all the data are added up and then divided by the number of figures in the data in order to ascertain the mean class or assumed mean method and step deviation method, the data of the given class is reduced into smaller units which makes it easy to do calculation and ascertain the mean of the class. 

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

What needs to be done for calculating mean for a continuous series?

  1. Mid-points of various class intervals are taken

  2. Lower class limits are taken

  3. Upper class limits are taken

  4. A or B or C
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To calculate the mean of a continuous series, mid points of the various class intervals is taken. For example, if the class is like 10-20 then before calculating the mean mid point that is 15 is calculated for the whole series which is added and divided by the number of terms in order to ascertain the mean. 

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

For grouped data, Arithmetic mean by Direct Method =

  1. sfX / sf

  2. sd / N

  3. sX / N

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

Arithmetic mean refers to the average amount in a given group of data. There are many ways to calculate arithmetic mean for grouped data like direct method where all the data are multiplied with their respective frequencies and then added up which are then divided by the summation of the frequencies or number of figures in the data in order to ascertain the mean. The formula is sfX/ sf where sfd is the summation of frequency multiplied by X for all figures and sf is the frequency or the number of element in the given data. 

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

For grouped data, Arithmetic mean by Assumed Mean Method =

  1. A + sd/N

  2. A + sfd/sf

  3. sfX/sf

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

In assumed mean method, any value can be taken as assumed mean whether it is there in the data or not but it should be centrally located in the data so that to simply the big figures in the data in order to ascertain mean of the given data through easy calculations. For grouped data, the formula for assumed mean is A+ sfd/sf where A is the assumed mean, sfd is the summation of frequency multiplied with X-A for all figures and sf is the summation of frequency or the number of element in the given data. 

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

The arithmetic mean of the first 100 natural numbers is _____.

  1. 50

  2. 52

  3. 51

  4. 50.5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Arithmetic mean refers to the average amount in a given group of data. So arithmetic mean can be calculated by adding the first term and the last term of the series and then dividing the sum by 2. In the given series the first term is 1 and  the last term is 100, so the 

Arithmetic mean = ( 1+100 ) /2 
                             = 101 /2 
                             = 50.5 

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

The mean of a sample of size 10 is 15. If the value of each item is doubled, the mean of the sample will be _______.

  1. 15

  2. 30

  3. 11

  4. 22
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Mean refers to the average amount in a given group of data. So arithmetic mean can be calculated by adding the first term and the last term of the series and then dividing the sum by 2. In the given series the first term 'a' is doubled and  the last term 'b' is also doubled , so the 

Mean = {(a+a)+ (b+b)}  /2 

          = (2a+2b) /2 

          = 2 (a+b) /2 

          = 2 [ (a+b)/2} 

Therefore, the mean is also doubled. So,if the mean was 15 then now it will be 30.  

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

If the frequency of observations $X _{i}$ is $f _{i} (i = 1, 2, ..... n)$.

  1. $\overline {X} = \dfrac {X _{1}\times X _{2} \times X _{3}\times ......X _{n}}{N}$
  2. $\overline {X} = \dfrac {X _{1} + X _{2} + X _{3} + ......X _{n}}{N}$
  3. $\overline {X} = \dfrac {\sum X _{i}}{N}$
  4. $\overline {X} = \dfrac {\sum f _{i}X _{i}}{\sum f}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The arithmetic mean for grouped data is defined as the sum of the products of each value and its corresponding frequency, divided by the total frequency.

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

Consider following frequency distribution.

Class Intervals $0-10$ $10-20$ $20-30$ $30-40$
Frequency $8$ $10$ $12$ $15$

Arithmetic mean $=$?

  1. $39.65$
  2. $22.55$
  3. $32.55$
  4. $23.56$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

0−10 | 10−20 | 20−30 | 30−40 | | --- | --- | --- | --- | --- | | Frequency(f) | 8 | 10 | 12 | 15 | | Class mark ( mid points= x) | 5 | 15 | 25 | 35 | | Fx | 40 | 150 | 300 | 525 |

Mean = summation of fx / summation of f

          = 1015/ 45

          = 22.55

Multiple choice mean and median mean maths assumed mean method assumed mean method of finding mean measure of central tendency

The mean of a sample of size $10$ is $15$. If the value of each item is reduced by $2$, the mean of the sample will be_____.

  1. $15$
  2. $13$
  3. $11$
  4. $22$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Mean refers to the average amount in a given group of data. So arithmetic mean can be calculated by adding the first term and the last term of the series and then dividing the sum by 2. In the given series the first term 'a' is decreased by 2 and  the last term 'b' is also decreased by 2 , so the 

Mean = {(a-2)+ (b-2)}  /2 

          = (a+b-4) /2 

          = {(a+b)/2} - 4/2

          = {(a+b)/2} - 2 

Therefore, the mean is also decreased by 2. So,if the mean was 15 then now it will be 13.