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 shortcut method for calculating mean deviation about mean mean deviation about mean and median range and mean deviation

In distribution $25\%$ of the observations are less than $46$ and $25\%$ of the observations are more than $54$. The quartile deviation of the distribution is _______.

  1. $3$
  2. $7$
  3. $4$
  4. $6$
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 . 

Q1= 46 and Q3=54

Quartile deviation = (Q3-Q1) /2

                             = (54-46)/2

                             = 8/2

                             = 4

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

If the standard deviation of $x _{1},x _{2},.....x _{n}$ is 3.5, then the standard deviatiuon of $-2x _{1}-3,-2x _{2}-3....,-2x _{n}-3$ is

  1. -7

  2. -4

  3. 7

  4. 1.75
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Standard deviation is independent of the change of origin (adding or subtracting a constant) and is scaled by the absolute value of the change of scale (multiplying by a constant). If the original SD is 3.5, the new SD is | -2 | * 3.5 = 7.

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

The variance of the data $6,\ 8,\ 10,\ 12\,,14\,,\ 16,\ 18,\ 20,\ 22,\ 24$ is

  1. $15$
  2. $20$
  3. $30$
  4. $33$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Mistake :$14$ is not given
Mean $\bar x=\dfrac{6+8+10+12+14+16+18+20+22+24}{10}=\dfrac{150}{10}=15$
Variance$=\dfrac{1}{n} \sum\limits _{i=1}^n(x _{i}-\bar x)^2$
$\implies \dfrac{1}{10}((6-15)^2+(8-15)^2+(10-15)^2+(12-15)^2+(14-15)^{2}+(16-15)^2+(18-15)^2+(20-15)^2$
$+(22-15)^2+(24-15)^2$

$\implies \dfrac{81+49+25+9+1+1+9+25+49+81}{10}$

$\implies \dfrac{330}{10}=33$
Multiple choice maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

Standard deviation of a collection of data is $2\sqrt{2}$. If each value in a data set  is multipled by $3$, then the standard deviation of the new data is.

  1. $\sqrt{12}$
  2. $4\sqrt{2}$
  3. $6\sqrt{2}$
  4. $9\sqrt{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
The standard deviation would also be multiplied by $3$.
Because the mean would also be $3x$ larger, the differences from the mean would be $3x$ larger too.
It is the same idea as if you were looking at your data set through an enlarging lens- everything would be $3x$ bigger, not only the data values, but also the mean, the differences from the mean, but just everything!
$\therefore$ the standard deviation becomes $2\sqrt{2}\times 3=6\sqrt{2}$
Multiple choice maths statistics and probability coefficient of variance variance and standard deviation measures of dispersion range and mean deviation

If the standard deviation of $x _1, x _2, .., x _n$ is $3.5$, then the standard deviation of $-2x _1-3, -2x _2-3$,....., -2x_n-3$ is?

  1. $-7$
  2. $-4$
  3. $7$
  4. $1.75$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The Standard Deviation of a set remains unchanged if each data is increased or decreased by a constant however changes similarly when data is multiplied or divided by a constant.
$\therefore $ The SD for the new data set will be $=-2\times 3.5=-7$
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

Coefficient of variation of a distribution is $60$ and its standard deviation is $21$, then its arithmetic mean is?

  1. $36$
  2. $37$
  3. $35$
  4. $38$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The coefficient of variation (CV) is calculated as (Standard Deviation / Mean) * 100. Rearranging the formula gives Mean = (Standard Deviation / CV) * 100. Substituting the given values: Mean = (21 / 60) * 100 = 0.35 * 100 = 35.

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.