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 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

If a constant value $40$ is subtracted from each observation of a set, the mean of the set is __________.

  1. increased by $40$
  2. decreased by $40$

  3. is not affected

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

If a constant is subtracted from every observation in a set, the mean of the set is also decreased by that same constant.

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

The mode of the following observations is $10, 19, 22, 16, 15, 18, 20, 18, 14, 18, 23$

  1. $17.55$
  2. $18$
  3. $15$
  4. $16.5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The mode is the value that appears most frequently. In the set 10, 19, 22, 16, 15, 18, 20, 18, 14, 18, 23, the number 18 appears three times, which is more than any other number.

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

The second quartile of the following observations is
$10, 19, 22, 16, 15, 18, 20, 18, 14, 18, 23$.

  1. $17.55$
  2. $18$
  3. $15$
  4. $20$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The second quartile (Q2) is the median. First, sort the data: 10, 14, 15, 16, 18, 18, 18, 19, 20, 22, 23. With 11 observations, the median is the 6th value, which is 18.

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

If Md, Q, D and P Stands for median, quartile, decile and percentile respectively, then which of the following relation between them is true?

  1. $M _d = Q _2 = D _6 = P _50$
  2. $M _d = Q _3 = D _6 = P _75$
  3. $M _d = Q _5 = D _5 = P _50$
  4. $M _d = Q _2 = D _5 = P _50$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The median (Md) divides the data into two halves (50%), the second quartile (Q2) is the median, the 5th decile (D5) is the median, and the 50th percentile (P50) is the median. Thus, Md = Q2 = D5 = P50.

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 a discrete series having $(2n+1)$ observations, median is ______.

  1. nth observation
  2. $(n+1)$th observation
  3. $[(n+2)/2]th$ observation
  4. $[(2n+1)/2]th$ observation
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a series with N observations, if N is odd, the median is the ((N+1)/2)th observation. Substituting N = 2n+1 gives ((2n+1+1)/2)th = (2n+2)/2 = (n+1)th observation.

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 a moderately skewed distribution the arithmetic mean is $10$ units and the mode is $7$ units, the median is ______.

  1. $9$
  2. $5$
  3. $8$
  4. $6$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a moderately skewed distribution, the empirical relationship is: Mean - Mode = 3(Mean - Median). Given Mean = 10 and Mode = 7, we have 10 - 7 = 3(10 - Median), so 3 = 3(10 - Median), which means 1 = 10 - Median, so Median = 9.

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 business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

Given the marginal cost function $\dfrac{2x}{3}+3-\dfrac{16}{x^2}$, find  average cost function.

  1. $\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$
  2. $\dfrac{1}{2}x^2+3x-7+\dfrac{16}{x}$
  3. $\dfrac{1}{4}x^2+3x-7+\dfrac{16}{x}$

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

$\Rightarrow$  We have $MC=\dfrac{2x}{3}+3-\dfrac{16}{x^2}$


$\Rightarrow$  $Average\,\,cost=\int (MC)dx$ 


$\Rightarrow$   $Average\,\,cost=\int (\dfrac{2x}{3}+3-\dfrac{16}{x^2})dx$


$\therefore$   $Average\,\,cost=\dfrac{2x^2}{2\times 3}+3x-7+\dfrac{16}{x}$

$\Rightarrow$   $Average\,\,cost=\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$

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 $=-2\times 3.5=-7$