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 to find variance and standard deviation variance and standard deviation measures of dispersion

The variance of $20$ observations is $5$. If each observation is multiplied by $2$, then what is the new variance of the resulting observations?

  1. $5$
  2. $10$
  3. $20$
  4. $40$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If each observation is multiplied by $2$, then mean$(\mu )$ will get doubled.

Now, variance $=\sigma ^{ 2 }=\dfrac { (X-\mu )^{ 2 } }{ n } $
Now here both $X$ and $\mu $ will be doubled, so the variance will become four times.
Thus, the new variance will be $4$ times the older variance which is $4\times 5=20$.
Hence, C is correct.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

T-distribution is symmetrical like normal distribution and its mean value is____________.

  1. zero

  2. -1

  3. 1

  4. 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The mean of t distribution is always equal to 0 and also the variance is always greater than 1 and it is symmetrical like normal distribution.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

A simple formula to calculate the standard error is___________.

  1. $S _{yx} =$ ${\sigma} _{y}{\sqrt{1 - {r}^2}}$
  2. $S _{xy} =$ ${\sigma} _{x}{\sqrt{1 - {r}^2}}$
  3. $S _{yx} = $ S.E.

  4. Both (A) and (B)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Standard error can be calculated by applying two formulas mentioned in option A and option B that shows how a sample mean deviates from the mean of whole population.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The value of $X^2$ describes the magnitude of the difference between___________.

  1. two normal distributions

  2. expected and observed frequency

  3. both (A) and (B)

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

Chi square distribution is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The coefficient correlation between x and y is 0.8, the covariance being 20. If the standard deviation of x is 4 find the standard deviation of y___.

  1. 6.25

  2. 10

  3. 11.25

  4. 9.10
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for correlation coefficient r is r = Cov(x,y) / (SDx * SDy). Plugging in the values: 0.8 = 20 / (4 * SDy). Thus, 0.8 = 5 / SDy, so SDy = 5 / 0.8 = 6.25.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

If Mean deviation is 16 then the standard deviation of the data is____.

  1. 18.90

  2. 20.0

  3. 19.90

  4. 19
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a normal distribution, the relationship between standard deviation (SD) and mean deviation (MD) is approximately SD = 1.25 * MD. Here, 1.25 * 16 = 20.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

What will be the relative range, if the spread of items in a given distribution lies between $100$ and $180$kg?

  1. $0.5$
  2. $0.4$
  3. $0.3$
  4. $0.55$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Relative range is calculated as (Max - Min) / (Max + Min). Here, (180 - 100) / (180 + 100) = 80 / 280 = 8 / 28 = 2 / 7, which is approximately 0.2857, rounding to 0.3.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The standard deviation of $5$ items is found to be $15$. What will be the standard deviation if the values of all the items are increased by $5$?

  1. $15$
  2. $20$
  3. $10$

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

Standard deviation is independent of the change of origin. Adding a constant to all items does not change the spread of the data.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The mean of $100$ observations is $18.4$ and sum of sqares of deviations from mean is $1444$, the Co-efficient of variation is ______.

  1. $30.6$
  2. $35.6$
  3. $20.6$
  4. $10.6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Standard deviation = sqrt(Sum of squared deviations / N) = sqrt(1444 / 100) = sqrt(14.44) = 3.8. Coefficient of Variation (CV) = (SD / Mean) * 100 = (3.8 / 18.4) * 100 = 20.65.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The first 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
C Correct answer
Explanation

Sorted data: 10, 14, 15, 16, 18, 18, 18, 19, 20, 22, 23. N=11. First quartile (Q1) position = (N+1)/4 = 12/4 = 3rd value. The 3rd value is 15.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The median for the following distribution is

X 2 3 4 5 6 7 8 9 10 11
Y 3 6 9 18 20 14 10 10 7 2



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

Total frequency = 3+6+9+18+20+14+10+10+7+2 = 99. Median is the (99+1)/2 = 50th value. Cumulative frequencies: 3, 9, 18, 36, 56. The 50th value falls in the category where X=6.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The mean of $25$ observations is $73.408$. If one observation $64$ is removed, the revised mean is ______.

  1. $72.8$
  2. $73.8$
  3. $80.8$
  4. $76.8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Original sum = 25 * 73.408 = 1835.2. New sum = 1835.2 - 64 = 1771.2. New mean = 1771.2 / 24 = 73.8.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

The correct relation between variance and standard deviation (S.D) of a variable X is _______.

  1. S.D = Var
  2. $S.D =[ Var(X)^{\frac{1}{2}}]$
  3. $S.D = [Var(x)]^2$

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

Standard deviation is the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data.

Variance is the mean of the squares of the deviations from the mean. 
Standard deviation is the square root of variance or variance is the square of standard deviation. 
S.D = {Var(X)}1/2 

or 
Var(X) = (S.D)2 

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

Given mean = $70.2$ and mode = $70.5$, find median using empirical relationship among them.

  1. $70.3$
  2. $70.5$
  3. $70.6$
  4. $70.4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The empirical relationship is Mode = 3*Median - 2*Mean. 70.5 = 3*Median - 2*(70.2). 70.5 = 3*Median - 140.4. 3*Median = 210.9. Median = 70.3.