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

If each observation of set is divided by $10$, the S.D of the new observations is ______.

  1. $10$ times of S.D of original obs.
  2. $\frac{1}{100}$th

  3. Not changed
  4. $\frac{1}{10}$th
Reveal answer Fill a bubble to check yourself
D 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. So the deviations are affected by division and multiplication. Therefore, if each observation of the set id divided by 10 then the whole standard deviation also becomes 1/10 th of the prior standard deviation. 

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 C.V of a distribution is $80\%$ and the mean of the distribution is $40$, the S.D of the distribution is ________.

  1. $33$
  2. $32$
  3. $35$
  4. $0.30$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Coefficient of variation is the coefficient of dispersion based on the standard deviation of the statistical series.

Coefficient of variation = ( standard deviation / mean )

=> 80 /100 = S.D / 40 

=> S.D = 32 

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?

  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 the square root of the arithmetic mean of the squares of the deviations measured from the arithmetic mean of the data. Standard deviation is not affected by the increase of decrease of observations in the series. 

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

Standard deviation = { (sum of the squares of the observations/ number of observations ) – (sum of observations/ number of observations ) } ½

                               = { (1444) –(18.4) } ½

                                = (1425.6) ½

                                = 37.75 

Coefficient of variation is the coefficient of dispersion based on the standard deviation of the statistical series.

Coefficient of variation = ( standard deviation / mean )x 100 

                                     = ( 37.75/ 18.4)x100

                                     = 20.51 


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 relation between variance and standard deviation is ________.

  1. variance is the square root of standard Deviation

  2. square of the standard deviation is equal to Variance

  3. variance is equal to standard deviation

  4. standard deviation is the square of the variance
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

A student obtained the mean and the standard deviation of 100 observations as 40 and 5.1. It was later found that one observation was wrongly copied as 50, the correct figure being 40. Find the correct mean and the S. D.

  1. Mean = 38.8, S. D. = 5

  2. Mean = 39.9, S. D. = 5

  3. Mean = 39.9, S. D. = 4

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
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 the coefficient of correlation between $x,y$ is $0.7$ and covariance is $35$ find the standard deviation of $y$ if the standard deviation of $x$ is $5$____.

  1. $10$
  2. $9$
  3. $8$
  4. $11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for covariance is Cov(x,y) = r * SD(x) * SD(y). Plugging in the values: 35 = 0.7 * 5 * SD(y). Thus, 35 = 3.5 * SD(y), which means SD(y) = 35 / 3.5 = 10.

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 sample of 30 observations is 1532. If the value of each item of the observation is increased by 5. The new standard deviation will be ______.

  1. same as original

  2. increased by 5

  3. reduced by 5

  4. reduced by 5 times.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Standard deviation is a measure of dispersion. Adding a constant to every observation shifts the data but does not change the spread or distance between points, so the standard deviation remains unchanged.

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

From the following data calculate the coefficient of concurrent deviation:
Number of pairs of observations=94
Number of pairs of concurrent deviation=34

  1. -0.5

  2. 0.06

  3. 1.5

  4. 0.09
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the coefficient of concurrent deviation is r_c = +/- sqrt(+/- (2c - n) / n). Here, n = 94 and c = 34. The number of deviations is n-1 = 93. The number of concurrent deviations is 34. The formula is r_c = +/- sqrt(abs((2c - n) / (n - 1))). Calculation: (2*34 - 93) / 93 = (68 - 93) / 93 = -25 / 93 approx -0.268. The square root of 0.268 is approx 0.51. Given the options, -0.5 is the intended answer.

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

For a distribution, coefficient of variation is 22.5% and mean is 7.5 find the standard deviation___.

  1. 1.9975

  2. 1.6875

  3. 1.243

  4. 1.0943
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Coefficient of Variation (CV) = (SD / Mean) * 100. Given CV = 22.5% and Mean = 7.5, then 22.5 = (SD / 7.5) * 100. SD = (22.5 * 7.5) / 100 = 168.75 / 100 = 1.6875.

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 a population has a standard deviation of $2.25$, how large the sample should be to allow a maximum error of $0.33$ with $95\%$ confidence level?

  1. $252$
  2. $201$
  3. $220$
  4. $178$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The formula for sample size n is n = (z * sigma / E)^2. For a 95% confidence level, z = 1.96. Given sigma = 2.25 and E = 0.33, n = (1.96 * 2.25 / 0.33)^2 = (4.41 / 0.33)^2 = (13.3636)^2 = 178.58. Rounding up gives 179, but 178 is the closest option.

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 $\sigma$ of the first $N$ natural numbers can be obtained using which one of the following formula?

  1. $\sigma =\cfrac { { N }^{ 2 }-1 }{ 12 } $
  2. $\sigma =\sqrt { \cfrac { { N }^{ 2 }-1 }{ 12 } } $
  3. $\sigma =\sqrt { \cfrac { N-1 }{ 12 } } $
  4. $\sigma =\sqrt { \cfrac { { N }^{ 2 }-1 }{ 6N } } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sigma^2=\dfrac{1}{n}\sum _{i=1}^{n}i^2-\left(\dfrac{1}{n}\sum _{i=1}^{n}i\right)^2$


     $=\dfrac{1}{n}(1^2+2^2+...+n^2)-\left(\dfrac{1}{n}(1+2+...+n)\right)^2$

     $=\dfrac{1}{n} \times \dfrac{n(n+1)(2n+1)}{6}-(\dfrac{n+1}{2})^2=\dfrac{n^2-1}{12}$

$\therefore \sigma=\sqrt{\dfrac{n^2-1}{12}}$

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 researcher has collected the following sample data. The mean of the sample is 5.
3  5  12  3  2
The standard deviation is...............

  1. 8.944

  2. 4.062

  3. 13.2

  4. 16.5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Data: 3, 5, 12, 3, 2. Mean = (3+5+12+3+2)/5 = 25/5 = 5. Variance = [(3-5)^2 + (5-5)^2 + (12-5)^2 + (3-5)^2 + (2-5)^2] / 5 = [4 + 0 + 49 + 4 + 9] / 5 = 66 / 5 = 13.2. SD = sqrt(13.2) = 3.63. If using sample SD (divide by n-1=4), Variance = 66/4 = 16.5, SD = sqrt(16.5) = 4.062.