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

Which of the following is a common regularization technique used in linear regression?

  1. L1 regularization (Lasso)

  2. L2 regularization (Ridge)

  3. Elastic Net regularization

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

L1 regularization (Lasso), L2 regularization (Ridge), and Elastic Net regularization are all commonly used regularization techniques in linear regression. L1 regularization adds a penalty term to the sum of absolute coefficients, L2 regularization adds a penalty term to the sum of squared coefficients, and Elastic Net regularization is a combination of both L1 and L2 regularization.

Multiple choice

The mean of a binomial distribution is given by:

  1. \(np\)
  2. \(nq\)
  3. \(np + nq\)
  4. \(np - nq\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The mean of a binomial distribution is given by (np).

Multiple choice

In a simple random sample, each member of the population has an:

  1. Equal chance of being selected

  2. Unequal chance of being selected

  3. No chance of being selected

  4. Chance of being selected that depends on their characteristics
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Simple random sampling ensures that every member of the population has an equal opportunity to be included in the sample, eliminating bias and increasing the representativeness of the sample.

Multiple choice

What is the mean of a normal distribution with a mean of 10 and a standard deviation of 2?

  1. 10

  2. 12

  3. 8

  4. 14
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The mean of a normal distribution is the center of the distribution, and it is equal to the expected value of the random variable. In this case, the mean is given as 10, so the mean of the normal distribution is 10.

Multiple choice

What is the standard deviation of a normal distribution with a mean of 10 and a variance of 4?

  1. 2

  2. 4

  3. 8

  4. 16
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The standard deviation of a normal distribution is the square root of the variance. In this case, the variance is given as 4, so the standard deviation is √4 = 2.

Multiple choice

What is the probability of getting a value between 8 and 12 from a normal distribution with a mean of 10 and a standard deviation of 2?

  1. 0.3413

  2. 0.6826

  3. 0.9545

  4. 0.9973
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The probability of getting a value between 8 and 12 from a normal distribution with a mean of 10 and a standard deviation of 2 can be calculated using the standard normal distribution (z-distribution). First, we need to convert the values 8 and 12 to z-scores using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Plugging in the values, we get z = (8 - 10) / 2 = -1 and z = (12 - 10) / 2 = 1. Then, we can use a standard normal distribution table or calculator to find the probability of getting a z-score between -1 and 1. The probability is 0.6826.

Multiple choice

What is the significance of the Doob's Martingale Convergence Theorem?

  1. It provides conditions for the convergence of Martingales

  2. It establishes the existence of a unique solution to a stochastic differential equation

  3. It relates Martingales to Brownian motion

  4. It characterizes the asymptotic behavior of Martingales
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Doob's Martingale Convergence Theorem provides sufficient conditions for the convergence of Martingales, ensuring that they have well-defined limits.

Multiple choice

What is the significance of the square-integrable condition in the context of Martingales?

  1. It ensures the existence of a unique solution to a stochastic differential equation

  2. It provides conditions for the convergence of Martingales

  3. It characterizes the asymptotic behavior of Martingales

  4. It allows for the application of certain mathematical techniques
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The square-integrable condition allows for the application of powerful mathematical techniques, such as stochastic calculus, to study the properties of Martingales.

Multiple choice

What is the mathematical function used in Linear Regression to model the relationship between input and output variables?

  1. Sigmoid Function

  2. Cost Function

  3. Hypothesis Function

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

In Linear Regression, the hypothesis function is a linear equation that represents the relationship between the input variables and the output variable.

Multiple choice

What is the expected value of the sum of two fair dice?

  1. 7

  2. 8

  3. 9

  4. 10
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expected value of a random variable X is given by the formula E(X) = ΣxP(X = x), where x is the value of the random variable and P(X = x) is the probability of X taking the value x. In this case, the random variable X is the sum of two fair dice. The possible values of X are 2, 3, 4, ..., 12. The probability of getting each value is 1/36. Therefore, the expected value of X is E(X) = 2(1/36) + 3(1/36) + 4(1/36) + ... + 12(1/36) = 7.

Multiple choice

What is the median of the following set of numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19?

  1. 9

  2. 10

  3. 11

  4. 12
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The median of a set of numbers is the middle value when the numbers are arranged in ascending order. In this case, the numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. The middle value is the 5th value. Therefore, the median of the set of numbers is 10.

Multiple choice

What is the standard deviation of the following set of numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19?

  1. 5.66

  2. 6.66

  3. 7.66

  4. 8.66
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The standard deviation of a set of numbers is a measure of how spread out the numbers are. It is calculated by taking the square root of the variance. The variance is calculated by finding the average of the squared differences between each number and the mean. In this case, the mean of the set of numbers is 10. The variance is calculated as follows: [(1 - 10)^2 + (3 - 10)^2 + (5 - 10)^2 + (7 - 10)^2 + (9 - 10)^2 + (11 - 10)^2 + (13 - 10)^2 + (15 - 10)^2 + (17 - 10)^2 + (19 - 10)^2] / 10 = 44.44. The standard deviation is the square root of the variance, which is √44.44 = 6.66.

Multiple choice

Which of the following is NOT a common type of forecasting conference?

  1. International Symposium on Forecasting

  2. European Conference on Forecasting

  3. Asia-Pacific Symposium on Forecasting

  4. American Statistical Association Annual Meeting
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The American Statistical Association Annual Meeting is a general statistics conference, not a forecasting conference.

Multiple choice

In a normal distribution, what is the area under the curve between the mean and one standard deviation?

  1. 0.3413

  2. 0.6826

  3. 0.9545

  4. 0.9973
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a normal distribution, the area under the curve between the mean and one standard deviation is approximately 0.3413.

Multiple choice

In a data set {1, 2, 3, 4, 5}, what is the mean?

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Mean is the sum of all values divided by the number of values in a dataset. In this case, (1+2+3+4+5) / 5 = 3.