Test: Moments, Skewness and Kurtosis - SSC CGL MCQ

To find the standard deviation (SD) of the difference between two independent random variables X and Y, you can use the following formula:

SD(X - Y) = sqrt[SD(X)^2 + SD(Y)^2]

Given that SD(X) = 6 and SD(Y) = 8, you can substitute these values into the formula:

SD(X - Y) = sqrt[(6^2) + (8^2)]
SD(X - Y) = sqrt[36 + 64]
SD(X - Y) = sqrt(100)
SD(X - Y) = 10

So, the standard deviation of X - Y is 10.

Therefore, the correct answer is B: 10.

The Coefficient of Variation is a relative measure of variation that standardizes the standard deviation of a dataset by dividing it by the mean of that dataset, expressed as a percentage. It is a useful tool for comparing the relative variation or dispersion of different datasets, especially when the datasets have different units or scales.

The moments about the mean are called "Central moments." These moments are calculated by raising the deviations of data points from the mean to various powers. Central moments provide information about the shape and variability of a probability distribution or dataset relative to its mean.

This is because the third moment about the mean is related to skewness, and if it is zero, it means that there is no skewness present in the distribution. In other words, the data is evenly distributed on both sides of the mean, which is a characteristic of a symmetrical distribution.

In a mesokurtic distribution (which is a normal distribution), the fourth central moment is equal to 3 times the standard deviation to the fourth power. That is:

Fourth Central Moment (µ4) = 3 * (Standard Deviation)^4

Given that µ4 = 243, we can solve for the standard deviation:

243 = 3 * (Standard Deviation)^4

Now, divide both sides by 3:

81 = (Standard Deviation)^4

To find the standard deviation, take the fourth root of both sides:

Standard Deviation = (81)^(1/4)

Standard Deviation = 3^(4/4)

Standard Deviation = 3^1

Standard Deviation = 3

So, the correct answer is indeed D: 3.

When all the scores on an examination cluster around the mean, it indicates that the dispersion of the scores is small. In such a case, the data points are closely packed around the mean, and there is little variability or spread in the scores.

To find the range of a set of scores, you simply subtract the minimum score from the maximum score.

In this case, the minimum score is 3, and the maximum score is 143.

Range = Maximum score - Minimum score
Range = 143 - 3
Range = 140

So, the correct answer is A: 140.

The ratio of the standard deviation to the arithmetic mean expressed as a percentage is called the "Coefficient of Variation."

The formula for the Coefficient of Variation (CV) is:

CV = (Standard Deviation / Mean) * 100%

So, the correct answer is D: Coefficient of Variation.

To find the standard deviation of a set of values, you can use the formula:

Standard Deviation = √[Σ(xi - μ)² / N]

Where:

xi represents each individual value in the dataset.
μ represents the mean of the dataset.
N is the number of values in the dataset.
First, let's calculate the mean of the values 4, 8, 12, and 16:

Mean = (4 + 8 + 12 + 16) / 4
Mean = 40 / 4
Mean = 10

Now, we can calculate the standard deviation of the values 4, 8, 12, and 16 using the same formula:

Standard Deviation = √[Σ(xi - μ)² / N]

Standard Deviation = √[((4 - 10)² + (8 - 10)² + (12 - 10)² + (16 - 10)²) / 4]

Standard Deviation = √[(36 + 4 + 4 + 36) / 4]

Standard Deviation = √[80 / 4]

Standard Deviation = √20

Standard Deviation ≈ 4.472

So, the correct answer is B: 4.472.

Moment ratios β1 and β2 are unitless quantities. This means that they are not dependent on the origin or scale of measurement and are expressed in the original units of the data. Moment ratios are dimensionless statistical measures used to describe the shape and characteristics of a probability distribution. They provide information about skewness (β1) and kurtosis (β2) and are independent of the units in which the data is measured.

So, the correct answer is C: Unitless quantities.

If the mean is less than the median in a distribution and the standard deviation is relatively large, the distribution is considered to be "Negatively skewed." In a negatively skewed distribution, the tail of the distribution extends more to the left, which means that there are some lower values pulling the mean to the left of the median.

So, the correct answer is C: Negatively skewed.

When all the observations in a dataset are increased by a constant percentage, the variance of the increased observations will become the square of the original variance multiplied by the square of the constant percentage increase.

In this case, all the observations are increased by 100%, which is equivalent to multiplying each observation by 2 (100% as a decimal is 1, so 100% increase means multiplying by 1 + 1 = 2).

So, the variance of the increased observations will be:

New Variance = (Original Variance) * (Percentage Increase)^2
New Variance = 50 * (2^2)
New Variance = 50 * 4
New Variance = 200

Therefore, the correct answer is B: 200.

To determine the skewness of a distribution based on its moments about the mean, we can use the following criteria:

If the first moment about the mean (μ1) is equal to 0, the distribution is symmetrical.
If the third moment about the mean (μ3) is greater than 0, the distribution is positively (right) skewed.
If the third moment about the mean (μ3) is less than 0, the distribution is negatively (left) skewed.
In this case, you mentioned that the first three moments about the mean are 1, 4, and 0, respectively.

μ1 = 1, which is not equal to 0.
μ3 = 0, which is equal to 0.
Since the first moment is not equal to 0, it suggests that the distribution is not perfectly symmetrical. However, since the third moment (μ3) is equal to 0, it indicates that there is no skewness in the distribution.

Therefore, based on the given moments about the mean, the distribution is symmetrical (option A).

Bowley's coefficient of skewness is calculated using the following formula:

Bowley's Skewness = (Q1 + Q3 - 2*Median) / (Q3 - Q1)

Where:

Q1 is the first quartile (25th percentile).
Q3 is the third quartile (75th percentile).
Median is the second quartile (50th percentile).
Bowley's coefficient of skewness measures the skewness of a distribution. It can take both positive and negative values, depending on whether the distribution is positively skewed (tail on the right) or negatively skewed (tail on the left).

Typically, the coefficient of skewness is considered positive if the distribution is positively skewed and negative if the distribution is negatively skewed. The value lies between -1 and +1 for most practical datasets. Values closer to -1 indicate stronger negative skewness, while values closer to +1 indicate stronger positive skewness.

So, the correct answer is B: 1 and +1, as Bowley's coefficient of skewness can take values in this range.