৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

Explanation: 
CV is a relative measure of dispersion, expressed as a percentage.
It allows comparison between datasets with different units or means.

Source: Business Statistics, Md. Abdul Aziz.

Explanation: 
In a positively skewed distribution, the tail is on the right side, so the mean is greater than the median. 
A positively skewed distribution, also known as a right-skewed distribution, is characterized by most of the data being concentrated on the left side, with a long tail extending to the right. 
This type of distribution is asymmetrical and often occurs in real-world scenarios such as income levels, stock returns, and hospital stay durations. 



Source: Analytics Yogi, Analytics Vidya.

Explanation: 
Leptokurtic distributions have sharper peaks and heavier tails compared to a normal distribution, indicating more extreme values.

Here are the definitions of leptokurtic, mesokurtic, and platykurtic distributions:

Leptokurtic: Distributions with high kurtosis (fat tails) that are more outlier-prone than a normal distribution.
They have a sharper peak and heavier tails, indicating more values in the tails and closer to the mean.

Mesokurtic: Distributions with medium kurtosis (normal distribution). 
They have a moderate peak and tails, similar to the normal distribution.

Platykurtic: Distributions with low kurtosis (thin tails) that are less outlier-prone than a normal distribution. 
They have a flatter peak and lighter tails, indicating fewer values in the tails.

                           

Source: Investopedia

Explanation: 
Range is calculated as the difference between the highest and lowest values.
It's very sensitive to outliers. For a dataset 1,2,4,8,50. 
Range= 50-1=49 Range is hugely affected by outlier 50 here.

Explanation: 
The coefficient of variation is used to compare two or more data sets. 
A lower CV indicates less relative variability and therefore more consistency.

Source: Investopedia 

Explanation:
For any constant a, 
​Var⁡(aX) = a²Var⁡(X). If $2$, 
​variance becomes $=4$ times the original (quadrupled).

Explanation:
When the moment kurtosis β₂ is greater than 3, the distribution is leptokurtic — it has heavier tails and more probability in extremes than a normal distribution. 

Source: Business Statistics, Md. Abdul Aziz

Explanation :
Chebyshev rule uses the formula : 1−1/k² to find the minimum proportion of data within k standard deviation. (where k must be greater than 1)

Here,
k=1.5
 so, k²=1.5×1.5=2.25.
Then, 1−1/2.25=1−0.444…=0.555…→ rounded = 55.56%.

Explanation:
In a normal (bell-shaped) distribution, the empirical (68–95–99.7) rule says about 68% of observations lie within ±1 SD of the mean.
Here, 90–110 is mean ±10 (±1 SD), so ≈68%.
​          

Explanation:
CV (%) = (SD / Mean) × 100.
SD / Mean = 5 ÷ 50 = 0.1.
0.1 × 100 = 10%.

Explanation: 
Range, mean deviation, and standard deviation are measures of dispersion (they describe spread).
The mean is a measure of central tendency, not dispersion. 

Explanation:
The median depends only on order; extreme values (very large/small) do not change the middle rank much, so median is robust to outliers.
That is why median is preferred over to mean for skewed data and outliers.

Source: Live MCQ lecture.

Explanation:
β₂ = μ₄/μ₂^2 
​= 150/4^2
​= 150/16 
​= 9.375

Since β2 > 3, the distribution is Leptokurtic (heavier tails than normal).
Basic definition of Kurtosis states that a distribution is leptokurtic if β₂ > 3.

Explanation:
Formula:
Skewness = 3(Mean−Median)/σ

Explanation:
β2 is called the coefficient of kurtosis.
For a normal distribution, β2 = 3.
γ2 = β2 - 3 → γ2 = 0 (i.e., mesokurtic distribution.)

Explanation:
Multiplying by 3 → mean becomes 60, SD becomes 15.
Adding 10 → mean becomes 70, SD unchanged at 15.
CV = (SD / Mean) × 100 = (15 / 70) × 100 = 21.428...%

Explanation: 
Dispersion measures the variability or spread of data points in a dataset.
It shows how far the data points lie from the mean or median, providing more insight than central tendency alone. 
Measures include range, standard deviation, variance, etc.
Dispersion is also affected by outliers or extreme values. (Range)

Explanation: 
Chebyshev’s Rule applies to any distribution, regardless of shape, and states that at least 1−(1/k^2) proportion of the data lies within k standard deviations of the mean (for k>1). 
It is especially useful when the data is not normally distributed.

source: Live MCQ class lecture 

Explanation: 
Variance measures the average squared deviation from the mean. 
Standard deviation is simply the square root of the variance, making it expressed in the same units as the original data.

Explanation: 
CV = (Standard Deviation ÷ Mean) × 100%. 
​It is a relative measure of dispersion, allowing comparison between datasets with different scales or units.

Explanation: 
A boxplot briefly reveals a dataset’s center (median), spread (IQR), range, skewness pattern, and outliers — all in one visual.

​Source: Business Statistics, S.P. Gupta, M.P. Gupta.

Explanation: 
The third central moment (μ₃) measures skewness. If μ₃ = 0, it means the distribution is symmetric about the mean (skewness = 0).

Source: Business Statistics, Md. Abdul Aziz.

Explanation: 
The box length is? 3 − ?1, which is the IQR — it shows the spread of the middle 50% of the data.

​Source: Business Statistics, M.A. Aziz, Investopedia

Explanation: 
A long upper whisker with the median shifted towards Q1 indicates a positive skew — data has a longer right tail.

                                       

Source: Mathsathome.com

Explanation:
Mean deviation (also called average deviation) measures the average distance between each data point and a central value, typically the mean or median. Importantly, it uses absolute values of the deviations to avoid negative results canceling out positive ones.

Option C is correct: It accurately describes how mean deviation is computed.

On the other hand,
Option A is incorrect: Mean deviation can be calculated from either the mean or median, though using the median often gives a smaller value.
Option B is incorrect: Because it uses absolute values, mean deviation is always non-negative.
Option D is incorrect: Mean deviation and standard deviation are different measures and are not generally equal.