1-Math-Obj-5.4

5.4.1 - Introduction to Skewness.

1. What does skewness measure in a dataset?

(A) Central tendency
(B) The symmetry of data distribution
(C) The variability of data
(D) The relationship between mean and median

Correct Answer: (B) The symmetry of data distribution

2. A distribution with a right tail longer than the left is called:

(A) Symmetrical
(B) Positively skewed
(C) Negatively skewed
(D) Normally distributed

Correct Answer: (B) Positively skewed

3. When the mean is greater than the median in a dataset, the distribution is likely to be:

(A) Negatively skewed
(B) Positively skewed
(C) Symmetrical
(D) Uniform

Correct Answer: (B) Positively skewed

4. In a negatively skewed distribution, the tail is:

(A) On the left side
(B) On the right side
(C) Symmetrical
(D) Absent

Correct Answer: (A) On the left side

5. If a dataset has no skewness, it is referred to as:

(A) Symmetrical
(B) Positively skewed
(C) Negatively skewed
(D) Uniformly distributed

Correct Answer: (A) Symmetrical

6. A perfectly symmetrical distribution has a skewness value of:

(A) +1
(B) -1
(C) 0
(D) Undefined

Correct Answer: (C) 0

7. Skewness is particularly important for analyzing:

(A) Variability
(B) Data symmetry and tail behavior
(C) Central tendency
(D) Correlation between variables

Correct Answer: (B) Data symmetry and tail behavior

8. Which of the following is an example of a negatively skewed distribution?

(A) Income distribution in a developed country
(B) Age of retirement among adults
(C) Distribution of exam scores where most students score above average
(D) Life expectancy in a population

Correct Answer: (C) Distribution of exam scores where most students score above average

9. The skewness of a normal distribution is:

(A) 1
(B) 0
(C) -1
(D) Infinity

Correct Answer: (B) 0

10. A positively skewed distribution is often observed in:

(A) Income distributions
(B) Normally distributed data
(C) Height data for adults
(D) Exam scores where most students fail

Correct Answer: (A) Income distributions

11. If the skewness of a distribution is -0.5, it is:

(A) Symmetrical
(B) Positively skewed
(C) Negatively skewed
(D) Normal

Correct Answer: (C) Negatively skewed

12. The skewness coefficient is typically measured by:

(A) Pearson’s coefficient of skewness
(B) Variance
(C) Standard deviation
(D) Mode

Correct Answer: (A) Pearson’s coefficient of skewness

13. Which statement is true about a symmetrical distribution?

(A) The mean, median, and mode are all different
(B) The mean, median, and mode are equal
(C) The mode is greater than the mean
(D) The median is greater than the mean

Correct Answer: (B) The mean, median, and mode are equal

14. In a positively skewed distribution, which is the correct order of the measures of central tendency?

(A) Mean > Median > Mode
(B) Mode > Mean > Median
(C) Median > Mean > Mode
(D) Mean = Median = Mode

Correct Answer: (A) Mean > Median > Mode

15. Skewness is an indicator of:

(A) The spread of data
(B) The direction of data symmetry
(C) The central value of the data
(D) The highest value in a dataset

Correct Answer: (B) The direction of data symmetry

5.4.2 - Karl Persion's Coefficient of Skewness.

16. What is Karl Pearson's Coefficient of Skewness used to measure?

(A) The central tendency of a dataset
(B) The spread of data points
(C) The asymmetry of a data distribution
(D) The relationship between variables

Correct Answer: (C) The asymmetry of a data distribution

17. Karl Pearson's Coefficient of Skewness formula is:

(A) 𝑀𝑒𝑎𝑛 − 𝑀𝑒𝑑𝑖𝑎𝑛 / 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
(B) 𝑀𝑒𝑑𝑖𝑎𝑛 − 𝑀𝑜𝑑𝑒 / 𝑅𝑎𝑛𝑔𝑒
(C) 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 / 𝑀𝑒𝑎𝑛 − 𝑀𝑒𝑑𝑖𝑎𝑛
(D) 𝑀𝑒𝑎𝑛 + 𝑀𝑜𝑑𝑒 / 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

Correct Answer: (A) 𝑀𝑒𝑎𝑛 − 𝑀𝑒𝑑𝑖𝑎𝑛 / 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

18. Karl Pearson's Coefficient of Skewness helps in understanding:

(A) Whether data is skewed to the left or right
(B) The central value of the dataset
(C) The data variability
(D) The distribution of quartiles

Correct Answer: (A) Whether data is skewed to the left or right

19. If Karl Pearson’s Coefficient of Skewness is positive, the distribution is:

(A) Symmetrical
(B) Positively skewed
(C) Negatively skewed
(D) Normally distributed

Correct Answer: (B) Positively skewed

20. When Karl Pearson's Coefficient of Skewness is negative, it indicates that the distribution has:

(A) More values in the right tail
(B) More values in the left tail
(C) A symmetrical distribution
(D) No skewness

Correct Answer: (B) More values in the left tail

21. What does it imply if Karl Pearson's Coefficient of Skewness equals 0?

(A) The distribution is positively skewed
(B) The distribution is negatively skewed
(C) The distribution is symmetrical
(D) The data set has no variability

Correct Answer: (C) The distribution is symmetrical

22. In a positively skewed distribution, the order of mean, median, and mode is generally:

(A) Mean < Median < Mode
(B) Mean > Median > Mode
(C) Mode > Median > Mean
(D) Median = Mode > Mean

Correct Answer: (B) Mean > Median > Mode

23. The value of Karl Pearson’s Coefficient of Skewness typically lies between:

(A) -3 and +3
(B) 0 and 1
(C) -1 and +1
(D) -5 and +5

Correct Answer: (A) -3 and +3

24. If a dataset has a mean of 50, a median of 40, and a standard deviation of 10, what is the Pearson’s Coefficient of Skewness?

(A) +1.0
(B) -1.0
(C) 0.0
(D) +0.5

Correct Answer: (A) +1.0

25. What does a large positive value of Pearson’s Coefficient of Skewness indicate?

(A) Extreme left skew
(B) Extreme right skew
(C) Symmetry in distribution
(D) No relationship between mean and median

Correct Answer: (B) Extreme right skew

26. Pearson’s Coefficient of Skewness is sensitive to which of the following?

(A) Outliers in the dataset
(B) Changes in sample size
(C) Changes in mode
(D) Variability in data quartiles

Correct Answer: (A) Outliers in the dataset

27. For which types of distributions is Karl Pearson's Coefficient of Skewness most appropriate?

(A) Symmetrical distributions
(B) Normal distributions
(C) Moderately skewed distributions
(D) Distributions with a high number of outliers

Correct Answer: (C) Moderately skewed distributions

28. A coefficient of skewness close to -3 indicates:

(A) A highly negatively skewed distribution
(B) A highly positively skewed distribution
(C) A symmetrical distribution
(D) No skewness

Correct Answer: (A) A highly negatively skewed distribution

29. Which measure of central tendency is most affected by extreme values in a distribution, causing skewness?

(A) Mean
(B) Median
(C) Mode
(D) Range

Correct Answer: (A) Mean

30. If the mean and median of a dataset are equal, Karl Pearson’s Coefficient of Skewness will be:

(A) Positive
(B) Negative
(C) Zero
(D) Undefined

Correct Answer: (C) Zero

5.4.3 - Bowley's Coefficient of Skewness.

31. What does Bowley's Coefficient of Skewness measure?

(A) The central tendency of a dataset
(B) The asymmetry of a data distribution
(C) The variability of data
(D) The strength of correlation

Correct Answer: (B) The asymmetry of a data distribution

32. Bowley’s Coefficient of Skewness is based on which of the following measures?

(A) Mean and variance
(B) Median and quartiles
(C) Mode and range
(D) Standard deviation and mean

Correct Answer: (B) Median and quartiles

33. The formula for Bowley’s Coefficient of Skewness is:

(A) 𝑄₃ + 𝑄₁ − 2 × 𝑀𝑒𝑑𝑖𝑎𝑛 / 𝑄₃ − 𝑄₁
(B) 𝑀𝑒𝑎𝑛 − 𝑀𝑒𝑑𝑖𝑎𝑛 / 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
(C) 𝑄₃ − 𝑀𝑒𝑑𝑖𝑎𝑛 / 𝑄₃ − 𝑄₁
(D) 𝑀𝑒𝑑𝑖𝑎𝑛 − 𝑄₁ / 𝑄₃ − 𝑀𝑒𝑑𝑖𝑎𝑛

Correct Answer: (A) 𝑄₃ + 𝑄₁ − 2 × 𝑀𝑒𝑑𝑖𝑎𝑛 / 𝑄₃ − 𝑄₁

34. A Bowley’s Coefficient of Skewness value greater than 0 indicates a distribution that is:

(A) Symmetrical
(B) Negatively skewed
(C) Positively skewed
(D) Uniform

Correct Answer: (C) Positively skewed

35. If Bowley’s Coefficient of Skewness equals 0, the distribution is:

(A) Positively skewed
(B) Negatively skewed
(C) Symmetrical
(D) Bimodal

Correct Answer: (C) Symmetrical

36. A Bowley’s Coefficient of Skewness less than 0 indicates a distribution that is:

(A) Symmetrical
(B) Positively skewed
(C) Negatively skewed
(D) Uniformly distributed

Correct Answer: (C) Negatively skewed

37. In the context of Bowley's Coefficient, what do 𝑄₁ and 𝑄₃ represent?

(A) Mean and median
(B) First and third quartiles
(C) Minimum and maximum values
(D) Mode and range

Correct Answer: (B) First and third quartiles

38. Which of the following distributions is most appropriate for calculating Bowley’s Coefficient of Skewness?

(A) Uniform distribution
(B) Discrete distribution with extreme values
(C) Continuous distribution with outliers
(D) Any distribution with quartiles available

Correct Answer: (D) Any distribution with quartiles available

39. Bowley’s Coefficient of Skewness is particularly useful because it:

(A) Is less affected by outliers than the mean
(B) Requires only the mean and standard deviation
(C) Is based solely on the total number of observations
(D) Utilizes the mode for calculation

Correct Answer: (A) Is less affected by outliers than the mean

40. Which of the following values indicates a perfectly symmetrical distribution using Bowley's Coefficient?

(A) 1
(B) 0
(C) -1
(D) Undefined

Correct Answer: (B) 0

41. Bowley’s Coefficient is particularly beneficial in the analysis of:

(A) Normally distributed data
(B) Skewed data
(C) Multimodal distributions
(D) Bivariate data

Correct Answer: (B) Skewed data

42. Which scenario is best suited for the application of Bowley's Coefficient of Skewness?

(A) Data with a normal distribution
(B) Income distribution data
(C) Height data in a controlled environment
(D) Data from a random sample with equal variance

Correct Answer: (B) Income distribution data

43. In Bowley’s Coefficient of Skewness, which quartile is subtracted from 𝑄₃?

(A) Median
(B) 𝑄₂
(C) 𝑄₁
(D) Mode

Correct Answer: (C) 𝑄₁

44. What does a Bowley's Coefficient of Skewness close to +1 indicate?

(A) Strong positive skewness
(B) Strong negative skewness
(C) No skewness
(D) Symmetrical distribution

Correct Answer: (A) Strong positive skewness

45. The interpretation of Bowley’s Coefficient of Skewness allows analysts to:

(A) Predict future data trends
(B) Assess the risk of extreme values
(C) Understand data symmetry and distribution characteristics
(D) Calculate the central value of data

Correct Answer: (C) Understand data symmetry and distribution characteristics

5.4.4 - Introduction to Kurtosis.

46. What does kurtosis measure in a statistical distribution?

(A) The spread of the data
(B) The symmetry of the distribution
(C) The peakedness of the distribution
(D) The central tendency of the data

Correct Answer: (C) The peakedness of the distribution

47. How many types of kurtosis are generally identified?

(A) Two types
(B) Three types
(C) Four types
(D) Five types

Correct Answer: (B) Three types

48. Which of the following describes a distribution with positive kurtosis?

(A) More spread out with lower peaks
(B) Less spread out with higher peaks
(C) Symmetrical with a flat peak
(D) Uniform distribution

Correct Answer: (B) Less spread out with higher peaks

49. What does a kurtosis value of zero indicate?

(A) Normal distribution
(B) Uniform distribution
(C) Heavy-tailed distribution
(D) Light-tailed distribution

Correct Answer: (A) Normal distribution

50. Which term is used to describe a distribution with a kurtosis greater than three?

(A) Platykurtic
(B) Mesokurtic
(C) Leptokurtic
(D) Normal kurtosis

Correct Answer: (C) Leptokurtic

51. What is the kurtosis value associated with a platykurtic distribution?

(A) Less than 3
(B) Exactly 3
(C) Greater than 3
(D) Undefined

Correct Answer: (A) Less than 3

52. The excess kurtosis is calculated by subtracting which value from the kurtosis?

(A) 1
(B) 2
(C) 3
(D) 4

Correct Answer: (C) 3

53. In statistical analysis, why is kurtosis important?

(A) It helps to identify trends over time
(B) It aids in understanding the likelihood of extreme values
(C) It provides insight into the central tendency of data
(D) It determines the sample size needed for analysis

Correct Answer: (B) It aids in understanding the likelihood of extreme values

54. Which of the following is NOT a characteristic of a leptokurtic distribution?

(A) Higher peak
(B) Thinner tails
(C) Increased probability of extreme values
(D) Higher kurtosis

Correct Answer: (B) Thinner tails

55. When analyzing financial returns, why is kurtosis particularly significant?

(A) It indicates average returns
(B) It shows volatility
(C) It measures risk of extreme losses or gains
(D) It determines the return on investment

Correct Answer: (C) It measures risk of extreme losses or gains

56. In the context of kurtosis, what does a high kurtosis value imply about the distribution?

(A) It is uniformly distributed
(B) It has a high probability of outliers
(C) It is normally distributed
(D) It has a flat peak

Correct Answer: (B) It has a high probability of outliers

57. Which statistical software can be used to calculate kurtosis?

(A) Microsoft Word
(B) Microsoft Excel
(C) Adobe Photoshop
(D) Any word processor

Correct Answer: (B) Microsoft Excel

58. How does kurtosis differ from skewness?

(A) Kurtosis measures peak while skewness measures symmetry
(B) Kurtosis is only used for continuous data
(C) Skewness is not a measure of distribution shape
(D) There is no difference

Correct Answer: (A) Kurtosis measures peak while skewness measures symmetry

59. Which of the following distributions is an example of a platykurtic distribution?

(A) Normal distribution
(B) Uniform distribution
(C) Exponential distribution
(D) Binomial distribution

Correct Answer: (B) Uniform distribution

60. What is the implication of a low kurtosis value in data analysis?

(A) The data has a high probability of extreme values
(B) The data is concentrated around the mean
(C) The data has a flat peak with less variability
(D) The data is perfectly symmetrical

Correct Answer: (C) The data has a flat peak with less variability