Measures of Dispersion

1. What is the range of a dataset?

A) The difference between the largest and smallest values
B) The average of all values
C) The midpoint of the dataset
D) The most frequently occurring value

Correct Answer: (A) The difference between the largest and smallest values

2. Which of the following formulas represents how to calculate the range?

A) Range = Maximum value - Minimum value
B) Range = Mean - Mode
C) Range = Median - Mode
D) Range = Maximum value + Minimum value

Correct Answer: (A) Range = Maximum value - Minimum value

3. In a dataset of {3, 7, 5, 10, 2}, what is the range?

A) 5
B) 8
C) 7
D) 10

Correct Answer: (B) 8

4. Which of the following statements is true regarding the range?

A) It provides information about the central tendency of the data.
B) It is sensitive to outliers.
C) It can be negative.
D) It is a measure of the average distance of values from the mean.

Correct Answer: (B) It is sensitive to outliers.

5. When calculating the range of a dataset, what happens if the smallest and largest values are the same?

A) The range is zero.
B) The range is one.
C) The range is infinite.
D) The range is undefined.

Correct Answer: (A) The range is zero.

6. Which of the following datasets has the largest range?

A) {1, 2, 3, 4, 5}
B) {10, 15, 20, 25, 30}
C) {5, 10, 15, 20, 100}
D) {8, 12, 16, 20}

Correct Answer: (C) {5, 10, 15, 20, 100}

7. Why is the range considered a limited measure of dispersion?

A) It considers all data points.
B) It ignores the distribution of values.
C) It is always positive.
D) It is easy to calculate.

Correct Answer: (B) It ignores the distribution of values.

8. If the range of a dataset is 20 and the minimum value is 15, what is the maximum value?

A) 35
B) 25
C) 20
D) 30

Correct Answer: (A) 35

9. In which situation would the range be a less informative measure of dispersion?

A) When the dataset has outliers
B) When the data is normally distributed
C) When the data has a large sample size
D) When all values are unique

Correct Answer: (A) When the dataset has outliers

10. Which of the following datasets has a range of 0?

A) {2, 2, 2, 2, 2}
B) {1, 2, 3, 4, 5}
C) {5, 10, 15, 20}
D) {7, 8, 9, 10, 11}

Correct Answer: (A) {2, 2, 2, 2, 2}

11. What is quartile deviation?

A) The difference between the maximum and minimum values
B) Half the difference between the first quartile (Q1) and the third quartile (Q3)
C) The average of the dataset
D) The median of the dataset

Correct Answer: (B) Half the difference between the first quartile (Q1) and the third quartile (Q3)

12. How is the quartile deviation calculated?

A) QD = Q3 - Q1
B) QD = (Q3 - Q1) / 2
C) QD = Q1 + Q3
D) QD = (Q1 + Q3) / 2

Correct Answer: (B) QD = (Q3 - Q1) / 2

13. If the first quartile (Q1) is 10 and the third quartile (Q3) is 20, what is the quartile deviation?

A) 5
B) 10
C) 15
D) 20

Correct Answer: (A) 5

14. Which of the following statements about quartile deviation is true?

A) It is affected by extreme values.
B) It provides a measure of the central tendency of the data.
C) It is a measure of variability.
D) It cannot be calculated for grouped data.

Correct Answer: (C) It is a measure of variability.

15. When is the quartile deviation considered a useful measure of dispersion?

A) When the data has a normal distribution
B) When the data contains outliers
C) When the data is skewed
D) When there are few data points

Correct Answer: (C) When the data is skewed

16. What does a small quartile deviation indicate about a dataset?

A) The values are widely spread apart.
B) The values are closely clustered around the median.
C) There are significant outliers in the data.
D) The dataset is uniform.

Correct Answer: (B) The values are closely clustered around the median.

17. Which of the following is NOT a characteristic of quartile deviation?

A) It is always a positive value.
B) It can be calculated using both ungrouped and grouped data.
C) It is sensitive to extreme values.
D) It is a measure of the spread of the middle 50% of the data.

Correct Answer: (C) It is sensitive to extreme values.

18. If a dataset has a quartile deviation of 0, what does this imply?

A) The data has high variability.
B) All values in the dataset are identical.
C) The dataset is normally distributed.
D) The dataset contains outliers.

Correct Answer: (B) All values in the dataset are identical.

19. In a dataset, if Q1 = 15 and Q3 = 30, what is the interquartile range (IQR)?

A) 10
B) 15
C) 30
D) 45

Correct Answer: (B) 15

20. How does quartile deviation relate to interquartile range (IQR)?

A) Quartile deviation is the same as IQR.
B) Quartile deviation is half of the IQR.
C) Quartile deviation is twice the IQR.
D) Quartile deviation and IQR are unrelated.

Correct Answer: (B) Quartile deviation is half of the IQR.

21. What is mean deviation?

A) The average of all values in a dataset.
B) The average of the absolute differences between each data point and the mean.
C) The difference between the maximum and minimum values.
D) The median of the dataset.

Correct Answer: (B) The average of the absolute differences between each data point and the mean.

22. How is mean deviation calculated?

A) MD = Σ|x - mean| / n.
B) MD = Σ(x - mean) / n.
C) MD = Σ|x - median| / n.
D) MD = Σ(x - mode) / n.

Correct Answer: (A) MD = Σ|x - mean| / n.

23. If the dataset is {5, 10, 15}, what is the mean deviation?

A) 5.
B) 10.
C) 15.
D) 0.

Correct Answer: (A) 5.

24. Which of the following statements about mean deviation is true?

A) It is affected by outliers.
B) It can be negative.
C) It is a measure of central tendency.
D) It considers all data points in the calculation.

Correct Answer: (D) It considers all data points in the calculation.

25. Why is mean deviation considered a useful measure of dispersion?

A) It is easy to compute.
B) It provides a precise measure of central tendency.
C) It is robust to outliers.
D) It focuses only on extreme values.

Correct Answer: (A) It is easy to compute.

26. How does mean deviation differ from standard deviation?

A) Mean deviation is always greater than standard deviation.
B) Mean deviation uses absolute values, while standard deviation uses squared values.
C) Mean deviation cannot be used for grouped data.
D) Mean deviation considers only the largest and smallest values.

Correct Answer: (B) Mean deviation uses absolute values, while standard deviation uses squared values.

27. If the mean of a dataset is 50 and the absolute deviations from the mean are {5, 10, 15}, what is the mean deviation?

A) 10.
B) 15.
C) 5.
D) 20.

Correct Answer: (C) 5.

28. In which situation is mean deviation preferred over standard deviation?

A) When dealing with small datasets.
B) When data contains outliers.
C) When data is normally distributed.
D) When calculating probabilities.

Correct Answer: (B) When data contains outliers.

29. If the mean deviation of a dataset is 0, what does this imply?

A) All values in the dataset are the same.
B) The dataset has high variability.
C) There are extreme outliers present.
D) The dataset is normally distributed.

Correct Answer: (A) All values in the dataset are the same.

30. Which of the following datasets will likely yield the highest mean deviation?

A) {10, 10, 10, 10}.
B) {1, 2, 3, 4}.
C) {0, 50, 100}.
D) {20, 20, 20, 20}.

Correct Answer: (C) {0, 50, 100}.

31. When calculating the mean deviation, what should be done with the absolute differences?

A) They should be squared.
B) They should be summed and divided by the number of observations.
C) They should be ignored.
D) They should be multiplied by the mean.

Correct Answer: (B) They should be summed and divided by the number of observations.

32. How does mean deviation provide insights about a dataset?

A) It identifies the most common value.
B) It indicates how closely the data points cluster around the mean.
C) It reveals the shape of the distribution.
D) It determines the largest value in the dataset.

Correct Answer: (B) It indicates how closely the data points cluster around the mean.

33. If a dataset has high variability, what can be inferred about its mean deviation?

A) It will be low.
B) It will be high.
C) It cannot be determined.
D) It will always be zero.

Correct Answer: (B) It will be high.

34. Mean deviation is most suitable for which type of data?

A) Categorical data.
B) Ordinal data.
C) Continuous data.
D) Qualitative data.

Correct Answer: (C) Continuous data.

35. If a dataset consists of {4, 8, 12, 16}, what is the mean deviation?

A) 2.
B) 3.
C) 4.
D) 5.

Correct Answer: (B) 3.

36. What is standard deviation?

A) The average of the dataset.
B) A measure of the spread of data points around the mean.
C) The difference between the maximum and minimum values.
D) The median of the dataset.

Correct Answer: (B) A measure of the spread of data points around the mean.

37. How is variance defined?

A) The average of absolute deviations from the mean.
B) The average of the squared deviations from the mean.
C) The difference between the highest and lowest values.
D) The sum of all data points divided by the number of points.

Correct Answer: (B) The average of the squared deviations from the mean.

38. The formula for variance (σ²) in a population is:

A) σ² = Σ(x - μ)² / N.
B) σ² = Σ(x - x̄)² / n.
C) σ² = Σ|x - μ| / N.
D) σ² = Σ(x - x̄) / n.

Correct Answer: (A) σ² = Σ(x - μ)² / N.

39. The relationship between variance and standard deviation is:

A) Standard deviation is the square of the variance.
B) Variance is the square of the standard deviation.
C) They are always equal.
D) They are unrelated.

Correct Answer: (B) Variance is the square of the standard deviation.

40. If the standard deviation of a dataset is 0, what does this imply?

A) The data points are widely spread apart.
B) The data has high variability.
C) All data points are identical.
D) There are significant outliers in the data.

Correct Answer: (C) All data points are identical.

41. How do you calculate the sample variance (s²)?

A) s² = Σ(x - x̄)² / n.
B) s² = Σ(x - x̄)² / (n - 1).
C) s² = Σ|x - x̄| / n.
D) s² = Σ(x - μ)² / N.

Correct Answer: (B) s² = Σ(x - x̄)² / (n - 1).

42. Which of the following is a characteristic of standard deviation?

A) It cannot be negative.
B) It can be negative.
C) It is always greater than the mean.
D) It ignores outliers.

Correct Answer: (A) It cannot be negative.

43. When would you use variance instead of standard deviation?

A) When you want a measure that is easier to interpret.
B) When dealing with large datasets.
C) When performing statistical analysis that requires squared values.
D) When the dataset contains outliers.

Correct Answer: (C) When performing statistical analysis that requires squared values.

44. If a dataset has a mean of 10 and the data points are {6, 8, 12, 14}, what is the standard deviation?

A) 2.
B) 4.
C) 3.
D) 5.

Correct Answer: (C) 3.

45. How does increasing the spread of a dataset affect the standard deviation?

A) It decreases the standard deviation.
B) It has no effect on the standard deviation.
C) It increases the standard deviation.
D) It changes the mean.

Correct Answer: (C) It increases the standard deviation.

46. If a dataset has a high variance, what does it imply about the data?

A) The data points are clustered closely around the mean.
B) The data points are widely spread out from the mean.
C) The data is uniform.
D) The data has no variability.

Correct Answer: (B) The data points are widely spread out from the mean.

47. Which of the following statements is true regarding variance?

A) It is always expressed in the same units as the data.
B) It can be expressed in squared units of the data.
C) It cannot be calculated for grouped data.
D) It is less useful than mean deviation.

Correct Answer: (B) It can be expressed in squared units of the data.

48. What is the effect of outliers on standard deviation?

A) It has no effect.
B) It decreases the standard deviation.
C) It can significantly increase the standard deviation.
D) It is not affected by outliers.

Correct Answer: (C) It can significantly increase the standard deviation.

49. Which of the following data sets has the highest variance?

A) {2, 2, 2, 2}.
B) {1, 2, 3, 4}.
C) {10, 20, 30, 40}.
D) {5, 5, 5, 5}.

Correct Answer: (C) {10, 20, 30, 40}.

50. When calculating the standard deviation, what is the first step?

A) Calculate the mean.
B) Find the median.
C) Identify the maximum value.
D) Determine the mode.

Correct Answer: (A) Calculate the mean.

51. What does the coefficient of variation (CV) measure?

A) The central tendency of a dataset.
B) The relative variability of a dataset.
C) The maximum value in a dataset.
D) The average value of a dataset.

Correct Answer: (B) The relative variability of a dataset.

52. The coefficient of variation is calculated as:

A) CV = (Standard Deviation / Mean) × 100.
B) CV = (Mean / Standard Deviation) × 100.
C) CV = (Variance / Mean) × 100.
D) CV = (Mean / Variance) × 100.

Correct Answer: (A) CV = (Standard Deviation / Mean) × 100.

53. A higher coefficient of variation indicates:

A) Less relative variability.
B) More relative variability.
C) No variability.
D) The same variability as other datasets.

Correct Answer: (B) More relative variability.

54. When is the coefficient of variation particularly useful?

A) When comparing datasets with different units.
B) When datasets are small.
C) When measuring only central tendency.
D) When calculating mean deviation.

Correct Answer: (A) When comparing datasets with different units.

55. If the mean of a dataset is zero, what happens to the coefficient of variation?

A) It is zero.
B) It is undefined.
C) It is infinite.
D) It is always positive.

Correct Answer: (B) It is undefined.

56. The coefficient of variation is expressed as:

A) A fraction.
B) A percentage.
C) A decimal.
D) An integer.

Correct Answer: (B) A percentage.

57. Which of the following statements is true regarding the coefficient of variation?

A) It can be negative.
B) It is a measure of dispersion that is independent of the units of measurement.
C) It is only applicable to normally distributed data.
D) It is calculated using the median.

Correct Answer: (B) It is a measure of dispersion that is independent of the units of measurement.

58. If Dataset A has a mean of 50 and a standard deviation of 10, and Dataset B has a mean of 100 and a standard deviation of 20, which dataset has a higher coefficient of variation?

A) Dataset A.
B) Dataset B.
C) Both have the same CV.
D) Cannot be determined without additional information.

Correct Answer: (A) Dataset A.

59. The coefficient of variation is particularly useful in which field?

A) Pure mathematics.
B) Financial analysis.
C) Geography.
D) Historical research.

Correct Answer: (B) Financial analysis.

60. What does a CV of 0% indicate about a dataset?

A) The dataset has high variability.
B) The dataset has low variability.
C) All data points are identical.
D) The dataset has negative values.