Measures of dispersion variance standard deviation and mean deviation - Question Bank

1. In the context of probability distributions, the variance measures:
A) The central tendency of the distribution
B) The asymmetry of the distribution
C) The spread or dispersion of the random variable around its expected value
D) The peak of the distribution
2. Which measure of dispersion is defined as \(\frac{Q_3 - Q_1}{2}\)?
A) Range
B) Mean Deviation
C) Variance
D) Quartile Deviation
3. If a dataset has a mean of 25 and a variance of 9, what is its standard deviation?
A) 3
B) 6
C) 9
D) 25
4. For the dataset {10, 20, 30, 40, 50}, what is the variance?
A) 100
B) 150
C) 200
D) 250
5. For a dataset {10, 20, 30, 40, 50}, what is the mean deviation from the mean?
A) 10
B) 12
C) 15
D) 20
6. Which of the following statements about measures of dispersion is TRUE?
A) Mean deviation is always greater than standard deviation.
B) Standard deviation is always greater than mean deviation.
C) Variance is the square of standard deviation.
D) Range is the most robust measure of dispersion.
7. What is the primary disadvantage of using variance as a measure of dispersion?
A) It is not sensitive to outliers
B) Its units are squared, making interpretation difficult
C) It does not consider the mean
D) It is difficult to calculate
8. If mean deviation is calculated using the median, it is called:
A) Standard Deviation
B) Median Absolute Deviation (MAD)
C) Quartile Deviation
D) Range
9. The standard deviation of a dataset {3, 3, 3, 3, 3} is:
A) 0
B) 1
C) 3
D) 9
10. Which measure of dispersion is based on the difference between the first and third quartiles?
A) Range
B) Mean Deviation
C) Variance
D) Interquartile Range
11. If the variance of a dataset is 0, it implies:
A) The mean is 0
B) All data points are identical
C) The data is perfectly normally distributed
D) The median is equal to the mean
12. For a continuous random variable X, the variance is given by:
A) \(\int_{-\infty}^{\infty} x f(x) dx\)
B) \(\int_{-\infty}^{\infty} |x - \mu| f(x) dx\)
C) \(\int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx\)
D) \(\int_{-\infty}^{\infty} x^2 f(x) dx\)
13. If two datasets have the same mean, the one with the smaller standard deviation is considered:
A) More dispersed
B) Less dispersed
C) Equally dispersed
D) Skewed
14. The coefficient of variation is calculated as:
A) (Standard Deviation / Mean) * 100
B) (Mean / Standard Deviation) * 100
C) (Variance / Mean) * 100
D) (Mean / Variance) * 100
15. Which measure of dispersion is most sensitive to outliers?
A) Mean Deviation
B) Variance
C) Range
D) Interquartile Range
16. What is the standard deviation for the dataset {1, 3, 5, 7, 9}?
A) 2
B) \(\sqrt{5}\)
C) \(\sqrt{8}\)
D) 4
17. For the dataset {1, 3, 5, 7, 9}, what is the variance?
A) 4
B) 5
C) 6
D) 8
18. For a dataset {1, 3, 5, 7, 9}, calculate the mean deviation from the mean.
A) 1.8
B) 2.0
C) 2.2
D) 2.4
19. The formula \(E[X^2] - (E[X])^2\) is equivalent to:
A) Mean Deviation
B) Standard Deviation
C) Variance
D) Root Mean Square
20. Which of the following properties does variance NOT possess?
A) Non-negativity
B) Additivity (for independent variables)
C) Invariance under addition of a constant
D) Homogeneity of degree 2
21. Mean deviation from the median is calculated using:
A) Squared differences from the median
B) Absolute differences from the mean
C) Absolute differences from the median
D) Squared differences from the median
22. If a dataset has a mean of 50 and a variance of 100, what is its standard deviation?
A) 5
B) 10
C) 50
D) 100
23. The term 's' typically represents:
A) Population Variance
B) Population Standard Deviation
C) Sample Variance
D) Sample Standard Deviation
24. The term '\(\sigma\)' typically represents:
A) Population Variance
B) Population Standard Deviation
C) Sample Variance
D) Sample Standard Deviation
25. Which of the following is a common way to calculate standard deviation for grouped data?
A) Sum of midpoints divided by the number of classes
B) Square root of the sum of (frequency * (midpoint - mean)^2) divided by the total frequency
C) Sum of frequencies divided by the range
D) Average of the class boundaries
26. If the mean of a dataset is 10 and the standard deviation is 0, what can be concluded about the data?
A) All values are 10
B) The data is normally distributed
C) The data is highly spread out
D) The median is also 10
27. The formula for variance of a discrete random variable X with probability mass function P(X=xi) is:
A) \(E[(X - E[X])^2]\)
B) \(E[|X - E[X]|]\)
C) \(E[X]^2\)
D) \(E[X^2]\)
28. Which measure of dispersion is least affected by extreme values?
A) Range
B) Variance
C) Standard Deviation
D) Interquartile Range
29. Mean deviation is also known as:
A) Average Absolute Deviation
B) Root Mean Square Deviation
C) Standard Error
D) Coefficient of Variation
30. If the variance of a dataset is 16, what is its standard deviation?
A) 2
B) 4
C) 8
D) 16
31. What is the variance of the dataset {2, 2, 2, 2}?
A) 0
B) 1
C) 2
D) 4
32. For a symmetric distribution, the mean deviation from the mean and the standard deviation are:
A) Equal
B) Mean deviation is always greater than standard deviation
C) Standard deviation is always greater than mean deviation
D) Unrelated
33. Which of the following best describes the standard deviation?
A) The average distance of data points from the median.
B) The difference between the maximum and minimum values.
C) A measure of the spread of data around the mean.
D) The average of the absolute deviations from the mean.
34. Variance is always:
A) Negative
B) Zero or positive
C) Positive
D) Equal to the mean
35. Which measure of dispersion is calculated as the difference between the largest and smallest values?
A) Variance
B) Standard Deviation
C) Mean Deviation
D) Range
36. If every observation in a dataset is multiplied by a constant 'k' (where k > 0), how does the standard deviation change?
A) It remains unchanged
B) It is multiplied by k^2
C) It is multiplied by k
D) It is divided by k
37. If a constant 'c' is added to every observation in a dataset, how does the standard deviation change?
A) It increases by 'c'
B) It decreases by 'c'
C) It remains unchanged
D) It becomes 'c' times the original standard deviation
38. The formula for mean deviation from the mean (MD) for a dataset x1, x2, ..., xn with mean \(\bar{x}\) is:
A) \(\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\)
B) \(\sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}\)
C) \(\frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|\)
D) \(\frac{1}{n} \sum_{i=1}^{n} (x_i - \text{median})\)
39. Which measure of dispersion is sensitive to every value in the dataset?
A) Range
B) Interquartile Range
C) Standard Deviation
D) All of the above
40. Mean deviation is generally preferred over standard deviation when:
A) The data is skewed
B) The data contains outliers
C) A simpler interpretation is needed
D) Mathematical tractability is paramount
41. If the standard deviation of a dataset is 5, what is its variance?
A) 5
B) 10
C) 25
D) \(\sqrt{5}\)
42. For a sample of size n, the unbiased sample variance (s^2) is often calculated as:
A) \(\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\)
B) \(\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\)
C) \(\frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|\)
D) \(\frac{1}{n-1} \sum_{i=1}^{n} |x_i - \bar{x}|\)
43. Which of the following is NOT a measure of dispersion?
A) Variance
B) Standard Deviation
C) Mean Deviation
D) Arithmetic Mean
44. The mean deviation is calculated using the:
A) Squared differences from the mean
B) Absolute differences from the median
C) Absolute differences from the mean
D) Squared differences from the median
45. If all values in a dataset are the same, what is the variance?
A) Undefined
B) Maximum possible value
C) Zero
D) One
46. What is the unit of standard deviation compared to the unit of the data?
A) Squared
B) Same
C) Half
D) Cubed
47. For a dataset with observations x1, x2, ..., xn and mean \(\bar{x}\), the formula for variance (\(\sigma^2\)) is:
A) \(\frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|\)
B) \(\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\)
C) \(\sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}\)
D) \(\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\)
48. Which measure of dispersion considers the absolute differences from the mean?
A) Variance
B) Standard Deviation
C) Mean Deviation
D) Interquartile Range
49. What is the square root of the variance called?
A) Mean Deviation
B) Quartile Deviation
C) Standard Deviation
D) Range
50. Which measure of dispersion is the average of the squared differences from the mean?
A) Standard Deviation
B) Variance
C) Mean Deviation
D) Range
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