Question 1

In the calculation of standard deviation, why is the square root of variance taken?

Accepted Answer

Correct Option: B. Solution: The square root of variance is taken to bring the units back to the original scale of data, as variance is in squared units.

Question 2

For a dataset with a mean of 50 and a variance of 25, what would be the new mean and variance if 5 is added to each data point?

Accepted Answer

Correct Option: A. Solution: Adding a constant to each data point shifts the mean by that constant but does not affect the variance. Therefore, the new mean is $50 + 5 = 55$ and the variance remains 25.

Question 3

For a data set with a mean of 10, which of the following statements is true if all data points are increased by 5?

Accepted Answer

Correct Option: C. Solution: Adding a constant to all data points does not change the variance.

Question 4

If the variance of a data set is 16, what is the standard deviation?

Accepted Answer

Correct Option: A. Solution: The standard deviation is the square root of the variance, so $$\sigma = \sqrt{16} = 4$$.

Question 5

In a continuous frequency distribution, how is the standard deviation calculated?

Accepted Answer

Correct Option: A. Solution: In a continuous frequency distribution, the standard deviation is calculated by using the mid-point of each class and applying the formula for discrete frequency distributions.

Question 6

When calculating the mean deviation for a continuous frequency distribution, what assumption is made about the frequencies in each class?

Accepted Answer

Correct Option: C. Solution: For a continuous frequency distribution, it is assumed that the frequency in each class is centered at its mid-point.

Question 7

Which of the following is a limitation of mean deviation?

Accepted Answer

Correct Option: B. Solution: Mean deviation is calculated using absolute values of deviations, thus it does not consider the direction of deviations.

Question 8

Which of the following measures of dispersion is most sensitive to extreme values in a data set?

Accepted Answer

Correct Option: A. Solution: The range is the difference between the maximum and minimum values and is highly sensitive to extreme values.

Question 9

Given a dataset, the mean deviation from the mean is calculated as 3.5. If each data point in the dataset is increased by 2, what will be the new mean deviation from the mean?

Accepted Answer

Correct Option: A. Solution: The mean deviation remains unchanged when each data point is increased by a constant value.

Question 10

What is the standard deviation of a data set if the variance is given as 36?

Accepted Answer

Correct Option: A. Solution: The standard deviation is the positive square root of the variance. Therefore, if the variance is 36, the standard deviation is $\sqrt{36} = 6$.

Question 11

If the variance of a dataset is 16, what will be the variance if each data point is multiplied by 3?

Accepted Answer

Correct Option: C. Solution: When each data point is multiplied by a constant $k$, the variance becomes $k^2$ times the original variance. Hence, the new variance is $3^2 	imes 16 = 144$.

Question 12

Two batsmen, A and B, have the same mean score of 53 over ten matches. Batsman A's scores range from 0 to 117, while Batsman B's scores range from 46 to 60. What can be inferred about the dispersion of their scores?

Accepted Answer

Correct Option: A. Solution: The range of Batsman A's scores is 117, while Batsman B's range is 14. A larger range indicates greater dispersion, so Batsman A's scores are more dispersed.

Question 13

If the range of a dataset is 50 and the minimum value is 20, what is the maximum value?

Accepted Answer

Correct Option: A. Solution: The range is calculated as the maximum value minus the minimum value. Therefore, maximum value = 20 + 50 = 70.

Question 14

In a discrete frequency distribution, the variance is calculated as 45.8. What is the standard deviation?

Accepted Answer

Correct Option: A. Solution: The standard deviation is the square root of the variance. Given the variance as 45.8, the standard deviation is $$\sqrt{45.8} = 6.77$$.

Question 15

What does a lower coefficient of variation (C.V.) indicate about a data set?

Accepted Answer

Correct Option: C. Solution: A lower coefficient of variation indicates more consistency in the data set.

Question 16

In a continuous frequency distribution, what is the first step to calculate the standard deviation?

Accepted Answer

Correct Option: B. Solution: In a continuous frequency distribution, the standard deviation is calculated by first finding the mid-point of each class.

Question 17

What is the shortcut method for calculating variance and standard deviation?

Accepted Answer

Correct Option: B. Solution: The shortcut method involves using an assumed mean and step-deviations to simplify calculations.

Question 18

If each observation in a dataset is multiplied by a constant, how does this affect the variance of the dataset?

Accepted Answer

Correct Option: C. Solution: When each observation is multiplied by a constant $k$, the variance of the resulting observations becomes $k^2$ times the original variance.

Question 19

How does multiplying each observation in a data set by a constant $k$ affect the mean deviation?

Accepted Answer

Correct Option: B. Solution: Multiplying each observation by a constant $k$ scales the deviations by $k$, thus the mean deviation becomes $k$ times the original mean deviation.

Question 20

If the standard deviation of a dataset is 4, what will be the standard deviation if each data point is decreased by 2?

Accepted Answer

Correct Option: A. Solution: Subtracting a constant from each data point does not affect the standard deviation, so the standard deviation remains 4.

Question 21

What is the formula for calculating the mean deviation about the mean for a discrete frequency distribution?

Accepted Answer

Correct Option: A. Solution: The mean deviation about the mean for a discrete frequency distribution is calculated using $$M.D.(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{N}$$ where $f_i$ is the frequency of each observation, $x_i$ is the observation, and $N$ is the total number of observations.

Question 22

Two batsmen, A and B, have the same mean score of 53 over 10 matches. If Batsman A's scores vary from 0 to 117 and Batsman B's scores vary from 46 to 60, which batsman is more consistent based on the coefficient of variation?

Accepted Answer

Correct Option: B. Solution: Batsman B is more consistent because his scores have a smaller range, indicating a lower coefficient of variation.

Question 23

For the data set: 6, 7, 10, 12, 13, 4, 8, 12, what is the mean deviation about the mean?

Accepted Answer

Correct Option: A. Solution: The mean deviation about the mean is calculated as 2.75 using the formula $$M.D.(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}$$.

Question 24

Given two data sets with the same mean, which measure of dispersion would you use to determine which set is more consistent?

Accepted Answer

Correct Option: D. Solution: The coefficient of variation is used to compare the relative variability of two data sets with the same mean.

Question 25

What is the formula for calculating the variance of a set of observations $x_1, x_2, ..., x_n$ with mean $\bar{x}$?

Accepted Answer

Correct Option: A. Solution: The variance is calculated as the mean of the squares of the deviations from the mean, which is $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$.

Question 26

If the variance of a dataset is 9, what will be the variance if each data point is increased by 5?

Accepted Answer

Correct Option: A. Solution: Variance remains unchanged when a constant is added to each data point. Therefore, the variance is still 9.

Question 27

In the shortcut method for calculating variance using step deviations, if the assumed mean is 50 and the common factor for step deviations is 5, what is the formula for calculating the mean of the dataset?

Accepted Answer

Correct Option: C. Solution: The formula for calculating the mean using step deviations is $x = A + h 	imes \frac{\sum f_i d_i}{N}$, where $A$ is the assumed mean, $h$ is the common factor, and $d_i$ are the step deviations.

Question 28

What is the range of runs scored by Batsman A in his last ten matches?

Accepted Answer

Correct Option: A. Solution: The range is calculated as the maximum value minus the minimum value. For Batsman A, the range is $117 - 0 = 117$.

Question 29

Which of the following is a limitation of using mean deviation as a measure of dispersion?

Accepted Answer

Correct Option: B. Solution: Mean deviation is not reliable when the degree of variability is high because it does not adequately represent the data's dispersion.

Question 30

For a dataset with a mean of 30 and a variance of 16, what would be the new variance if each data point is divided by 2?

Accepted Answer

Correct Option: A. Solution: When each data point is divided by a constant, the variance is divided by the square of that constant. So, the new variance is $\frac{16}{2^2} = 4$.

Question 31

A set of observations has a mean of 30 and a variance of 80. If each observation is multiplied by 3, what is the new variance?

Accepted Answer

Correct Option: A. Solution: When each observation is multiplied by a constant, the variance is multiplied by the square of that constant. Here, the new variance is $3^2 	imes 80 = 720$.

Question 32

Which of the following statements about mean deviation is true?

Accepted Answer

Correct Option: B. Solution: Mean deviation may be obtained from any measure of central tendency, such as mean or median, and it involves the mean of the absolute values of deviations.

Question 33

A dataset has a mean of 40 and a range of 60. If the minimum value is increased by 10, how does this affect the range and mean?

Accepted Answer

Correct Option: C. Solution: Increasing the minimum value by 10 does not affect the range, which is the difference between maximum and minimum values, but it increases the mean as it affects the sum of the dataset.

Question 34

What is the formula for calculating the mean deviation about a central value $a$ for ungrouped data?

Accepted Answer

Correct Option: A. Solution: The mean deviation about a central value $a$ is calculated as the mean of the absolute values of the deviations of the observations from $a$, which is $\frac{\sum |x_i - a|}{n}$.

Question 35

Consider a dataset where each observation is multiplied by a constant $k$. How does this transformation affect the variance of the dataset?

Accepted Answer

Correct Option: C. Solution: When each observation in a dataset is multiplied by a constant $k$, the variance of the dataset is multiplied by $k^2$.

Question 36

Why is standard deviation preferred over mean deviation for further algebraic treatment?

Accepted Answer

Correct Option: A. Solution: Standard deviation is preferred because it involves squaring deviations, allowing for algebraic operations, unlike mean deviation which uses absolute values.

Question 37

If each observation in a dataset is increased by a constant $a$, what will be the effect on the variance of the dataset?

Accepted Answer

Correct Option: B. Solution: Adding a constant to each observation in a dataset does not affect the variance, as variance measures the spread of the data relative to the mean, not the absolute values.

Question 38

Which of the following is a measure of dispersion?

Accepted Answer

Correct Option: C. Solution: Range is a measure of dispersion, calculated as the difference between the maximum and minimum values.

Question 39

If the mean deviation about the mean for a dataset is 5 and the number of observations is doubled without changing the data values, what happens to the mean deviation?

Accepted Answer

Correct Option: B. Solution: The mean deviation is independent of the number of observations as long as the data values remain unchanged. Therefore, it remains the same.

Question 40

Which batsman shows more consistency based on the coefficient of variation (C.V.)?

Accepted Answer

Correct Option: B. Solution: Batsman B shows more consistency as the scores are less scattered compared to Batsman A, indicating a lower coefficient of variation.

Question 41

In a continuous frequency distribution, the assumed mean is 30 and the class width is 10. If the step deviations are given as $d_i = -2, -1, 0, 1, 2$, what is the mean of the distribution?

Accepted Answer

Correct Option: B. Solution: The mean is calculated using the formula $x = A + h 	imes \frac{\sum f_i d_i}{N}$, where $A = 30$, $h = 10$, and the sum of $d_i$ is zero. Therefore, the mean is $30$.

Question 42

What is the effect on the variance of a dataset if a constant is added to each observation?

Accepted Answer

Correct Option: C. Solution: Adding a constant to each observation does not affect the variance because variance measures the spread of the data, which remains the same.

Question 43

Consider a dataset where the mean deviation about the mean is 6. If the dataset is transformed such that each value is divided by 2, what will be the mean deviation of the transformed dataset?

Accepted Answer

Correct Option: A. Solution: The mean deviation is divided by the absolute value of the constant factor, which is 2.

Question 44

Why is standard deviation preferred over mean deviation for further algebraic treatment?

Accepted Answer

Correct Option: B. Solution: Mean deviation is calculated using absolute values of deviations, which limits its use in algebraic operations, whereas standard deviation uses squared deviations, allowing for further algebraic manipulation.

Question 45

Given a dataset with values: 5, 10, 15, 20, and 25, what is the variance of this dataset?

Accepted Answer

Correct Option: A. Solution: The mean of the dataset is 15. The squared deviations from the mean are: $(5-15)^2 = 100$, $(10-15)^2 = 25$, $(15-15)^2 = 0$, $(20-15)^2 = 25$, $(25-15)^2 = 100$. The variance is the mean of these squared deviations: $\frac{100 + 25 + 0 + 25 + 100}{5} = 50$.

Question 46

If the coefficient of variation for series X is 15% and for series Y is 10%, which series is more consistent?

Accepted Answer

Correct Option: B. Solution: Series Y is more consistent because it has a lower coefficient of variation, indicating less relative variability.

Question 47

For a discrete frequency distribution, the mean deviation about the mean is calculated using which of the following formulas?

Accepted Answer

Correct Option: A. Solution: The mean deviation about the mean for a discrete frequency distribution is calculated using the formula $$M.D.(x) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - \bar{x}|$$, where $N$ is the sum of frequencies.

Question 48

If the variance of a data set is 25, what is the standard deviation?

Accepted Answer

Correct Option: A. Solution: The standard deviation is the square root of the variance, so it is 5.

Question 49

For a grouped dataset, the mean deviation from the mean is 8. If each frequency is doubled, what will be the new mean deviation from the mean?

Accepted Answer

Correct Option: B. Solution: Doubling the frequencies does not change the mean deviation from the mean.

Question 50

How does the step-deviation method simplify the calculation of variance?

Accepted Answer

Correct Option: B. Solution: The step-deviation method simplifies calculations by using a common factor to scale deviations, reducing the size of the numbers involved.

Question 51

Which measure of dispersion involves calculating the average of the absolute deviations from a central value?

Accepted Answer

Correct Option: B. Solution: Mean deviation is calculated as the mean of the absolute values of the deviations from a central value.

Question 52

If the variance of a data set is 16, what is the standard deviation?

Accepted Answer

Correct Option: A. Solution: The standard deviation is the positive square root of the variance. Thus, $\sqrt{16} = 4$.

Question 53

In a continuous frequency distribution, how is the mean deviation about the median calculated?

Accepted Answer

Correct Option: A. Solution: In a continuous frequency distribution, the mean deviation about the median is calculated by taking the absolute deviations from the median and dividing by the number of observations.

Question 54

Which measure of dispersion is most affected by extreme values?

Accepted Answer

Correct Option: A. Solution: Range is the difference between the maximum and minimum values and is most affected by extreme values.

Question 55

A dataset has a mean deviation from the median of 5. If a constant value of 10 is subtracted from each observation, what will be the new mean deviation from the median?

Accepted Answer

Correct Option: A. Solution: Subtracting a constant from each observation does not change the mean deviation from the median.

Question 56

Given the scores of Batsman B: 53, 46, 48, 50, 53, 53, 58, 60, 57, 52, what does the range tell us about the dispersion of his scores?

Accepted Answer

Correct Option: B. Solution: The range for Batsman B is $60 - 46 = 14$. A small range indicates that the scores are closely clustered around the mean.

Question 57

For a given ungrouped dataset, the mean deviation about the mean is 4. If the dataset is transformed by multiplying each observation by 3, what is the mean deviation of the transformed dataset?

Accepted Answer

Correct Option: B. Solution: The mean deviation is multiplied by the absolute value of the constant factor, which is 3.

Question 58

A dataset consists of the numbers: 5, 10, 15, 20, and 25. If each number is multiplied by 4, what is the new standard deviation?

Accepted Answer

Correct Option: B. Solution: The standard deviation is multiplied by the constant factor. If the original standard deviation is $s$, the new standard deviation is $4s$. Given that the original standard deviation is 5, the new standard deviation is $4 	imes 5 = 20$.

Question 59

What is the standard deviation of the data set: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24?

Accepted Answer

Correct Option: A. Solution: The standard deviation is calculated as the square root of the variance, which is 5.74 for this data set.

Question 60

In the step-deviation method, what is the role of the assumed mean?

Accepted Answer

Correct Option: B. Solution: The assumed mean serves as a reference point for calculating deviations in the step-deviation method.

Question 61

What is the variance of a data set defined as?

Accepted Answer

Correct Option: B. Solution: Variance is defined as the mean of squared deviations from the mean.

Question 62

A dataset consists of the numbers: 2, 4, 6, 8, and 10. What is the standard deviation of this dataset?

Accepted Answer

Correct Option: C. Solution: The mean is 6. The squared deviations from the mean are: $(2-6)^2 = 16$, $(4-6)^2 = 4$, $(6-6)^2 = 0$, $(8-6)^2 = 4$, $(10-6)^2 = 16$. The variance is $\frac{16 + 4 + 0 + 4 + 16}{5} = 8$. The standard deviation is $\sqrt{8} = 2.83$.

Question 63

Consider a frequency distribution with the following class intervals and frequencies: 10-20 (2), 20-30 (3), 30-40 (8), 40-50 (14), 50-60 (8), 60-70 (3), 70-80 (2). Calculate the standard deviation using the mid-point method.

Accepted Answer

Correct Option: B. Solution: The mid-point for each class is used to calculate the mean and variance. The standard deviation is the square root of the variance. Using the given data, the standard deviation is calculated as 6.77.

Question 64

A dataset has a variance of 10. If each value in the dataset is multiplied by 3, what will be the new variance?

Accepted Answer

Correct Option: B. Solution: Multiplying each observation by 3 will result in the variance being multiplied by $3^2 = 9$. Therefore, the new variance will be $10 	imes 9 = 90$.

Statistics

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Measures of Dispersion

Mean Deviation

Variance and Standard Deviation

Shortcut Method for Variance and Standard Deviation

Effect of Linear Transformations on Variance

Range and Basic Interpretation of Dispersion

Coefficient of Variation and Consistency

Limitations and Choice of Dispersion Measure

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Measures of Dispersion

Mean Deviation

Variance and Standard Deviation

Shortcut Method for Variance and Standard Deviation

Effect of Linear Transformations on Variance

Range and Basic Interpretation of Dispersion

Mean Deviation for Grouped Data

Standard Deviation for Frequency Distributions

Coefficient of Variation and Consistency

Limitations and Choice of Dispersion Measure

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Multiple Choice Questions

Q1.In the calculation of standard deviation, why is the square root of variance taken?

Correct Answer: B

Q2.For a dataset with a mean of 50 and a variance of 25, what would be the new mean and variance if 5 is added to each data point?

Correct Answer: A

Q3.For a data set with a mean of 10, which of the following statements is true if all data points are increased by 5?

Correct Answer: C

Q4.If the variance of a data set is 16, what is the standard deviation?

Correct Answer: A

Q5.In a continuous frequency distribution, how is the standard deviation calculated?

Correct Answer: A

Q6.When calculating the mean deviation for a continuous frequency distribution, what assumption is made about the frequencies in each class?

Correct Answer: C

Q7.Which of the following is a limitation of mean deviation?

Correct Answer: B

Q8.Which of the following measures of dispersion is most sensitive to extreme values in a data set?

Correct Answer: A

Q9.Given a dataset, the mean deviation from the mean is calculated as 3.5. If each data point in the dataset is increased by 2, what will be the new mean deviation from the mean?

Correct Answer: A

Q10.What is the standard deviation of a data set if the variance is given as 36?

Correct Answer: A

Q11.If the variance of a dataset is 16, what will be the variance if each data point is multiplied by 3?

Correct Answer: C

Q12.Two batsmen, A and B, have the same mean score of 53 over ten matches. Batsman A's scores range from 0 to 117, while Batsman B's scores range from 46 to 60. What can be inferred about the dispersion of their scores?

Correct Answer: A

Q13.If the range of a dataset is 50 and the minimum value is 20, what is the maximum value?

Correct Answer: A

Q14.In a discrete frequency distribution, the variance is calculated as 45.8. What is the standard deviation?

Correct Answer: A

Q15.What does a lower coefficient of variation (C.V.) indicate about a data set?

Correct Answer: C

Q16.In a continuous frequency distribution, what is the first step to calculate the standard deviation?

Correct Answer: B

Q17.What is the shortcut method for calculating variance and standard deviation?

Correct Answer: B

Q18.If each observation in a dataset is multiplied by a constant, how does this affect the variance of the dataset?

Correct Answer: C

Q19.How does multiplying each observation in a data set by a constant kkk affect the mean deviation?

Correct Answer: B

Q20.If the standard deviation of a dataset is 4, what will be the standard deviation if each data point is decreased by 2?

Correct Answer: A

Q21.What is the formula for calculating the mean deviation about the mean for a discrete frequency distribution?

Correct Answer: A

Q22.Two batsmen, A and B, have the same mean score of 53 over 10 matches. If Batsman A's scores vary from 0 to 117 and Batsman B's scores vary from 46 to 60, which batsman is more consistent based on the coefficient of variation?

Correct Answer: B

Q23.For the data set: 6, 7, 10, 12, 13, 4, 8, 12, what is the mean deviation about the mean?

Correct Answer: A

Q24.Given two data sets with the same mean, which measure of dispersion would you use to determine which set is more consistent?