Chapter 13 — Statistics Question Bank

A. MCQs

1. The midpoint of a class interval is called its:
A. Frequency
B. Class mark
C. Cumulative frequency
D. Deviation

2. If $x_i$ represents a class mark and $f_i$ its frequency, the direct-method mean is:
A. $\frac{\sum x_i}{\sum f_i}$
B. $\frac{\sum f_i}{\sum x_i}$
C. $\frac{\sum f_ix_i}{\sum f_i}$
D. $\sum f_ix_i$

3. In the assumed-mean method, $d_i$ is defined as:
A. $a+x_i$
B. $x_i-a$
C. $a-x_i$
D. $x_i/a$

4. In the step-deviation method,$$u_i=$$

A. $x_i+a$
B. $x_i-a$
C. $\frac{x_i-a}{h}$
D. $\frac{a-x_i}{h}$

5. The class having the highest frequency is called the:
A. Median class
B. Modal class
C. Mean class
D. Central class

6. To locate the median class, we compare cumulative frequencies with:
A. $n$
B. $n/2$
C. $2n$
D. $n-1$

7. If $n=80$, the value used to locate the median class is:
A. 20
B. 30
C. 40
D. 80

8. In the grouped-data median formula, $cf$ represents:
A. Frequency of median class
B. Total frequency
C. Cumulative frequency of the class preceding the median class
D. Class frequency after median class

9. Which measure is generally most affected by extreme observations?
A. Median
B. Mode
C. Mean
D. Class mark

10. The measure most suitable for identifying the most popular item is:
A. Mean
B. Median
C. Mode
D. Range

11. A cumulative frequency distribution formed using successive upper limits is called:
A. More-than type
B. Less-than type
C. Equal type
D. Modal type

12. In a more-than cumulative frequency distribution, the values used are generally:
A. Upper class limits
B. Class marks
C. Lower class limits
D. Frequencies

13. A graph representing cumulative frequency is called a/an:
A. Histogram
B. Frequency polygon
C. Ogive
D. Bar graph

14. Before applying the grouped-data median formula, class intervals should be:
A. Unequal
B. Continuous
C. Unordered
D. Open-ended only

15. If the class interval is $20-30$, its class mark is:
A. 10
B. 20
C. 25
D. 30

16. If the frequencies are $5,8,12,7$, the cumulative frequency up to the third class is:
A. 12
B. 20
C. 25
D. 32

17. If the frequencies are $6,10,14,8$, the modal class is the class having frequency:
A. 6
B. 8
C. 10
D. 14

18. In the formula for grouped-data mode, $f_1$ denotes the:
A. Frequency before modal class
B. Frequency of modal class
C. Frequency after modal class
D. Total frequency

19. The empirical relationship given in the chapter is:
A. $3M_d=M_o+2\bar{x}$
B. $M_d=3M_o+2\bar{x}$
C. $3M_o=M_d+2\bar{x}$
D. $M_d=M_o+\bar{x}$

20. If a distribution contains significant extreme values and a typical observation is required, the more appropriate measure is generally:
A. Mean
B. Median
C. Mode
D. Class mark

Answer Key — MCQs

Q	Ans	Q	Ans
1	B	11	B
2	C	12	C
3	B	13	C
4	C	14	B
5	B	15	C
6	B	16	C
7	C	17	D
8	C	18	B
9	C	19	A
10	C	20	B

The definitions and formula choices above follow the chapter’s treatment of grouped mean, mode, median and cumulative frequency.

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B. Fill in the Blanks

1. The midpoint of a class interval is called its ________.

2. The sum of all frequencies is denoted by ________.

3. In the direct method, mean $=\frac{\sum f_ix_i}{\_\_\_\_\_\_}$.

4. In the assumed-mean method, $d_i=x_i-\_\_\_\_\_\_$.

5. In step deviation, $u_i=\frac{x_i-a}{\_\_\_\_\_\_}$.

6. The class with the highest frequency is called the ________ class.

7. The class containing the median is called the ________ class.

8. To locate the median class, we use ________.

9. Cumulative frequency is obtained by progressively ________ frequencies.

10. A cumulative frequency curve is known as an ________.

11. A less-than cumulative frequency distribution uses successive ________ limits.

12. A more-than cumulative frequency distribution uses successive ________ limits.

13. In the median formula, $h$ represents the ________.

14. In the median formula, $f$ represents the frequency of the ________ class.

15. The measure that identifies the most frequent observation is called the ________.

16. The mean is particularly affected by ________ values.

17. Before using the grouped median and mode formulas, class intervals should be made ________.

18. In the mode formula, $f_1$ is the frequency of the ________ class.

Answers

1. Class mark
2. $\sum f_i$
3. $\sum f_i$
4. $a$
5. $h$
6. Modal
7. Median
8. $n/2$
9. Adding
10. Ogive
11. Upper
12. Lower
13. Class size/width
14. Median
15. Mode
16. Extreme
17. Continuous
18. Modal

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C. True or False

1. The class mark is the midpoint of a class interval.

2. The modal class has the smallest frequency.

3. Mean takes all observations into account.

4. Extreme values can affect the mean.

5. Median is always equal to mode.

6. Cumulative frequency is obtained by adding frequencies successively.

7. An ogive is a cumulative frequency curve.

8. For grouped median, the median class is located using $n/2$.

9. The frequency before the median class is used as $cf$.

10. $h$ in the median formula represents total frequency.

11. In a less-than cumulative frequency distribution, upper limits are used.

12. In a more-than cumulative frequency distribution, lower limits are used.

13. The modal class is identified by the highest frequency.

14. The direct, assumed-mean and step-deviation methods are methods for finding grouped mean.

15. Continuous class intervals are important when constructing an ogive.

Answers

1. True
2. False
3. True
4. True
5. False
6. True
7. True
8. True
9. True
10. False
11. True
12. True
13. True
14. True
15. True

These points are directly supported by the chapter’s summary and discussion of cumulative frequency and continuous classes.

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D. Assertion–Reason

Choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

1.

Assertion: Median can be preferable to mean when extreme observations are present.
Reason: Extreme observations can strongly influence the mean.

2.

Assertion: The class with the highest frequency is called the modal class.
Reason: Mode is associated with the most frequently occurring value.

3.

Assertion: The median class is located using $n/2$.
Reason: The median divides the distribution into two approximately equal parts.

4.

Assertion: A less-than cumulative frequency table uses upper class limits.
Reason: It counts observations below successive upper limits.

5.

Assertion: An ogive represents cumulative frequency.
Reason: Cumulative frequencies are plotted to form a cumulative frequency curve.

6.

Assertion: The assumed mean method can simplify mean calculations.
Reason: Deviations are measured from a convenient assumed value.

Answers

1. A
2. A
3. A
4. A
5. A
6. A

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E. Match the Following

Column A	Column B
1. Class mark	A. Most frequent value
2. Mode	B. Midpoint of class
3. Median	C. Running total
4. Cumulative frequency	D. Middle/central value
5. Ogive	E. Cumulative frequency curve
6. Modal class	F. Class with maximum frequency

Answers

1 → B
2 → A
3 → D
4 → C
5 → E
6 → F

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F. Very Short Answer Questions

1. What is a class mark?

2. What does $f_i$ represent?

3. What is cumulative frequency?

4. What is a modal class?

5. What is a median class?

6. What is an ogive?

7. What is an assumed mean?

8. What is meant by class size?

9. Why can median be preferable to mean in the presence of extreme values?

10. Which measure is useful for finding the most popular item?

11. What value is used to locate the median class?

12. What does $cf$ represent in the grouped median formula?

13. What does $h$ represent in the median formula?

14. Name the three methods of finding the grouped mean.

15. What should be checked about class intervals before using grouped median/mode formulas?

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G. Formula-Based Objective Questions

1.

For classes $10-20,20-30,30-40$, the class marks are:

A. 10, 20, 30
B. 15, 25, 35
C. 20, 30, 40
D. 5, 15, 25

Answer: B

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2.

If $a=50$ and $x_i=62$, then $d_i$ is:

A. 12
B. 112
C. −12
D. 50

Answer: A

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3.

If $x_i=70,\ a=50,\ h=10$, then:$$u_i=\frac{x_i-a}{h}$$

is:

A. 1
B. 2
C. 7
D. 20

Answer: B

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4.

If total frequency $n=60$, the median position for locating the median class is based on:

A. 15
B. 20
C. 30
D. 60

Answer: C

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5.

Suppose cumulative frequencies are:$$8,\ 17,\ 29,\ 43,\ 60$$

and $n=60$. The median class corresponds to the cumulative frequency:

A. 8
B. 17
C. 29
D. 43

Answer: D

Because $n/2=30$, and 43 is the first cumulative frequency greater than 30.

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6.

If the frequencies are:$$4,\ 7,\ 15,\ 9,\ 5$$

the modal class is the class corresponding to:

A. 4
B. 7
C. 15
D. 9

Answer: C

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H. Numerical MCQs

1. Mean

A grouped distribution has:$$x_i: 10,20,30$$$$f_i: 2,3,5$$

The mean is:

A. 20
B. 23
C. 24
D. 25

Answer: C

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2. Class Mark

For the class $40-60$, the class mark is:

A. 20
B. 40
C. 50
D. 60

Answer: C

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3. Cumulative Frequency

If frequencies are $7, 11, 8, 14$, the cumulative frequency up to the third class is:

A. 18
B. 26
C. 29
D. 40

Answer: B

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4. Median Class

A distribution has total frequency 50 and cumulative frequencies:$$6,\ 15,\ 27,\ 38,\ 50$$

The median class is the class corresponding to:

A. 15
B. 27
C. 38
D. 50

Answer: B

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5. More-than Cumulative Frequency

If total frequency is 70 and the frequency of the first class is 12, the more-than cumulative frequency at the beginning of the second class is:

A. 12
B. 58
C. 70
D. 82

Answer: B

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I. Identify the Correct Method

1. A question gives class marks and frequencies and asks for mean. Which general formula can be used?

Answer:$$\bar{x}=\frac{\sum fx}{\sum f}$$

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2. The class marks are large and subtracting a convenient value would make the arithmetic easier. Which method is suitable?

Answer: Assumed mean method.

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3. The deviations from the assumed mean are convenient multiples of the class size. Which method is especially efficient?

Answer: Step-deviation method.

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4. The question asks for the most frequent value in grouped data. Which measure should be found?

Answer: Mode.

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5. The question asks for the value dividing the distribution into two approximately equal parts. Which measure is required?

Answer: Median.

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J. Case-Based Questions

Case 1 — Mean

A teacher records marks of students in different groups. Each group is represented by its class mark and has a corresponding frequency.

Q1. Which value represents the midpoint of a group?
Answer: Class mark.

Q2. Which product is required in the direct method?
Answer: $fx$.

Q3. Which expression gives the mean?
Answer:$$\frac{\sum fx}{\sum f}$$

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Case 2 — Median

A frequency table contains 80 observations.

Q1. What value is used to locate the median class?
Answer: $80/2=40$.

Q2. Suppose the cumulative frequencies are $12,28,43,61,80$. Which cumulative frequency identifies the median class?
Answer: 43.

Q3. Why?
Answer: It is the first cumulative frequency greater than 40.

Q4. Which frequency is used as $cf$ in the formula?
Answer: The cumulative frequency of the class immediately preceding the median class.

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Case 3 — Mode

A distribution has class frequencies:$$5,\ 9,\ 17,\ 12,\ 6$$

Q1. Which class is the modal class?
Answer: The third class.

Q2. What is special about its frequency?
Answer: It is the highest frequency.

Q3. Which three frequencies are required in the mode formula?
Answer: The frequencies immediately before, within, and after the modal class: $f_0,f_1,f_2$.

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K. High-Value Conceptual Questions

1. Why is class mark used while calculating the mean of grouped data?

2. Why can the mean be a poor representative when extreme values are present?

3. Why is median often preferred for finding a typical value when extreme observations exist?

4. Why is mode appropriate when the most popular item is required?

5. Explain the difference between a modal class and a median class.

6. Differentiate between less-than and more-than cumulative frequency distributions.

7. Explain how an ordinary frequency distribution can be obtained from a less-than cumulative frequency table.

8. Why should class intervals be continuous before applying grouped-data median and mode formulas?

9. Explain the role of $n/2$ in finding the grouped median.

10. Why is cumulative frequency necessary for finding the median of grouped data?

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Questions

If you’re preparing for an exam, don’t try to solve every question in the chapter blindly. These are the highest-value patterns:

1. Mean using direct method
2. Mean using assumed mean
3. Mean using step deviation
4. Finding class marks
5. Finding modal class
6. Finding mode using formula
7. Constructing less-than cumulative frequency
8. Constructing more-than cumulative frequency
9. Finding median class
10. Finding median using formula
11. Converting cumulative frequency into ordinary frequency
12. Converting discontinuous classes into continuous classes
13. Choosing mean vs median vs mode
14. Using the empirical relation

$$3M_d=M_o+2\bar{x}$$

15. Reading/interpreting an ogive