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Frequency Distribution

Key Concepts

Solved Examples: 62

Practice Questions: 5

Solved Examples

Question 1: Which of the following represents raw data?
A. Marks and their frequencies arranged in a table
B. Marks arranged in ascending order with frequencies
C. Marks recorded exactly as collected without any arrangement
D. Marks grouped into class intervals

Raw data is data collected in its original form before it is organised or classified.

Therefore, the correct answer is Option C.

Question 2: A teacher records the marks of 12 students as follows:18, 20, 18, 22, 25, 20, 18, 25, 22, 20, 25, 18
Which type of frequency distribution is most suitable?
A. Grouped frequency distribution
B. Ungrouped frequency distribution
C. Cumulative frequency distribution
D. Relative frequency distribution

There are only four distinct values (18, 20, 22 and 25).
Since the number of distinct observations is small, an ungrouped frequency distribution is sufficient.

Therefore, the correct answer is Option B.

Question 3: Which of the following cannot be a frequency?
A. 0
B. 5
C. 12
D. 4.5

Frequency represents the number of observations. It must always be a whole number greater than or equal to 0.
Since 4.5 is not a whole number, it cannot represent a frequency.
Therefore, the correct answer is Option D.

Question 4: A grouped frequency distribution uses the following class intervals:
10–20
20–30
30–40
What is the problem with these class intervals?

The value 20 belongs to both the first and second class intervals.
Similarly, 30 belongs to both the second and third intervals.
This creates overlapping class intervals, causing ambiguity while classifying observations.
Therefore, the class intervals are not mutually exclusive and should be modified appropriately.

Question 5: The frequencies of five observations are:
3, 6, 8, 5 and 4.
Without knowing the actual observations, what is the total number of observations?

Total number of observations = Sum of all frequencies
= 3 + 6 + 8 + 5 + 4
= 26

Question 6: Can an observation have a frequency of 0? Explain.

Yes.
A frequency of 0 means that the observation does not occur in the data set.
For example, if the observations are 5, 6 and 8, then the frequency of 7 is 0 because it is absent from the data.
Therefore, a frequency of 0 is valid and indicates that the observation is not present.

Question 7: The frequency distribution given below shows the number of books read by students.

Books Read	1	2	3	4
Frequency	6	11	9	7

Which observation occurs most frequently?

The highest frequency is 11.
This corresponds to the observation 2.
Therefore, the observation that occurs most frequently is 2.

Question 8: The following frequency distribution shows the number of hours spent reading by a group of students.

Reading Hours	1	2	3
Frequency	4	4	7

Can two different observations have the same frequency?

Yes.
Frequency indicates how many times an observation occurs.
In the table above, the observations 1 and 2 both have a frequency of 4.
Therefore, two or more different observations can have the same frequency.

Question 9: The following data represents the number of goals scored by a football team in 15 matches.
2, 1, 3, 2, 0, 2, 1, 4, 3, 2, 1, 2, 3, 0, 2
Find the frequency of the observation 2.

The observation 2 appears in the 1st, 4th, 6th, 10th, 12th and 15th positions.
Therefore, Frequency of 2 = 6

Question 10: The following frequency distribution shows the favourite fruits of students.

Fruit	Apple	Banana	Orange	Mango
Frequency	15	18	11	18

Which fruits are equally popular?

Both Banana and Mango have a frequency of 18.
Therefore, these two fruits are equally popular.

Question 11: The total number of observations in a data set is 40.

Observation	A	B	C	D
Frequency	9	12	11	?

Find the missing frequency.

Sum of known frequencies
= 9 + 12 + 11
= 32
Missing frequency
= 40 − 32
= 8

Question 12: The following data shows the number of goals scored by a team in 12 matches:
2, 3, 1, 2, 4, 3, 2, 1, 3, 2, 4, 3
Prepare an ungrouped frequency distribution table.

Count the number of times each observation occurs.

Goals Scored	1	2	3	4
Frequency	2	4	4	2

The sum of frequencies = 2 + 4 + 4 + 2 = 12, which matches the total number of observations.

Question 13: The following table shows the number of pets owned by families in a locality.

Number of Pets	0	1	2	3
Frequency	6	10	8	3

Find the total number of families surveyed.

Total number of families = Sum of all frequencies
= 6 + 10 + 8 + 3
= 27

Question 14: The following tally table records the number of customers visiting a shop.

Day	Monday	Tuesday	Wednesday	Thursday
Tally Marks	\|\|\|\|/	\|\|\|\|/ \|\|	\|\|\|	\|\|\|\|

Find the total number of customers.

Monday = 5
Tuesday = 7
Wednesday = 3
Thursday = 4
Total customers = 5 + 7 + 3 + 4 = 19

Question 15: The marks obtained by 20 students are given below:
12, 18, 15, 22, 25, 27, 31, 34, 18, 24, 29, 35, 21, 17, 26, 38, 14, 19, 33, 28
Prepare a grouped frequency distribution using the class intervals:
10–19, 20–29, 30–39

Arrange the observations into the given class intervals.

Class Interval	10–19	20–29	30–39
Frequency	7	8	5

Question 16: The total number of observations is 40.

Class Interval	0–9	10–19	20–29	30–39
Frequency	8	12	?	9

Find the missing frequency.

Sum of known frequencies
= 8 + 12 + 9
= 29
Missing frequency
= 40 − 29
= 11

Question 17: Find the class width of the class interval 40–49.

Class Width = Upper Class Limit − Lower Class Limit + 1
= 49 − 40 + 1
= 10

Question 18: The following class intervals all have the same class width.
10–19, 20–29, 30–39, 40–?
Find the missing upper class limit.

Each class interval has a width of 10.
Therefore, the last class interval should be: 40–49
Hence, the missing upper class limit is 49.

Question 19: Study the following frequency table.

Number of Children	1	2	3	4
Frequency	5	13	9	7

By how many does the frequency of families with 2 children exceed that of families with 1 child?

Frequency of 2 children = 13
Frequency of 1 child = 5
Difference = 13 − 5 = 8

Question 20: A student prepared the following frequency table from the data:

Observation	5	6	7	8
Frequency	3	3	2	1

Identify the error.

Count the observations.
5 appears 4 times.
6 appears 3 times.
7 appears 2 times.
8 appears 1 time.
The frequency of 5 has been recorded incorrectly.
It should be 4, not 3.

Question 21: The following frequency distribution shows the number of books read by students in a month. How many students read at least 3 books?

Books Read	1	2	3	4	5
Frequency	5	8	10	6	3

Students who read at least 3 books read 3, 4 or 5 books.
Number of such students
= 10 + 6 + 3
= 19
∴ 19 students

Question 22: The following frequency distribution shows the number of goals scored by a team in different matches. How many matches did the team score fewer than 3 goals?

Goals Scored	0	1	2	3	4
Frequency	2	7	9	6	4

Fewer than 3 goals means 0, 1 or 2 goals.
Number of matches
= 2 + 7 + 9
= 18
∴ 18 matches

Question 24: The following frequency distribution shows the number of siblings of students in a class. How many students have at most 2 siblings?

Number of Siblings	0	1	2	3	4
Frequency	3	9	12	5	1

“At most 2” means 2 or fewer.
Number of students
= 3 + 9 + 12
= 24
∴ 24 students

Question 25: The following table shows the number of students choosing different modes of transport to school. Complete the missing frequencies.

Mode of Transport	Bus	Bicycle	Walk	Car
Tally Marks	\|\|\|\|/ \|\|\|	\|\|\|\|/ \|	\|\|\|\|	\|\|\|
Frequency	?	?	?	?

Bus = 5 + 3 = 8
Bicycle = 5 + 1 = 6
Walk = 4
Car = 3
∴ Frequencies = 8, 6, 4 and 3

Question 26: A student prepared the following frequency table from tally marks. Identify and correct the error.

Number of Pets	0	1	2	3
Tally Marks	\|\|\|	\|\|\|\|/ \|\|	\|\|\|\|/	\|\|\|\|
Recorded Frequency	3	7	4	4

For 2 pets, the tally marks represent 5
But the recorded frequency is 4.
∴ The error is in the frequency for 2 pets.
Correct frequency = 5.

Question 27: The following are the scores obtained by 20 students in a test:
12, 18, 24, 15, 31, 27, 19, 34, 22, 16, 29, 25, 13, 37, 21, 28, 17, 33, 26, 14
Prepare a grouped frequency distribution using the class intervals 10–19, 20–29 and 30–39.

Count the observations falling within each class interval.

Class Interval	10–19	20–29	30–39
Frequency	8	8	4

Sum of frequencies = 8 + 8 + 4 = 20
∴ The frequency distribution accounts for all 20 observations.

Question 28: The following are the daily temperatures (in °C) recorded over 24 days:
21, 24, 27, 23, 29, 31, 26, 22, 25, 28, 32, 24, 27, 30, 23, 26, 29, 21, 25, 33, 28, 24, 31, 27
Prepare a grouped frequency distribution using the class intervals 20–24, 25–29 and 30–34.

Count the observations in each class interval.

Temperature (°C)	20–24	25–29	30–34
Frequency	8	11	5

Sum of frequencies = 8 + 11 + 5 = 24
∴ The frequency distribution accounts for all 24 observations.

Question 29: The following are the numbers of pages read by 18 students during a week:
35, 42, 51, 27, 63, 48, 55, 39, 72, 44, 58, 31, 67, 52, 46, 75, 28, 61
Prepare a grouped frequency distribution using the class intervals 20–39, 40–59 and 60–79.

Count the observations falling within each class interval.

Pages Read	20–39	40–59	60–79
Frequency	4	8	6

Sum of frequencies = 4 + 8 + 6 = 18
∴ The frequency distribution accounts for all 18 observations.

Question 30: A teacher wants to group the marks of students ranging from 0 to 50 into five equal-width class intervals. Which of the following is the most appropriate choice?
A. 0–10, 11–20, 21–30, 31–40, 41–50
B. 0–9, 10–19, 20–29, 30–39, 40–49
C. 0–10, 10–20, 20–30, 30–40, 40–50
D. 1–10, 11–20, 21–30, 31–40, 41–50

The intervals should:
Cover the required range without gaps.
Have equal widths.
Avoid overlapping.
Option A covers 0 to 50 using five equal-width intervals of 10 marks.
∴ Correct answer = Option A

Question 31: The ages of participants in a survey range from 12 to 61 years. The investigator wants to divide the data into five equal-width class intervals. Which set of intervals is most suitable?
A. 12–21, 22–31, 32–41, 42–51, 52–61
B. 12–20, 21–30, 31–40, 41–50, 51–61
C. 10–20, 20–30, 30–40, 40–50, 50–60
D. 12–22, 23–33, 34–44, 45–55, 56–66

The intervals should cover the entire range from 12 to 61 and have equal widths without gaps or overlaps.
Option A gives five intervals, each containing 10 integer values:
12–21, 22–31, 32–41, 42–51, 52–61
∴ Correct answer = Option A

Question 32: The number of books read by students ranges from 1 to 30. A teacher wants to group the data into three equal-width class intervals. Which of the following is the most appropriate choice?
A. 1–10, 11–20, 21–30
B. 1–9, 10–19, 20–30
C. 1–10, 10–20, 20–30
D. 1–11, 12–21, 22–30

The intervals should cover all values from 1 to 30, have equal widths and have no gaps or overlaps.
Option A gives three equal-width intervals 1–10, 11–20, 21–30
∴ Correct answer = Option A

Question 33: A grouped frequency distribution uses the class intervals 10–19, 20–29, 30–39 and 40–49. What is the class width?

Each class contains 10 integer values.
For example, the first class contains 10, 11, 12, …, 19
∴ Class width = 10

Question 34: A data set contains values ranging from 1 to 50. If the observations are divided into class intervals of width 10, how many classes are required?

Suitable class intervals are:
1–10, 11–20, 21–30, 31–40, 41–50
∴ Number of classes = 5

Question 35: A grouped frequency distribution contains 6 classes, each having a class width of 5. If the first class is 10–14, what is the last class interval?

The class intervals are:
10–14
15–19
20–24
25–29
30–34
35–39
∴ Last class interval = 35–39

Question 36: Consider the following class intervals 10–19, 20–29, 30–39
Are these inclusive or exclusive class intervals?

In these intervals, both the lower and upper limits are included.
For example, 19 belongs to the class 10–19, and 20 belongs to the class 20–29.
∴ These are inclusive class intervals.

Question 37: Consider the following class intervals 10–20, 20–30, 30–40
Are these inclusive or exclusive class intervals?

In these intervals, the upper limit of one class is the lower limit of the next class. The common boundary value is assigned to only one class according to the interval convention.
For example, the intervals can be interpreted as:
10 ≤ x < 20 20 ≤ x < 30 30 ≤ x < 40 ∴ These are exclusive class intervals.[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row admin_label="SE_Question" _builder_version="4.27.6" _module_preset="default" custom_margin="-50px||||false|false" global_colors_info="{}"][et_pb_column type="4_4" _builder_version="4.27.6" _module_preset="default" global_colors_info="{}"][et_pb_text _builder_version="4.27.6" _module_preset="default" text_font="|300|||||||" text_font_size="18px" custom_padding="0.5%||0.5%|1%|false|false" hover_enabled="0" text_font_size_tablet="17px" text_font_size_phone="16px" text_font_size_last_edited="on|phone" border_radii="on|3px|3px|3px|3px" border_width_all="1px" border_color_all="#0d9488" border_width_all_tablet="1px" border_width_all_phone="2px" border_width_all_last_edited="on|phone" global_colors_info="{}" sticky_enabled="0"]Question 38: Convert the following inclusive class intervals into exclusive class intervals 20–29, 30–39, 40–49

The gap between consecutive limits is 30 − 29 = 1
Half of the gap = 1 ÷ 2 = 0.5
Subtract 0.5 from each lower limit and add 0.5 to each upper limit.

Inclusive Interval	20–29	30–39	40–49
Exclusive Interval	19.5–29.5	29.5–39.5	39.5–49.5

∴ Exclusive intervals = 19.5–29.5, 29.5–39.5, 39.5–49.5

Question 39: Convert the following inclusive class intervals into exclusive class intervals 15–24, 25–34, 35–44

The difference between the upper limit of one class and the lower limit of the next class is 25 − 24 = 1
Correction factor = 1 ÷ 2 = 0.5
Subtract 0.5 from each lower limit and add 0.5 to each upper limit.
∴ Exclusive class intervals 19.5–24.5, 24.5–34.5, 34.5–44.5

Question 40: Convert the following exclusive class intervals into inclusive class intervals 9.5–19.5, 19.5–29.5, 29.5–39.5

Add 0.5 to each lower limit and subtract 0.5 from each upper limit.
∴ Inclusive class intervals 10–19, 20–29, 30–39

Question 41: The following class intervals are given in exclusive form 20–30, 30–40, 40–50
Convert them into inclusive class intervals, assuming that the observations are recorded as whole numbers.

Since the observations are whole numbers, the boundary values can be adjusted by 1.
∴ Inclusive class intervals 20–29, 30–39, 40–49

Question 42: The following grouped frequency distribution shows the marks obtained by students in a test. How many students scored between 20 and 39 marks?

Marks	0–9	10–19	20–29	30–39	40–49
Frequency	3	7	12	9	4

Marks between 20 and 39 fall in the classes 20–29 and 30–39.
Number of students
= 12 + 9
= 21
∴ 21 students

Question 43: The following table shows the distribution of daily study hours of students. Which class interval has the highest frequency?

Study Hours	0–1	1–2	2–3	3–4	4–5
Frequency	4	8	15	11	6

The highest frequency is 15.
It corresponds to the class interval 2–3.
∴ The class interval 2–3 has the highest frequency.

Question 44: The following grouped frequency distribution shows the ages of participants in a competition. How many participants were younger than 30 years?

Age (years)	10–19	20–29	30–39	40–49
Frequency	6	14	9	5

Participants younger than 30 years belong to the classes 10–19 and 20–29.
Number of participants
= 6 + 14
= 20

Question 45: The following grouped frequency distribution shows the number of pages read by students in a week. How many students read at least 40 pages?

Pages Read	20–29	30–39	40–49	50–59	60–69
Frequency	5	9	12	8	6

At least 40 pages means 40 pages or more.
Relevant classes are 40–49, 50–59 and 60–69.
Number of students
= 12 + 8 + 6
= 26
∴ 26 students

Question 46: A teacher prepared the following frequency distribution from a data set containing 30 observations. Identify the error.

Class Interval	0–9	10–19	20–29	30–39
Frequency	6	8	7	11

Total frequency
= 6 + 8 + 7 + 11
= 32
But the data set contains only 30 observations.
∴ The frequency table is inconsistent because the total frequency is 32 instead of 30.

Question 47: A frequency table contains the following class intervals. Identify the problem.

Class Interval	10–20	20–30	30–40

The value 20 appears in both 10–20 and 20–30.
Similarly, 30 appears in both 20–30 and 30–40.
Thus, the class intervals overlap and an observation at a boundary cannot be assigned uniquely.
∴ The class intervals are not mutually exclusive.

Question 48: A student prepares the following frequency distribution for 25 observations. The frequencies are 4, 7, 6 and 9. The student concludes that the table is correct because all four frequencies are positive. Is the conclusion correct?

No.
The total frequency is 4 + 7 + 6 + 9 = 26
But the data set contains only 25 observations.
∴ The table is incorrect because the total frequency exceeds the number of observations by 1.

Question 49: The class marks of a grouped frequency distribution are 15, 25, 35, 45 and 55. Find the class width and reconstruct the class intervals.

The difference between consecutive class marks is
Class width = 25 − 15 = 10
Half of the class width 10 ÷ 2 = 5
For each class
Lower limit = Class mark − 5
Upper limit = Class mark + 5

Class Mark	15	25	35	45	55
Class Interval	10–20	20–30	30–40	40–50	50–60

So Class width = 10
∴ Class intervals = 10–20, 20–30, 30–40, 40–50 and 50–60

Question 50: The class marks of a grouped frequency distribution are 52, 62, 72, 82 and 92. Reconstruct the class interval corresponding to the class mark 72.

The difference between consecutive class marks is class width = 62 − 52 = 10
Half of the class width 10 ÷ 2 = 5
For class mark 72
Lower limit = 72 − 5 = 67
Upper limit = 72 + 5 = 77
∴ The class interval is 67–77.

Question 51: The class marks of a continuous grouped frequency distribution are 4.5, 9.5, 14.5 and 19.5. Reconstruct all the class intervals.

The difference between consecutive class marks is
Class width = 9.5 − 4.5 = 5
Half of the class width 5 ÷ 2 = 2.5
Subtract 2.5 from and add 2.5 to each class mark.

Class Mark	4.5	9.5	14.5	19.5
Class Interval	2–7	7–12	12–17	17–22

∴ Class intervals = 2–7, 7–12, 12–17 and 17–22

Question 52: A frequency distribution contains the following class intervals: Below 20, 20–40, 40–60, 60–80 and 80 and Above. Assuming equal class widths, reconstruct the two open-ended classes.

The width of the known classes is 40 − 20 = 20
For the first class Lower limit = 20 − 20 = 0
∴ Below 20 → 0–20
For the last class Upper limit = 80 + 20 = 100
∴ 80 and Above → 80–100
The complete distribution is 0–20, 20–40, 40–60, 60–80 and 80–100

Question 53: A grouped frequency distribution of ages contains the classes 30–45, 45–60, 60–75 and 75 and Over. Assuming equal class widths, find the upper limit of the last class.

Class width 45 − 30 = 15
The last class begins at 75.
Upper limit = 75 + 15 = 90
∴ Upper limit of the last class = 90 years

Question 54: A salary distribution contains the following classes
Less than ¤150, ¤150–¤250, ¤250–¤350 and ¤350–¤450
Assuming equal class widths, find the class mark of the first class after converting it into a bounded class.

Class width ¤250 − ¤150 = ¤100
The first class ends at ¤150.
Lower limit = ¤150 − ¤100 = ¤50
Thus, the first class is ¤50–¤150.
Class mark = (50 + 150) ÷ 2 = 100
∴ Class mark = ¤100

Question 55: A grouped frequency distribution has a total frequency of 100. The frequencies of four of its five classes are 18, 27, 25 and 14. Find the missing frequency.

Sum of known frequencies
= 18 + 27 + 25 + 14
= 84
Missing frequency
= 100 − 84
= 16
∴ Missing frequency = 16

Question 56: A grouped frequency distribution contains 40 observations.

Class Interval	0–10	10–20	20–30	30–40
Frequency	7	f	15	6

Find the value of the missing frequency f.

The sum of all frequencies must equal 40.
7 + f + 15 + 6 = 40
28 + f = 40
f = 40 − 28
f = 12
∴ f = 12

Question 57: A grouped frequency distribution contains 52 observations. The frequencies of the first three classes are 9, 12 and 15. The final two classes have equal frequencies. Find the frequency of each of the final two classes.

Sum of the first three frequencies
= 9 + 12 + 15
= 36
Remaining observations
= 52 − 36
= 16
Since the final two classes have equal frequencies
Frequency of each class = 16 ÷ 2 = 8
∴ The frequency of each of the final two classes is 8.

Question 58: A company records the test scores of 80 applicants in five consecutive classes. The frequencies are 12, x, 28, y and 10. If the frequency 28 is twice the frequency x, find the value of y.

Given 28 = 2x
or, x = 28 ÷ 2
or, x = 14
The frequencies are now
12, 14, 28, y, 10
Their total is 80.
12 + 14 + 28 + y + 10 = 80
or, 64 + y = 80
or, y = 80 − 64
or, y = 16
∴ y = 16

Question 59: Convert the following inclusive class intervals into continuous exclusive class intervals
11–20, 21–30, 31–40 and 41–50

The gap between consecutive classes is 21 − 20 = 1
Correction factor 1 ÷ 2 = 0.5
Subtract 0.5 from each lower limit and add 0.5 to each upper limit.

Inclusive Class Interval	11–20	21–30	31–40	41–50
Continuous Class Interval	10.5–20.5	20.5–30.5	30.5–40.5	40.5–50.5

∴ The continuous class intervals are 10.5–20.5, 20.5–30.5, 30.5–40.5 and 40.5–50.5.

Question 60: The inclusive class intervals of a distribution are 1–5, 6–10, 11–15 and 16–20. Find the continuous boundaries of the class 6–10.

The gap between consecutive classes is 6 − 5 = 1
Correction factor 1 ÷ 2 = 0.5
For the class 6–10
Lower boundary = 6 − 0.5 = 5.5
Upper boundary = 10 + 0.5 = 10.5
∴ The continuous boundaries are 5.5 and 10.5.

Question 61: An inclusive frequency distribution has class intervals 1.0–1.4, 1.5–1.9, 2.0–2.4 and 2.5–2.9. Convert the first class interval into its continuous form.

The gap between the first two classes is 1.5 − 1.4 = 0.1
Correction factor 0.1 ÷ 2 = 0.05
For the first class
Lower boundary = 1.0 − 0.05 = 0.95
Upper boundary = 1.4 + 0.05 = 1.45
∴ The continuous class interval is 0.95–1.45.

Question 62: Convert the following inclusive frequency distribution into a continuous frequency distribution.

Class Interval	30–39	40–49	50–59	60–69
Frequency	15	22	18	9

The gap between consecutive classes is 40 − 39 = 1
Correction factor 1 ÷ 2 = 0.5
Adjust each class by subtracting 0.5 from its lower limit and adding 0.5 to its upper limit.

Continuous Class Interval	29.5–39.5	39.5–49.5	49.5–59.5	59.5–69.5
Frequency	15	22	18	9

∴ The continuous class intervals are 29.5–39.5, 39.5–49.5, 49.5–59.5 and 59.5–69.5.

What is Frequency Distribution?

Frequency Distribution is a method of organising raw data by showing how many times each value or group of values occurs in a data set. Instead of listing observations randomly, it arranges them in a systematic table, making the data easier to read, understand and analyse.
A frequency distribution helps identify patterns, compare values and summarise large amounts of data. It forms the foundation for many statistical concepts such as mean, median, mode, graphs and data interpretation.

Frequency and Frequency Distribution

The number of times a particular value appears in a data set is called its frequency. A table that lists each value (or group of values) along with its corresponding frequency is called a frequency distribution.
For example, if the marks of students are:
25, 30, 25, 40, 30, 25
then the frequency of 25 is 3, the frequency of 30 is 2 and the frequency of 40 is 1.
Frequency distributions make it easier to organise raw data and identify repeated values.

Types of Frequency Distribution

Frequency distributions are mainly of two types.
1. Ungrouped Frequency Distribution
In an ungrouped frequency distribution, every distinct value is listed separately along with its frequency.
Example:

Marks	10	15	20
Frequency	1	2	3

This type is suitable when the number of observations is small or when there are only a few distinct values.

2. Grouped Frequency Distribution
When the data contains many observations or a large range of values, similar values are grouped into class intervals.
Example:

Class Interval	0–10	10–20	20–30
Frequency	4	8	12

Grouped frequency distributions make large data sets easier to understand and analyse.

Tally Marks

Tally marks provide a quick method of counting observations while preparing a frequency table.

Each observation is represented by a vertical stroke.

The fifth observation is drawn across the previous four strokes to form a group of five.
Example:

Value	A	B	C	D
Tally Marks	\|\|\|\|	\|\|\|\|/ \|\|	\|\|	\|\|\|\|/
Frequency	4	7	2	5

Using tally marks reduces counting errors and makes it easier to prepare frequency distribution tables, especially when dealing with large sets of raw data.

Class Intervals

When a data set contains a large number of observations, it becomes difficult to list every individual value separately. In such cases, similar values are grouped together into ranges called class intervals.

A class interval represents a group of values within a specific range. Each observation is placed into the appropriate class interval, and the number of observations in each interval is recorded as its frequency.

For example, the marks of 40 students can be grouped as follows:

Marks	0–10	10–20	20–30	30–40	40–50
Frequency	3	7	12	10	8

A good class interval should:

Cover all observations in the data set.
Have a suitable width so that the data remains easy to interpret.
Avoid unnecessary repetition or excessive grouping.

The difference between the upper and lower limits of a class interval is called the class width.

Formula:

Class Width = Upper Class Limit − Lower Class Limit

For example:

Class Interval = 20–30

Class Width = 30 − 20 = 10

Parts of a Frequency Distribution Table

A frequency distribution table contains different components that help organise and analyse data effectively.
The main parts of a frequency distribution table are:

1. Class Interval
A class interval is a range of values grouped together.
Example:
10–19, 20–29, 30–39
Each observation is placed into one of these intervals.

2. Tally Marks
Tally marks are used to count observations quickly while preparing the frequency table.
The fifth tally mark crosses the previous four marks, making groups of five.

3. Frequency
Frequency represents the number of observations present in a particular class interval.
Example:

Class Interval	10–19	20–29	30–39	40–49
Tally Marks	\|\|\|\|/ \|\|	\|\|\|\|/ \|\|\|\|	\|\|\|\|/ \|	\|\|\|
Frequency	7	9	6	3

A well-prepared frequency distribution table allows large amounts of data to be summarised clearly and forms the basis for further statistical analysis.

Advantages of Frequency Distribution

A frequency distribution offers several advantages in organising and analysing data effectively.

It organises raw data into a systematic and meaningful form, making large amounts of information easier to understand.
It helps identify patterns, trends and repeated values within a data set.
It makes comparison between different groups of data easier by presenting information in a structured table.
It helps in calculating statistical measures such as mean, median and mode.
It provides the foundation for preparing statistical graphs such as bar graphs, histograms and frequency polygons.
It reduces complex data into a simple format, allowing quick interpretation and analysis.

A properly prepared frequency distribution is an important step in statistical analysis because it transforms unorganised data into a form that can be easily studied and interpreted.

Summary of Frequency Distribution

Concept	Description
Raw Data	Data collected in its original, unorganised form before it is arranged into a table.
Frequency	The number of times a particular value or observation occurs in a data set.
Frequency Distribution	A systematic arrangement of data showing each value or class interval together with its frequency.
Ungrouped Frequency Distribution	Lists every distinct value separately along with its corresponding frequency.
Grouped Frequency Distribution	Groups observations into class intervals and records the frequency of each interval.
Tally Marks	A quick counting method used while preparing a frequency table. Every fifth observation is represented by a diagonal stroke across the previous four tally marks.
Class Interval	A range of values grouped together in a grouped frequency distribution.
Lower Class Limit	The smallest value included in a class interval.
Upper Class Limit	The largest value included in a class interval.
Class Width	The size of a class interval. Formula: Class Width = Upper Class Limit − Lower Class Limit
Uses of Frequency Distribution	Organises raw data, simplifies analysis, identifies patterns and forms the basis for statistical calculations and graphs.

Common Mistakes

Confusing Frequency with Observation. Students often write the observation itself as the frequency. Remember that an observation is the recorded value, while the frequency is the number of times that value occurs.
Counting Frequencies Incorrectly. Students sometimes miss repeated observations or count the same observation more than once. Count each observation carefully before preparing the frequency table.
Preparing an Incorrect Tally Table. Students often forget that every fifth tally mark should be represented by a diagonal stroke across the previous four tally marks. This makes counting easier and reduces errors.
Ignoring the Total Frequency. The sum of all frequencies should always be equal to the total number of observations in the given data set. Always verify the total after completing the table.
Confusing Grouped and Ungrouped Frequency Distributions. Students often use class intervals even when the data contains only a few distinct observations. Ungrouped frequency distributions are used for individual observations, whereas grouped frequency distributions are used for large data sets.
Creating Incorrect Class Intervals. Students sometimes create overlapping class intervals or leave gaps between consecutive intervals. Every observation should belong to exactly one class interval.
Calculating the Class Width Incorrectly. Students often subtract the lower class limit from the upper class limit without adding 1. For inclusive class intervals, use:
Class Width = Upper Class Limit − Lower Class Limit + 1
Confusing Class Limits with Class Intervals. Students sometimes identify the entire class interval as the class limit. The lower class limit is the smallest value in the interval, while the upper class limit is the largest value.
Misinterpreting Frequency Tables. Students often assume that the observation with the largest value has the highest frequency. Always compare the frequency values, not the observations themselves.
Ignoring Missing Observations. While preparing a frequency table, students sometimes omit observations that appear only once. Every distinct observation must be included in the table.
Using Unequal Class Widths. Students often prepare grouped frequency distributions with inconsistent class widths, making the data difficult to interpret. Unless specified otherwise, all class intervals should have the same width.
Failing to Verify the Completed Frequency Table. After preparing a frequency table, students often forget to check whether the frequencies add up to the total number of observations and whether every observation has been included exactly once.

Practice Questions

Question 1: The number of pets owned by 15 families is given below:
1, 2, 1, 3, 2, 1, 4, 2, 3, 1, 2, 2, 3, 1, 4
Prepare a frequency table and identify the observation with the highest frequency.

Question 2: The total number of observations is 40.

Observation	A	B	C	D
Frequency	9	12	?	8

Find the missing frequency.

Question 3: Study the grouped frequency distribution.

Marks	0–9	10–19	20–29	30–39
Frequency	4	10	9	7

Find:
a) The class interval with the highest frequency.
b) The total number of observations.

Question 4: Consider the class interval 60–69.
Find:
a) Lower class limit
b) Upper class limit
c) Class width

Question 5: Study the grouped frequency distribution.

Age (Years)	10–19	20–29	30–39	40–49
Frequency	6	11	8	5

Find:
a) The total number of observations.
b) The class interval containing the greatest number of observations.

Frequency Distribution

Key Concepts

Solved Examples: 62

Practice Questions: 5

Solved Examples

What is Frequency Distribution?

Frequency and Frequency Distribution

Types of Frequency Distribution

Tally Marks

Class Intervals

Parts of a Frequency Distribution Table

Advantages of Frequency Distribution

Summary of Frequency Distribution

Common Mistakes

Practice Questions

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