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Frequency Distribution

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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.

Launch 09 Summary

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.
Launch 6 SolvedExample

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.
Launch 7 CommonMistakes

Common Mistakes

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
Launch 3 PracticeQuestions

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.

Launch 3 PracticeQuestions

Related Topics

Introduction to Statistics

Measures of Central Tendency

Polygons

Standard Trigonometric Ratios


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