Frequency Distribution in Statistics: Purpose, Importance, Types, Tables, Basic Terms, Examples
This article explains frequency distribution in statistics from the basics, including frequency tables, grouped and ungrouped distributions, class intervals, class boundaries, relative frequency, cumulative frequency, histograms, frequency polygons, frequency curves and ogives.
The terminology and graphical conventions used here follow standard introductory-statistics textbook treatment. Where different classroom conventions exist, such as inclusive versus exclusive class intervals, the convention is stated explicitly.
What is Frequency in Statistics?
In statistics, frequency means the number of times a particular value, category or class occurs in a data set.
1. For example, if the marks obtained by students include the value 75 five times, the frequency of 75 is 5.2. For example, the number 8 has a frequency of 4 in the following list of numbers, because it occurs 4 times:
1, 2, 3, 4, 6, 8, 9, 8, 5, 1, 1, 8, 0, 6, 8.
In simple terms:
Frequency = Number of observations having a particular value or belonging to a particular class.
What is a Frequency Distribution?
A frequency distribution is a systematic arrangement of data showing the frequency associated with each value or class of values.
For small discrete data sets, individual values may be listed separately. For larger or continuous data sets, observations are usually grouped into class intervals.
In statistics, a frequency distribution is a tabular representation, that shows the number of observations within a given interval. In other words, the frequency distribution is simply a table in which the data is grouped into classes based on similar characteristics, and the number of observations in each class is recorded. It displays the frequency of occurrence of different values of a single variable.
Frequency distributions are especially useful for normal distributions, which show the observations of probabilities divided among standard deviations. Traders use frequency distributions in finance to keep track of price activity and identify trends.
Hence, Frequency distribution is the method of organizing data in such a way that all of its characteristics are summarised in a table.
Consider the following example to better explain this: The following are the results of ten students in an exam:
85, 77, 50, 85, 50, 77, 77, 94, 94, 50.
Exam Marks | No. of Students |
50 | 3 |
77 | 3 |
85 | 2 |
| 94 | 2 |
Important distinction: A frequency distribution is fundamentally a tabular organization of frequencies. A histogram, frequency polygon or ogive is a graphical representation based on that distribution.
Purpose of constructing a frequency distribution
The purpose of constructing a frequency distribution is to achieve the following three objectives:
(i) to make data analysis easier.
(ii) to estimate the frequencies of an unknown population distribution from sample data distributions, and
(iii) to make the computation of various statistical measures easier.
Why is Frequency Distribution Important?
A frequency distribution is useful because it:
- Organizes a large set of raw data.
- Shows how observations are distributed.
- Makes important patterns easier to identify.
- Facilitates comparison between data sets.
- Provides a basis for graphical presentation.
- Helps in calculating and interpreting statistical measures.
- Provides a convenient basis for further statistical analysis.
Frequency distributions are useful for many kinds of data. They are not limited to normally distributed data.
Basic Terms Used in Frequency Distribution
1. Raw Data
Raw data are observations collected in their original form before they are organized or summarized.
Example:
12, 15, 11, 18, 15, 14, 12, 20, 15
2. Frequency
Frequency is the number of times an observation or class occurs.
3. Class Interval
A class interval is a range used to group observations in a grouped frequency distribution.
Examples:
10–19, 20–29, 30–39, 40–49
4. Class Limits
The smallest and largest values included in a class are called its lower class limit and upper class limit.
For the class 20–29:
- Lower class limit = 20
- Upper class limit = 29
5. Class Boundaries
Class boundaries are the actual boundaries used to make adjacent classes continuous when the observations are recorded to a specified precision.
For integer-valued observations grouped as 20–29 and 30–39, the corresponding boundaries can be:
19.5–29.5 and 29.5–39.5
This convention is particularly important when constructing histograms from integer-valued observations.
6. Class Mark or Midpoint
The class mark, also called the class midpoint, is the value halfway between the lower and upper class limits.
For a class interval with lower limit L and upper limit U:
\[ \text{Class Mark} = \frac{L+U}{2} \]
7. Class Width
Class width is the size of a class interval.
For continuous class boundaries:
\[ \text{Class Width} = \text{Upper Boundary} - \text{Lower Boundary} \]
For equal-width classes, the difference between successive lower class limits also gives the class width.
8. Relative Frequency
Relative frequency expresses a class frequency as a proportion of the total number of observations.
If f is the frequency and N is the total frequency:
\[ \text{Relative Frequency} = \frac{f}{N} \]
Percentage frequency is:
\[ \text{Percentage Frequency} = \frac{f}{N}\times100 \]
9. Cumulative Frequency
Cumulative frequency is the running total of frequencies.
For example, if the frequencies are 5, 8 and 7, the cumulative frequencies are:
5, 13, 20
Types of Frequency Distribution
The most common forms of frequency distribution are:
- Ungrouped or discrete frequency distribution
- Grouped frequency distribution
- Relative frequency distribution
- Cumulative frequency distribution
1. Ungrouped or Discrete Frequency Distribution
In an ungrouped frequency distribution, individual values are listed along with their frequencies.
This form is suitable when the number of distinct values is relatively small.
2. Grouped Frequency Distribution
In a grouped frequency distribution, observations are arranged into class intervals.
It is particularly useful when the data set is large or contains many different values.
3. Relative Frequency Distribution
A relative frequency distribution shows the proportion or percentage of observations in each category or class.
The sum of all relative frequencies is 1, apart from rounding differences.
4. Cumulative Frequency Distribution
A cumulative frequency distribution shows the accumulated frequency up to or from a particular class.
There are two common forms:
- Less-than cumulative frequency distribution
- More-than cumulative frequency distribution
These distributions are used to construct ogives, or cumulative frequency curves.
How to Construct a Frequency Distribution Table
To make a frequency distribution table, Tally marks are often used. A basic frequency distribution table can be constructed using the following steps.
- Collect the raw data. Identify the smallest and largest observations.
- Decide whether individual values or class intervals should be used. Choose suitable class intervals if grouping is required.
- Make the class intervals mutually exclusive and collectively exhaustive.
- Write the categories or class intervals in the first column.
- Count the observations belonging to each value or class. Tally the numbers in each category or class interval using tally marks.
- Record the counts as frequencies. Check that the total frequency equals the total number of observations.
- That is, count the tally marks and write the frequency in the last column. The frequency is simply the total.
What is Tally Marks
Tally marks are a convenient counting device used while constructing a frequency table. Each observation is represented by one tally mark, and groups of five are commonly used to make counting easier.
Tally marks (or Hash marks) are defined using the system of unary numerals. They are a type of numeral that is used to count. They're best for counting or tallying continuous results, like the score in a game or sport, because no intermediate results need to be rejected or erased.
Tally marks are usually written in a group of five lines. The first four lines are drawn vertically, while every fifth line runs diagonally over the first four lines, i.e. from the top of the first line to the bottom of the fourth line.
How to count Tally Marks:
Tally marks are the quickest way to keep track of numbers in groups of five. To get the logic, look at the diagram below. 
As you can see,
♦ The tally mark '|' is used to express the number 1.
♦ The tally mark '| |' is used to express the number 2.
♦ The tally mark '| | |' is used to express the number 3.
♦ The tally mark '| | | |' is used to express the number 4.
♦ But, the number 5 is not denoted by the tally marks '| | | | |'. We draw four vertical lines ( | | | | ) with a diagonal ( \ ) line through it for the number 5.
Example 1: Ungrouped Frequency Distribution
Solution:
Frequency Distribution
Note: Tally marks are not restricted to continuous data. They are simply a practical way of recording counts during the construction of a frequency table.
Inclusive and Exclusive Class Intervals
Inclusive Method
In the inclusive method, both class limits are included in the class.
Examples:
10–13, 14–17, 18–21, 22–25
This method is commonly used for discrete or integer-valued observations.
Exclusive Method
In the exclusive method, the upper limit of one class becomes the lower limit of the next class.
Example:
10–20, 20–30, 30–40, 40–50
This form is especially convenient for continuous measurements.
The important point is that class intervals must not overlap and every observation must belong to one and only one class.
Example 2: Grouped Frequency Distribution
Construct a frequency distribution from the following data by inclusive method taking 4 as the class interval:Solution:
Frequency Distribution
Because these are integer-valued inclusive classes, their corresponding class boundaries, when a continuous representation is needed, are:
| Class Interval | Class Boundaries |
|---|---|
| 10–13 | 9.5–13.5 |
| 14–17 | 13.5–17.5 |
| 18–21 | 17.5–21.5 |
| 22–25 | 21.5–25.5 |
| 26–29 | 25.5–29.5 |
| 30–33 | 29.5–33.5 |
| 34–37 | 33.5–37.5 |
| 38–41 | 37.5–41.5 |
Example 3: Weekly Wages of 100 Workers
Consider a data set containing the weekly wages of 100 workers. Prepare a statistical table from the following:
Weekly wages (Rs.) of 100 workers of Factory A
88, 23, 27, 28, 86, 96, 94, 93, 86, 99, 82, 24, 24, 55, 88, 99, 55, 86, 82, 36, 96, 39, 26, 54, 87, 100, 56, 84, 83, 46, 102, 48, 27, 26, 29, 100, 59, 83, 84, 48, 104, 46, 30, 29, 40, 101, 60, 89, 46, 49, 106, 33, 36, 30, 40, 103, 70, 90, 49, 50, 104, 36, 37, 40, 40, 106, 72, 94, 50, 60, 24, 39, 49, 46, 66, 107, 76, 96, 46, 67, 26, 78, 50, 44, 43, 46, 79, 99, 36, 68, 29, 67, 56, 99, 93, 48, 80, 102, 32, 51Solution:
If these whole-number wages are to be represented in a histogram, the corresponding class boundaries can be written as 19.5–29.5, 29.5–39.5, and so on.
Graphical Representation of a Frequency Distribution
A frequency distribution can be represented graphically in several ways.
The guiding principles that apply to the graphical representation of frequency distributions are exactly the same as those that apply to the graphic and diagrammatic representation of other types of data.
The data in a frequency distribution can be displayed as graphs, which highlight important characteristics and relationships that would be difficult to detect from a simple inspection of the frequency tables.
The following are the most widely used graphs for charting a frequency distribution for a general understanding of the details of the data:- Line Frequency Diagram
- Histogram
- Frequency polygon
- Smoothed frequency curves
- Ogives or cumulative frequency curves.
Relative Frequency Distribution
A relative frequency distribution is useful when we want to compare groups of different sizes or express frequencies as proportions or percentages.
The basic formula is:
\[ \text{Relative Frequency}=\frac{f}{N} \]
For example, if 20 out of 100 students belong to a particular category:
\[ \text{Relative Frequency}=\frac{20}{100}=0.20 \]
Therefore:
\[ \text{Percentage Frequency}=0.20\times100=20\% \]
Frequency Distribution: Important Formulae
| Concept | Formula |
|---|---|
| Class Mark | \(\frac{L+U}{2}\) |
| Class Width | Upper Boundary − Lower Boundary |
| Relative Frequency | \(\frac{f}{N}\) |
| Percentage Frequency | \(\frac{f}{N}\times100\) |
| Frequency Density | \(\frac{f}{\text{Class Width}}\) |
| Grouped Median | \(L+\left(\frac{N/2-c.f.}{f}\right)h\) |
How to Check a Frequency Distribution
Before using a frequency table for further analysis, check the following:
- The total frequency must equal the number of observations.
- Every observation should belong to exactly one class.
- Class intervals should not overlap.
- Class intervals should cover the required range of the data.
- Cumulative frequencies should be calculated correctly.
- Relative frequencies should add to approximately 1.
- Percentage frequencies should add to approximately 100%.
- For a histogram with unequal class widths, use frequency density.
Common Mistakes Students Make
1. Confusing Frequency with Cumulative Frequency
Frequency tells how many observations are in one value or class. Cumulative frequency is the running total.
2. Double-Counting Observations
Each observation must be counted exactly once.
3. Using Overlapping Class Intervals
Classes such as 20–30, 30–40 and 40–50 can cause ambiguity when the data are recorded as whole numbers. Use a clearly defined convention such as 20–29, 30–39, 40–49 for integer data, or use continuous classes with clearly defined boundaries.
4. Forgetting Class Boundaries
When a histogram requires continuous class intervals, students often plot inclusive class limits directly instead of using appropriate class boundaries.
5. Leaving Gaps Between Histogram Bars
Histogram rectangles normally touch because they represent adjacent numerical intervals.
6. Ignoring Class Width
When class widths are unequal, using frequency directly as the rectangle height can misrepresent the distribution. Frequency density should be used.
7. Using the Wrong Class Mark
For a class 20–29:
\[ \text{Class Mark}=\frac{20+29}{2}=24.5 \]
8. Mixing Less-Than and More-Than Ogive Methods
Remember:
- Less-than ogive → upper class boundaries
- More-than ogive → lower class boundaries
9. Assuming the Area Under Every Frequency Curve Is 1
This is incorrect. Area equal to 1 is a defining property of a probability density function after appropriate normalization, not of an ordinary frequency curve.
10. Making Arithmetic Errors in Cumulative Frequencies
Always verify the final cumulative frequency. For a complete distribution, the final cumulative frequency should equal the total number of observations.
11. Forgetting Axis Labels
Graphs should clearly identify the variable or class intervals on the horizontal axis and frequency, cumulative frequency or frequency density on the vertical axis, as appropriate.
Exam-Ready Definitions
Frequency
Frequency is the number of times a particular value or class occurs in a data set.
Frequency Distribution
A frequency distribution is a systematic arrangement of data showing the frequencies associated with individual values or class intervals.
Class Interval
A class interval is a range of values used to group observations in a grouped frequency distribution.
Histogram
A histogram is a graphical representation of a grouped quantitative frequency distribution using adjoining rectangles whose bases represent class intervals.
Frequency Polygon
A frequency polygon is a line graph obtained by plotting frequencies against class marks and joining successive points by straight line segments.
Frequency Curve
A frequency curve is a smooth graphical representation of the general shape of a frequency distribution.
Ogive
An ogive is a cumulative frequency curve obtained by plotting cumulative frequencies against suitable class boundaries or limits.
Quick Comparison of Important Graphs
| Graph | Main Purpose | Horizontal Axis | Vertical Axis |
|---|---|---|---|
| Histogram | Display grouped quantitative data | Class intervals | Frequency or frequency density |
| Frequency Polygon | Show and compare distribution patterns | Class marks | Frequency |
| Frequency Curve | Show smoothed distribution shape | Variable values | Frequency |
| Less-Than Ogive | Show observations below values | Upper class boundaries | Less-than cumulative frequency |
| More-Than Ogive | Show observations above values | Lower class boundaries | More-than cumulative frequency |
Real-Life Applications of Frequency Distribution
Frequency distributions are used in many practical situations, including:
- Education: distribution of examination marks.
- Business: distribution of sales or customer purchases.
- Economics: distribution of income or expenditure.
- Labour studies: distribution of wages or working hours.
- Health and science: distribution of measurements such as height, weight or waiting time.
- Research: summarizing observations before further statistical analysis.
Frequently Asked Questions About Frequency Distribution
What is frequency distribution in statistics?
A frequency distribution is a systematic arrangement of data showing how many observations occur at each value or within each class interval.
What are the main types of frequency distribution?
The main types are ungrouped frequency distribution, grouped frequency distribution, relative frequency distribution and cumulative frequency distribution.
What is the difference between frequency and cumulative frequency?
Frequency gives the number of observations in a particular value or class, whereas cumulative frequency gives the running total up to or from that class.
What is a histogram?
A histogram is a graph of grouped quantitative data in which adjoining rectangles represent class intervals and their frequencies or frequency densities.
What is a frequency polygon?
A frequency polygon is a line graph obtained by plotting frequencies against class marks and joining the points with straight line segments.
What is an ogive?
An ogive is a cumulative frequency curve. The two common types are the less-than ogive and the more-than ogive.
What is the difference between a histogram and a bar chart?
A histogram represents quantitative data grouped into numerical intervals and normally has adjoining bars. A bar chart generally represents categories and normally has gaps between the bars.
What is frequency density?
Frequency density is frequency divided by class width. It is especially important when histogram class intervals have unequal widths.
What is a class mark?
A class mark is the midpoint of a class interval and is calculated by adding the lower and upper class limits and dividing by 2.
Can an ogive be used to find the median?
Yes. A cumulative frequency graph can be used to obtain a graphical estimate of the median. The median can also be calculated for grouped data using the grouped-data median formula.
Finally,
Frequency distribution is a fundamental concept in statistics because it transforms raw data into an organized form that is easier to interpret and analyze. Depending on the nature and size of the data, we may use an ungrouped or grouped frequency distribution, relative frequencies or cumulative frequencies.
Graphs such as histograms, frequency polygons, frequency curves and ogives provide different ways of viewing the same underlying distribution. Understanding class intervals, class boundaries, class marks, cumulative frequency and frequency density is essential for constructing these tables and graphs correctly.
For examinations, the most important points to remember are simple: count each observation once, avoid overlapping classes, use appropriate class boundaries, calculate cumulative frequencies carefully, and choose the correct graph for the type of data.
Read More
- Objectives of Statistics
- Importance and Application of Statistics in Business and Management
- Characteristics of Statistics
- Nature of Statistics: Is Statistics a Science or an Art?
- जीवन समंक क्या है? जीवन समंकों का अर्थ और परिभाषा (What is Vital statistics? Meaning and definition of Vital statistics in Hindi)
- Characteristics of Statistics in Hindi - सांख्यिकी की विशेषताएं
Recommended Statistics Textbooks
The terminology and classical graphical methods covered in this article are consistent with standard introductory and university-level statistics texts, including:
- S. C. Gupta & V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons.
- S. P. Gupta, Statistical Methods.
- Murray R. Spiegel & Larry J. Stephens, Schaum's Outline of Statistics.
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