A frequency distribution is a way to organize data and see how often each value appears. It shows how many times each value or range of values occurs in a dataset. This helps us understand patterns, like which values are common and which are rare. Frequency distributions are often shown in tables or graphs, and make it easier to analyze and conclude data.
Frequency DistributionFrequency Distribution Graphs
To represent the Frequency Distribution, there are various methods such as a Histogram, Bar Graph, Frequency Polygon, and Pie Chart.
A brief description of all these graphs is as follows:
| Graph Type | Description | Use Cases |
|---|
| Histogram | Represents the frequency of each interval of continuous data using bars of equal width. | Continuous data distribution analysis. |
| Bar Graph | Represents the frequency of each interval using bars of equal width; it can also represent discrete data. | Comparing discrete data categories. |
| Frequency Polygon | Connects midpoints of class frequencies using lines, similar to a histogram but without bars. | Comparing various datasets. |
| Pie Chart | Circular graph showing data as slices of a circle, indicating the proportional size of each slice relative to the whole dataset. | Showing relative sizes of data portions. |
Frequency Distribution Table
A frequency distribution table is a way to organize and present data in a tabular form, showing which helps us summarize the large dataset into a concise table. In the frequency distribution table, there are two columns: one representing the data either in the form of a range or an individual data set and the other column showing the frequency of each interval or individual.
For example, let's say we have a dataset of students' test scores in a class.
Test Score | Frequency |
|---|
0-20 | 6 |
20-40 | 12 |
40-60 | 22 |
60-80 | 15 |
80-100 | 5 |
Check: Difference between Frequency Array and Frequency Distribution
Types of Frequency Distribution
There are four types of frequency distribution:
- Grouped Frequency Distribution
- Ungrouped Frequency Distribution
- Relative Frequency Distribution
- Cumulative Frequency Distribution
Grouped Frequency Distribution
In Grouped Frequency Distribution observations are divided between different intervals known as class intervals and then their frequencies are counted for each class interval. This Frequency Distribution is used mostly when the data set is very large.
Example: Make the Frequency Distribution Table for the ungrouped data given as follows:
23, 27, 21, 14, 43, 37, 38, 41, 55, 11, 35, 15, 21, 24, 57, 35, 29, 10, 39, 42, 27, 17, 45, 52, 31, 36, 39, 38, 43, 46, 32, 37, 25
Solution:
As there are observations in between 10 and 57, we can choose class intervals as 10-20, 20-30, 30-40, 40-50, and 50-60. In these class intervals, all the observations are covered and for each interval, there are different frequency which we can count for each interval.
Thus, the Frequency Distribution Table for the given data is as follows:
| Class Interval | Frequency |
|---|
10 - 20 | 5 |
20 - 30 | 8 |
30 - 40 | 12 |
40 - 50 | 6 |
50 - 60 | 3 |
Ungrouped Frequency Distribution
In Ungrouped Frequency Distribution, all distinct observations are mentioned and counted individually. This Frequency Distribution is often used when the given dataset is small.
Example: Make the Frequency Distribution Table for the ungrouped data given as follows:
10, 20, 15, 25, 30, 10, 15, 10, 25, 20, 15, 10, 30, 25
Solution:
As unique observations in the given data are only 10, 15, 20, 25, and 30 with each having a different frequency.
Thus, the Frequency Distribution Table of the given data is as follows:
| Value | Frequency |
|---|
10 | 4 |
15 | 3 |
20 | 2 |
25 | 3 |
30 | 2 |
Relative Frequency Distribution
This distribution displays the proportion or percentage of observations in each interval or class. It is useful for comparing different data sets or for analyzing the distribution of data within a set.
Relative Frequency is given by:
Relative Frequency = (Frequency of Event)/(Total Number of Events)
Example: Make the Relative Frequency Distribution Table for the following data:
| Score Range | 0-20 | 21-40 | 41-60 | 61-80 | 81-100 |
|---|
| Frequency | 5 | 10 | 20 | 10 | 5 |
|---|
Solution:
To Create the Relative Frequency Distribution table, we need to calculate Relative Frequency for each class interval. Thus Relative Frequency Distribution table is given as follows:
| Score Range | Frequency | Relative Frequency |
|---|
0-20 | 5 | 5/50 = 0.10 |
21-40 | 10 | 10/50 = 0.20 |
41-60 | 20 | 20/50 = 0.40 |
61-80 | 10 | 10/50 = 0.20 |
81-100 | 5 | 5/50 = 0.10 |
Total | 50 | 1.00 |
Cumulative Frequency Distribution
A cumulative frequency distribution shows the total number of observations up to and including a certain value or class. It helps in understanding how values accumulate across a dataset:
- Less than Type: We sum all the frequencies before the current interval.
- More than Type: We sum all the frequencies after the current interval.
Check:
Let's see how to represent a cumulative frequency distribution through an example.
Example: The table below gives the values of runs scored by Virat Kohli in the last 25 T-20 matches. Represent the data in the form of less-than-type cumulative frequency distribution:
| 45 | 34 | 50 | 75 | 22 |
| 56 | 63 | 70 | 49 | 33 |
| 0 | 8 | 14 | 39 | 86 |
| 92 | 88 | 70 | 56 | 50 |
| 57 | 45 | 42 | 12 | 39 |
Solution:
Since there are a lot of distinct values, we'll express this in the form of grouped distributions with intervals like 0-10, 10-20 and so. First let's represent the data in the form of grouped frequency distribution.
| Runs | Frequency |
|---|
0-10 | 2 |
10-20 | 2 |
20-30 | 1 |
30-40 | 4 |
40-50 | 4 |
50-60 | 5 |
60-70 | 1 |
70-80 | 3 |
80-90 | 2 |
90-100 | 1 |
Now we will convert this frequency distribution into cumulative frequency distribution by summing up the values of current interval and all the previous intervals.
| Runs scored by Virat Kohli | Cumulative Frequency |
|---|
Less than 10 | 2 |
Less than 20 | 4 |
Less than 30 | 5 |
Less than 40 | 9 |
Less than 50 | 13 |
Less than 60 | 18 |
Less than 70 | 19 |
Less than 80 | 22 |
Less than 90 | 24 |
Less than 100 | 25 |
This table represents the cumulative frequency distribution of less than type.
Runs scored by Virat Kohli | Cumulative Frequency |
|---|
More than 0 | 25 |
More than 10 | 23 |
More than 20 | 21 |
More than 30 | 20 |
More than 40 | 16 |
More than 50 | 12 |
More than 60 | 7 |
More than 70 | 6 |
More than 80 | 3 |
More than 90 | 1 |
This table represents the cumulative frequency distribution of more than type.
We can plot both the type of cumulative frequency distribution to make the Cumulative Frequency Curve.
Frequency Distribution Curve
A frequency distribution curve, also known as a frequency curve, is a graphical representation of a data set's frequency distribution. It is used to visualize the distribution and frequency of values or observations within a dataset. Let's understand its different types based on the shape of it, as follows:
Frequency Distribution Curve Types |
|---|
| Type of Distribution | Description |
|---|
| Normal Distribution | Symmetrical and bell-shaped; data concentrated around the mean. |
| Skewed Distribution | Not symmetric; can be positively skewed (right-tailed) or negatively skewed (left-tailed). |
| Bimodal Distribution | Two distinct peaks or modes in the frequency distribution, suggesting data from different populations. |
| Multimodal Distribution | More than two distinct peaks or modes in the frequency distribution. |
| Uniform Distribution | All values or intervals have roughly the same frequency, resulting in a flat, constant distribution. |
| Exponential Distribution | The rapid drop-off in frequency as values increase, resembling an exponential function. |
| Log-Normal Distribution | The logarithm of the data follows a normal distribution, often used for multiplicative data, positively skewed. |
Check: Grouping of Data
Various formulas can be learned in the context of Frequency Distribution, one such formula is the coefficient of variation. This formula for Frequency Distribution is discussed below in detail.
Frequency (f)= Number of times a value occurs/Total number of observations
Coefficient of Variation
We can use mean and standard deviation to describe the dispersion in the values. But sometimes while comparing the two series or frequency distributions becomes a little hard as sometimes both have different units.
The coefficient of Variation is defined as,
\bold{\frac{\sigma}{\bar{x}} \times 100}
Where,
- σ represents the standard deviation
- \bold{\bar{x}} represents the mean of the observations
Note: Data with greater C.V. is said to be more variable than the other. The series having lesser C.V. is said to be more consistent than the other.
Comparing Two Frequency Distributions with the Same Mean
We have two frequency distributions. Let's say \sigma_{1} \text{ and } \bar{x}_1are the standard deviation and mean of the first series and \sigma_{2} \text{ and } \bar{x}_2are the standard deviation and mean of the second series. The Coefficient of Variation(CV) is calculated as follows
C.V. of the first series = \frac{\sigma_1}{\bar{x}_1} \times 100
C.V. of second series = \frac{\sigma_2}{\bar{x}_2} \times 100
We are given that both series have the same mean, i.e.,
\bar{x}_2 = \bar{x}_1 = \bar{x}
So, now C.V. for both series are,
C.V. of the first series = \frac{\sigma_1}{\bar{x}} \times 100
C.V. of the second series = \frac{\sigma_2}{\bar{x}} \times 100
Notice that now, both series can be compared with the value of standard deviation only. Therefore, we can say that for two series with the same mean, the series with a larger deviation can be considered more variable than the other one.
Solved Examples of Frequency Distribution
Example 1: Suppose we have a series with mean of 20 and a variance of 100. Find out the Coefficient of Variation.
Solution:
We know the formula for Coefficient of Variation,
\frac{\sigma}{\bar{x}} \times 100
Given mean \bar{x} = 20 and variance \sigma^2 = 100.
Substituting the values in the formula,
\frac{\sigma}{\bar{x}} \times 100 \\ = \frac{20}{\sqrt{100}} \times 100 \\ = \frac{20}{10} \times 100 \\ = 200
Example 2: Given two series with Coefficients of Variation 70 and 80. The means are 20 and 30. Find the values of standard deviation for both series.
Solution:
In this question we need to apply the formula for CV and substitute the given values.
Standard Deviation of first series.
C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 70 = \frac{\sigma}{20} \times 100 \\ 1400 = \sigma \times 100 \\ 14 = \sigma
Thus, the standard deviation of first series = 14
Standard Deviation of second series.
C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 80 = \frac{\sigma}{30} \times 100 \\ 2400 = \sigma \times 100 \\ 24 = \sigma
Thus, the standard deviation of first series = 24
Example 3: Draw the frequency distribution table for the following data: 2, 3, 1, 4, 2, 2, 3, 1, 4, 4, 4, 2, 2, 2
Solution:
Since there are only very few distinct values in the series, we will plot the ungrouped frequency distribution.
| Value | Frequency |
1 | 2 |
2 | 6 |
3 | 2 |
4 | 4 |
Total | 14 |
Example 4: The table below gives the values of temperature recorded in Hyderabad for 25 days in summer. Represent the data in the form of less-than-type cumulative frequency distribution:
| 37 | 34 | 36 | 27 | 22 |
| 25 | 25 | 24 | 26 | 28 |
| 30 | 31 | 29 | 28 | 30 |
| 32 | 31 | 28 | 27 | 30 |
| 30 | 32 | 35 | 34 | 29 |
Solution:
Since there are so many distinct values here, we will use grouped frequency distribution. Let's say the intervals are 20-25, 25-30, 30-35. Frequency distribution table can be made by counting the number of values lying in these intervals.
| Temperature | Number of Days |
|---|
20-25 | 2 |
25-30 | 10 |
30-35 | 13 |
This is the grouped frequency distribution table. It can be converted into cumulative frequency distribution by adding the previous values.
| Temperature | Number of Days |
|---|
Less than 25 | 2 |
Less than 30 | 12 |
Less than 35 | 25 |
Example 5: Make a Frequency Distribution Table as well as the curve for the data:
{45, 22, 37, 18, 56, 33, 42, 29, 51, 27, 39, 14, 61, 19, 44, 25, 58, 36, 48, 30, 53, 41, 28, 35, 47, 21, 32, 49, 16, 52, 26, 38, 57, 31, 59, 20, 43, 24, 55, 17, 50, 23, 34, 60, 46, 13, 40, 54, 15, 62}
Solution:
To create the frequency distribution table for given data, let's arrange the data in ascending order as follows:
{13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62}
Now, we can count the observations for intervals: 10-20, 20-30, 30-40, 40-50, 50-60 and 60-70.
| Interval | Frequency |
|---|
| 10 - 20 | 7 |
| 20 - 30 | 10 |
| 30 - 40 | 10 |
| 40 - 50 | 10 |
| 50 - 60 | 10 |
| 60 - 70 | 3 |
From this data, we can plot the Frequency Distribution Curve as follows:
Read More:
Conclusion - Frequency Distribution
The frequency distribution provides a clear summary of how often each value or category occurs within a dataset. It allows us to see the distribution of values and understand the pattern or spread of data. By organizing data into groups and displaying their frequencies, we gain insights into the central tendency, variability, and shape of the data distribution.
This facilitates better understanding and interpretation of the dataset, aiding in decision-making, analysis, and communication of findings.
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice