# GSEB Class 11 Statistics Notes Chapter 2 Presentation of Data

This GSEB Class 11 Commerce Statistics Notes Chapter 2 Presentation of Data covers all the important topics and concepts as mentioned in the chapter.

## Presentation of Data Class 11 GSEB Notes

Meaning and need of Classification:
1. Numerical Variable:
A numeric characteristic that varies from unit to unit of a population or sample is called numeric variable. For example, age of a person, price of an item, profit of a firm, number of children per family, etc. are numeric variables.

2. Qualitative Variable or Attribute:
A qualitative characteristic that varies from unit to unit of a population or sample cannot be measured numerically but can be described is called qualitative variable or attribute. For example, profession of a person, efficiency of a worker, etc. are attributes.

3. Types of Numerical Variable:

• Discrete Variable: If a variable can assume definite or countable values within the specified range, then it is called discrete variable. For example, number of flowers per plant, number of accident on a road, etc. are discrete variables.
• Continuous Variable: If a variable can assume any value within the specified range, then it is called continuous variable. For example, age of a person (in years), weight of a student (in kg), salary of an employee (in ₹), temperature of a day (in Celsius), etc. are continuous variable.

4. Discrete Data and Continuous Data:
The data on discrete variable is discrete data, while the data on continuous variable is continuous data.

5. Raw Data:
Data obtained by population inquiry or sample inquiry are called original data or ungrouped data or raw data. For example, marks of 100 students in the subject statistics.

6. Classification:
A process of arranging ungrouped or raw data in systematic and short form is called classification.

7. Classified Data:
The data obtained by classification are called classified data or grouped data.

8. Need of Classification:

• To represent large data into simple, short and attractive manner,
• to compare the various characteristics of the data and
• to analyse the data by saving time, money and labour.

Types of Classification:
1. Quantitative Classification:

• Classification of data on numeric variable, discrete and continuous is called quantitative classification. It is also known as numerical classification or frequency distribution.
• Frequency Distribution: If discrete or continuous data are classified according to the value of the variable, then it is called frequency distribution.

Types of Frequency Distribution:
1. Discrete Frequency Distribution: When discrete data are classified according to the values of a variable showing how frequent each value occurs, then it is called discrete frequency distribution. Thus, a table showing various possible values of discrete variable with their respective frequencies is called discrete frequency distribution. For example, the table showing the number of families as per the number of children is discrete frequency distribution.

2. Continuous Frequency Distribution: When the variable of raw data is continuous or range of the data is large, then dividing the range of data into fix number of groups or classes, data are classified in the manner, how many values of the variable occur in each class. Thus, a table showing various classes with their respective frequencies is called continuous frequency distribution. For example, the table showing the number of companies according to various classes of the profit (in ₹) earned during a year.

Frequency:
A numeric value showing the repetition of value of variable is called the frequency (f) of that value. Similarly, the number of observations corresponding to each class is the frequency of that class.

Range:
The difference between the highest value and the lowest value of the data is called range.

Class or Class Interval:
The interval obtained by two fixed values of the variable is called a class. For example, 10- 14, 15-19, 20-24 etc.

Lower Limit:
The lowest value of a class is called the lower limit of that class. For example, the lower limit of the class 10- 14 is 10.

Upper Limit:
The highest value of a class is called the upper limit of that class. For example, the upper limit of class 10-14 is 14.

Exclusive Class:
If the upper limit of a class is not included in that class but is included in the next class, such a class is called exclusive class. In exclusive class, the upper limit of any class is the lower limit of the next class, e.g., the classes 10-20, 20-30, 30-40. … are exclusive classes. Here, the upper limit 20 of class 10-20 is not included in class 10-20, but it is included in the class 20 – 30. The frequency distribution having exclusive classes is called exclusive type frequency distribution. It is carried out for the continuous raw data.

Inclusive Class:
If the upper limit of a class is included in the class itself, such a class is called inclusive class. In inclusive class, the upper limit of any class and the lower limit of the next class are not equal, e.g., the classes 10-19, 20-29, 30-39, … are inclusive classes. Here, the upper limit 19 of class 10-19 is included in the class itself. The frequency distribution having inclusive classes is called inclusive type frequency distribution. It is carried out for the discrete raw data having large value of range.

Conversion of Inclusive Type Continuous Frequency Distribution into Exclusive Type Continuous Frequency Distribution:
For such conversion class limits are expressed as class boundary points.
Class Boundary Points:

• Lower Boundary Point: It is an average of lower limit of a class and the upper limit of previous class. For example, for classes 10-14, 15-19, 20-24, …, etc. lower boundary point of class 15 – 19 is $$\frac{15+14}{2}$$ = 14.5.
• Upper Boundary Point: It is an average of the upper limit of a class and the lower limit of succeeding class. For example, for classes 10- 14, 15- 19. 20-24, …, etc. upper boundary point of class 15-19 is $$\frac{19+20}{2}$$ = 19.5.

[Note; For exclusive classes class limits are the class boundary points. For example, for classes 10-20, 20 – 30, 30 – 40 etc. lower limit and lower boundary point of the class 20 – 30 are 20 and upper limit and upper boundary point of the class 20-30 are 30.)

Mid Value:
The value obtained by dividing the sum of the values of the lower limit and the upper limit of a class is called the middle value or mid value of the class, e.g., the mid value of the class
10-14 = $$\frac{10+14}{2}$$ = 12.

Class Length:
The difference between the values of the upper and lower boundary points of any class is called the class length of that class, e.g., the class length of class 15 – 19 = 19.5 – 14,5 = 5.

Cumulative Frequency:
The sum of the frequencies of values less than or equal to some specified value of the variable is called the cumulative frequency of that specified value of a discrete distribution. Similarly, the sum of frequencies of the classes preceding to the specified class and the frequency of the specified class is called the cumulative frequency of that specified class of a continuous frequency distribution.

• Less than’ Cumulative Frequency:
In continuous frequency distribution, the ‘less than’ cumulative frequency of a given class is the sum of frequencies of all classes which include all observations less than or equal to the upper boundary point of that class. Such cumulative frequencies are in ascending order.
• More than’ Cumulative Frequency:
In continuous frequency distribution, the ‘more than’ cumulative frequency of a given class is the sum of frequencies of all classes which include all observations more than or equal to the lower boundary point of that class. Such cumulative frequencies are in descending order.

Cumulative Frequency Distribution:
A table showing the cumulative frequency according to the value or a class of values is called cumulative frequency distribution.

• Discrete Cumulative Frequency Distribution: The cumulative frequency distribution obtained by considering the value of a discrete variable is called discrete cumulative frequency distribution.
• Continuous Cumulative Frequency Distribution: The cumulative fre¬quency distribution obtained by considering the boundary points is called continuous cumulative frequency distribution.

Types of Cumulative Frequency Distribution:

• ‘Less than’ Type Cumulative Frequency Distribution: The distribution obtained by each value of the variable and its corresponding cumulative frequency of a discrete frequency distribution is called a discrete cumulative frequency distribution of ‘less than’ type. Similarly, the distribution obtained by the upper boundary point of each class and its corresponding cumulative frequency is called a continuous cumulative frequency distribution of ‘less than’ type.
• ‘More than’ Type Cumulative Frequency Distribution: The distribution obtained by each value of the variable and its corresponding ‘more than’ cumulative frequency of a discrete frequency distribution is called a discrete cumulative frequency distribution of ‘more than’ type. Similarly, the distribution representing the lower boundary points and their corresponding cumulative frequencies is called ‘more than’ type continuous cumulative frequency distribution.

2. Qualitative Classification:
Classification of data according to the attributes of the information by arranging them in rows and columns is called qualitative classification. It is known as Tabulation.

Types of Qualitative Classification:

• Simple Classification: A classification on the basis of a single attribute is called simple classification or tabulation. For example, classification of the data of employees of a company on the basis of their status in the company.
• Manifold Classification: A classification of row data carried out by considering more than one attribute under study is called manifold, classification or tabulation. For example, classification of data of employees of a company on the basis of their sex and status of working in the company.

Tabulation:
Tabulation is a process of arranging in systematic manner the qualitative data into rows and columns on the basis of the attributes. In the tabulation title of the table, sources of the data and explanation of data are given.

Uses of Tabulation:

• Represents the extensive data in simple, organised and precise manner,
• required information can be obtained easily
• various characteristics to be compared are placed side by side. Hence comparison becomes easy
• row and/or column totals are found, hence errors can be rectified easily.
• unnecessary information is removed, hence the time, money and labour required for the study of data is saved and
• the analysis of the data becomes simple and convenient.

Rules of Tabulation:

• Appropriate title should be given
• there should be clear and simple captions to the rows and column
• size of the table should be proportionate to the space available
• the interrelated information should be placed adjacent to each other,
• large numbers should be represented in hundred, thousands, lakhs or crores,
• separate lines should be drawn to distinguish the main characteristics of the data
• provision for indicating the totals of primary and subsidiary characteristics should be there in a table
• large volume of data should be represented in different- tables instead of a single table
• source of the data must be mentioned at the end of the table and
• before preparing the final table, a rough table should be prepared.

Diagram:
Diagram is a tool to represent huge and complex data into simple and attractive manner in order to understand the data easily.

Importance and Limitations of Diagram:
Importance:

• Represents the data in attractive, simple and concise form
• the data expressed by diagrams are remembered for longer time
• saves time in representing the data
• the comparative study of the data becomes very simple
• easily understood by the illiterates, less educated or even by children
• easy to understand irrespective of language barriers.

Limitations:

• Lac of accuracy in drawing diagrams leads to wrong interpretation
• Illusionary effect of diagrams misleads the public opinion and
• there is a loss of accuracy of the data.

Types of Diagram:
1. One Dimensional Diagram: A diagram drawn by considering only one characteristic of the data is called one dimensional diagram.

• Bar Diagram: It is drawn considering only one characteristic such as different places, things or time.
• Multiple or Adjacent Bar Diagram: It is drawn, if the data about different places, things or times on more than one mutually related characteristics are given.
• Simple Divided Bar Diagram: It is drawn if the data about different places, things or times consists of several mutually related sub-data.
• Percentage Divided Bar Diagram: It is drawn when mutually related sub-data are to be compared effectively.

2. Two Dimensional Diagrams: When the volume of the data is large, then considering length and breadth, two-dimensional diagrams are drawn.

• Circle Diagram: It is drawn when the volume of data regarding two or more places, things or times is large.
• Pie Diagram: It is drawn, when the data related to different places, things or times consist of several mutually related sub-data on different components are numerically large.’

3. Pictogram: A diagram in which the data are represented by appropriate pictures is called pictorial diagram or pictogram. It has no barrier of language.

Important Formulae:
1. Class Boundary Points:

• Upper boundary point of any class = $$=\frac{\text { upper limit of the given class }+\text { lower limit of the succeeding class }}{2}$$
• Lower boundary point of any class = $$=\frac{\text { lower limit of the given class }+\text { upper limit of the succeeding class }}{2}$$

2. Mid Value:
Mid value = $$\frac{\text { lower limit }+\text { upper limit }}{2}$$

3. Class Length:
Class length = upper boundary point – lower boundary point range R
Class length = $$\frac{\text { range }+\text { no. of classes }}{2}$$ i.e., c = $$\frac{R}{k}$$, ck ≥ R
Where, c = class length
k = no. of classes
R = range = value of the highest observation – value of the lower observation

4. Class Limits:
Upper limit = Mid value + $$\frac{\text { class length }}{2}$$; Lower limit = Mid value – $$\frac{\text { class length }}{2}$$