CU-BCA-SEM-III-PROBABILITY AND STATICS- Second Draft-converted

BACHELOR OF COMPUTER APPLICATIONS SEMESTER III PROBABILITY AND STATISTICS BCA133

2 CU IDOL SELF LEARNING MATERIAL (SLM)

CHANDIGARH UNIVERSITY Institute of Distance and Online Learning Course Development Committee Prof. (Dr.) R.S.Bawa Pro Chancellor, Chandigarh University, Gharuan, Punjab Advisors Prof. (Dr.) Bharat Bhushan, Director – IGNOU Prof. (Dr.) Majulika Srivastava, Director – CIQA, IGNOU Programme Coordinators & Editing Team Master of Business Administration (MBA) Bachelor of Business Administration (BBA) Coordinator – Dr. Rupali Arora Coordinator – Dr. Simran Jewandah Master of Computer Applications (MCA) Bachelor of Computer Applications (BCA) Coordinator – Dr. Raju Kumar Coordinator – Dr. Manisha Malhotra Master of Commerce (M.Com.) Bachelor of Commerce (B.Com.) Coordinator – Dr. Aman Jindal Coordinator – Dr. Minakshi Garg Master of Arts (Psychology) Bachelor of Science (Travel &Tourism Management) Coordinator – Dr. Samerjeet Kaur Coordinator – Dr. Shikha Sharma Master of Arts (English) Bachelor of Arts (General) Coordinator – Dr. Ashita Chadha Coordinator – Ms. Neeraj Gohlan Academic and Administrative Management Prof. (Dr.) R. M. Bhagat Prof. (Dr.) S.S. Sehgal Executive Director – Sciences Registrar Prof. (Dr.) Manaswini Acharya Prof. (Dr.) Gurpreet Singh Executive Director – Liberal Arts Director – IDOL © No part of this publication should be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the authors and the publisher. SLM SPECIALLY PREPARED FOR 3 CU IDOL STUDENTS First Printed and Published by: TeamLease Edtech Limited www.teamleaseedtech.com CONTACT NO:- 01133002345 For: CHANDIGARCHU IUDONLISVELEFRLESAIRTNYING MATERIAL (SLM) Institute of Distance and Online Learning

Published in 2021 All rights reserved. No Part of this book may be reproduced or transmitted, in any form or by any means, without permission in writing from Chandigarh University. Any person who does any unauthorized act in relation to this book may be liable to criminal prosecution and civil claims for damages. This book is meant for educational and learning purpose. The authors of the book has/have taken all reasonable care to ensure that the contents of the book do not violate any existing copyright or other intellectual property rights of any person in any manner whatsoever. In the event the Authors has/ have been unable to track any source and if any copyright has been inadvertently infringed, please notify the publisher in writing for corrective action. CONTENTS 4 CU IDOL SELF LEARNING MATERIAL (SLM)

Unit 1: Statistics.........................................................................................................................6 Unit 2: Frequency Distribution & Mean..................................................................................18 Unit 3: Median .........................................................................................................................44 Unit 4: Mode And Mean Deviation .........................................................................................63 Unit 5: Measures Of Dispersion ..............................................................................................89 Unit 6: Correlation And Regression ......................................................................................122 Unit 7: Types Of Correlation, Meaning, Uses Of Regression Analysis ................................150 Unit 8: Relationship Between Correlation And Regression Analysis ...................................171 Unit 9: Probability..................................................................................................................180 Unit 10: Probability, Random Variable And Its Probability Distribution .............................206 Unit 11: Mean And Variance Of Haphazard Variable...........................................................224 5 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 1: STATISTICS Structure 1.0 Learning Objectives 1.1 Introduction 1.2 Data Collection 1.3 Techniques of Data Collection 1.4 Methods of Primary Data Collection 1.5 Sources of Secondary Data 1.6 Measures of Central Tendency 1.7 Summary 1.8 Keywords 1.9 Learning activity 1.10 Unit End Questions 1.11 References 1.0 LEARNING OBJECTIVES After studying this Unit students will be able to: • Explain about importance of data collection • Describe Techniques of data collection • Analyse about Secondary data and its sources • Outline about measures of central tendency 1.1 INTRODUCTION Statistics has been defined differently by different authors from time to time, the reason for a variety of definitions are primarily two, First, in modern times the field of utility of statistics has widened considerably. In ancient times statistics was confined only to the affairs of the state but now it embraces almost every sphere of human activity. Hence a number of odd definitions which were confined to a very narrow field of enquiry were replaced by new definitions which are much more comprehensive and exhaustive. Secondly Statistics has been defined in two ways. Some writer 6 CU IDOL SELF LEARNING MATERIAL (SLM)

s defines it as 'statistical data', that is numerical statement of facts, while others define it as ' statistical methods', i.e., complete body of the principles and techniques used in collecting and analysing statistics as 'statistical data'. Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory. Statisticians, people who do statistics, are particularly concerned with determining how to draw reliable conclusions about large groups and general phenomena from the observable characteristics of small samples that represent only a small portion of the large group or a limited number of instances of a general phenomenon. The two major areas of statistics are known as descriptive statistics, which describes the properties of sample and population data, and inferential statistics, which uses those properties to test hypotheses and draw conclusions. Some common statistical tools and procedures include the following: • Descriptive • Mean (average) • Variance • Skewness • Kurtosis • Inferential • Liner regression analysis • Analysis of variance (ANOVA) • Logit/Probit models • Null hypothesis testing Understanding Statistics Statistics are used in virtually all scientific disciplines such as the physical and social sciences, as well as in business, the humanities, government, and manufacturing. Statistics is fundamentally a branch of applied mathematics that developed from the application of mathematical tools including calculus and linear algebra to probability theory. In practice, statistics is the idea we can learn about the properties of large sets of objects or events (a population) by studying the characteristics of a smaller number of similar objects or events (a sample). Because in many cases gathering comprehensive data about an entire population is too 7 CU IDOL SELF LEARNING MATERIAL (SLM)

costly, difficult, or flat out impossible, statistics start with a sample that can conveniently or affordably be observed. Two types of statistical methods are used in analyzing data: descriptive statistics and inferential statistics. Statistician’s measure and gather data about the individuals or elements of a sample, then analyze this data to generate descriptive statistics. They can then use these observed characteristics of the sample data, which are properly called \"statistics,\" to make inferences or educated guesses about the unmeasured (or unmeasured) characteristics of the broader population, known as the parameters. Descriptive Statistics Descriptive statistics mostly focus on the central tendency, variability, and distribution of sample data. Central tendency means the estimate of the characteristics, a typical element of a sample or population, and includes descriptive statistics such as mean, median, and mode. Variability refers to a set of statistics that show how much difference there is among the elements of a sample or population along the characteristics measured, and includes metrics such as range, variance, and standard deviation. The distribution refers to the overall \"shape\" of the data, which can be depicted on a chart such as a histogram or dot plot, and includes properties such as the probability distribution function, skewness, and kurtosis. Descriptive statistics can also describe differences between observed characteristics of the elements of a data set. Descriptive statistics help us understand the collective properties of the elements of a data sample and form the basis for testing hypotheses and making predictions using inferential statistics. Inferential Statistics Inferential statistics are tools that statisticians use to draw conclusions about the characteristics of a population from the characteristics of a sample and to decide how certain they can be of the reliability of those conclusions. Based on the sample size and distribution of the sample data statisticians can calculate the probability that statistics, which measure the central tendency, variability, distribution, and relationships between characteristics within a data sample, provide an accurate picture of the corresponding parameters of the whole population from which the sample is drawn. Inferential statistics are used to make generalizations about large groups, such as estimating average demand for a product by surveying a sample of consumers' buying habits, or to attempt to predict future events, such as projecting the future return of a security or asset class based on returns in a sample period. Regression analysis is a common method of statistical inference that attempts to determine the strength and character of the relationship (or correlation) between one dependent variable (usually denoted by Y) and a series of other variables (known as independent variables). The output of a 8 CU IDOL SELF LEARNING MATERIAL (SLM)

regression model can be analyzed for statistical significance, which refers to the claim that a result from findings generated by testing or experimentation is not likely to have occurred randomly or by chance but are instead likely to be attributable to a specific cause elucidated by the data. Having statistical significance is important for academic disciplines or practitioners that rely heavily on analyzing data and research. Difference between descriptive and inferential statistics Descriptive statistics are used to describe or summarize the characteristics of a sample or data set, such as a variable's mean, standard deviation, or frequency. Inferential statistics, in contrast, employs any number of techniques to relate variables in a data set to one another, for example using correlation or regression analysis. These can then be used to estimate forecasts or infer causality. 1.2 DATA COLLECTION Statistical technique is the technique which is used in conducting the statistical enquiry pertaining to definite Phenomenon. They include all the statistical methods beginning from the collection of data till interpretation of collected data. Statistical techniques involve, Collection of data Statistical investigation requires systematic collection of data, so that all significant groups are represented in the data. To establish the potential market for a latest product, in this case the researcher might study 2500 consumers in a particular geographical area. It must be assured that the group contains people who represent the variables such as race, income level, education and neighbourhood. The data which we have collected from various sources will greatly affect the results and hence, highest importance should be given to this process and every possible safety measure should be taken to make sure the accuracy, while gathering and collecting data. Depending upon the sources of data utilized during collection of data that are used for normal purposes, the statistical data can be classified into two types i.e., primary and secondary. It is the process of assembling and evaluating the data from different sources for different parameters. Accuracy of data collected is necessary to maintain the integrity of research. The main objective of data collection is to assemble quality information so that it becomes easy for the retriever to access it. 1.3 TECHNIQUES OF DATA COLLECTION Data Collection starts with decisions of what, where, how, when and from where to collect the info from. So, there are two types to collect data: - 9 CU IDOL SELF LEARNING MATERIAL (SLM)

Primary Data This is raw form of information, directly collected from the source. It has to be original. There are two types of Primary data: - • Quantitative Data: - As the name suggest this type of data mainly expressed in numbers. It is easy to collect and much more reliable. It is a formal kind of approach which can be collected through questionnaire in short span of time. The data collected are from “closed questions” which means the respondent can reply to only pre-mentioned answers. • Qualitative Data: - These can mainly be collected from group discussions and personal interviews. It provides variety of data as there is no specific limit for its answers. These are much more descriptive in its kind. Secondary Data It consists of information which is already been mentioned somewhere, may be collected for some other purpose. It has already gone through various processes to authenticate it. 1.4 METHODS OF PRIMARY DATA COLLECTION 1. Observation Method This is a form of Qualitative approach in which the data is collected from the observer point of view. In the words of P.V Young, “Observation may be defined as systematic viewing, coupled with consideration of seen phenomenon.” • Types of Observations • Structured Observation: There are some standardized forms which are predetermined to do specific observation. • Unstructured Observation: When no specific thought is given prior to observation. • Controlled Observation: When the observation is done according to pre-determined procedure. • Uncontrolled Observation: It presents spontaneous picture of the situation which has taken place in natural conditions. • Advantages of Observation • No bias Information • Its current information • It is independent view of observer • Disadvantages of Observation • It’s expensive and consumes more time • Extracts limited information 10 CU IDOL SELF LEARNING MATERIAL (SLM)

• Unforeseen factors may interfere with observational tasks • Mostly people are not directly available to get the right observation 2. Interview Method In this the information is collected through enquiry. This is in the form of oral-verbal communication. In this the interviewer asks questions to extract information from the interviewee. • Types of Interview • Structured Interview: In this face-to-face communication is done. It consists of predetermined questions, standard technique to record. Interviewer asks questions from prescribed format. • Unstructured Interview: This type of interview is more conversational as there is flexibility to ask questions. Sometimes it consumes more time. Interviewer has freedom to ask questions without following sequence. • Advantages of Interview • Data collected is usable • Response can be assured • Interviewer can supervise and control • Flexibility to restructure the questions • Open-ended questions can any time be asked • Disadvantages of Interview • Consumes more time • Comparatively costly • Leads to biased results 3. Questionnaire Method Number of Questions is designed to extract information from the participants. These questionnaires can be mailed to the respondents, and asked them to reply back with their answers Types of Questionnaire • Close ended Questions: It consists of structured alternatives with yes or no as options. These are easy to answer and involves less time. Types of Close ended questions • Dichotomous questions: Respondent has to make a choice between two responses i.e., yes/no, male/female. • Multiple Choice Questions: This offers more than one choice • Rank order Questions: Respondent is asked to choose from the “most” to the “last” • Rating Questions: Respondents are asked to reply along ordered dimensions. 11 CU IDOL SELF LEARNING MATERIAL (SLM)

• Open Ended Questions: In this respondent is open to answer and reflect his feeling about the asked topics in the questionnaire. • Advantages of Questionnaire • Relatively simple method • Consumes less time • Data can be collected from widely scattered data • Disadvantages of Questionnaire • Respondent may ignore some of the questions • Printing may become costly • Responses may lack depth of answers 4. Survey Method: - This method is generally adopted to diagnose and solve the social problems. It consists of describing, recording, analyzing and interpreting conditions that exists. These are concerned with conditions that exists, process that are going on, trends that are developing. They mainly consider present happenings but sometimes refer past to relate to current conditions. • Types of Survey • Census survey • Sample Survey • Social Survey • Economic Survey • Public Opinion survey • Advantages of Survey • No biased opinions • Structured nature of questions reduces bias nature from researcher as well • Disadvantages of Survey • Respondent may show disinterest to answer as he does not know the purpose. • It’s difficult to check reliability and validity of results 1.5 SOURCES OF SECONDARY DATA 1. Publications of Central, state and local government 12 2. Technical and trade journals 3. Books, magazines newspapers 4. Reports and publications of industry, bank, stock exchange 5. Reports by researchers and economists CU IDOL SELF LEARNING MATERIAL (SLM)

6. Public records 1.6 MEASURES OF CENTRAL TENDENCY A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode. The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used. The central tendency measure is defined as the number used to represent the centre or middle of a set of data values. The three commonly used measures of central tendency are the mean, median, and mode. A statistic that tells us how the data values are dispersed or spread out is called the measure of dispersion. A simple measure of dispersion is the range. The range is equivalent to the difference between the highest and least data values. Another measure of dispersion is the standard deviation, representing the expected difference (or deviation) among a data value and the mean. 1.7 SUMMARY • Data collection is the process of gathering and measuring information on variables of interest, in an established systematic fashion that enables one to answer stated research questions, test hypotheses, and evaluate outcomes. • A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. • The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode. 1.8 KEYWORDS • Population: A collection of persons, things, or objects under study. • Sample: It must contain the characteristics of the population. • Statistic: A number that represents a property of the sample 13 CU IDOL SELF LEARNING MATERIAL (SLM)

• Parameter: A numerical characteristic of the whole population that can be estimated by a statistic. • Variable: Usually notated by capital letters such as X and Y, is a characteristic or measurement that can be determined for each member of a population. • Data: The actual values of the variable. 1.9 LEARNING ACTIVITY The random sample is drawn, observer forms are developed and distributed, and observers are recruited and trained. On Election Day, observers take up positions at assigned polling stations and get ready to collect and report the data. On Election Day, domestic observers usually make two reports. Form 1 report which contains information about whether proper procedures have been followed during the opening of polling stations. This first qualitative report is made after the polling stations have opened, usually immediately after the first voter in line has voted. The second report comes from a form similar to Form 2, This provides qualitative data on the voting procedures and the closing of the polling stations, as well as data on the vote count. The common practice is for observers to report these data immediately after polling stations have produced an official result. In most cases, a polling station result is “official” after the polling station officials and the party agents present at the count have signed the public document that records the vote totals for that particular polling station. This chapter focuses primarily on the official vote data (Form 2), but there are broad issues of data reporting that apply to all observer reports. So, the place to begin is with general guidelines that apply to both the first and second reports. For each report, observers make three separate calls; they report the same data to three different locations. • Call 1: Observers make the first call directly to the central data collection centre. • Call 2: Observers make the second call to their assigned regional coordinator. • Call 3: Observers make the third call to a back-up network of private telephones in the capital city. 1.10 UNIT END QUESTIONS A. Descriptive Questions 14 Short Questions 1. Define Descriptive Statistics 2. Define Inferential Statistics CU IDOL SELF LEARNING MATERIAL (SLM)

3. What are the three measures of central tendency used to summarize data? 4. What are the 4 measures of central tendency? 5. How do you explain central tendency? Long Answer Questions: 1. Explain the different types of Data Collection. 2. Explain the measure of central tendency. 3. Write is measure of central tendency and explain the role of central tendency in understanding data? 4. Explain the difference between Data Mining and Data Analysis? 5. Write the important steps in the data validation process? B. Multiple choice Questions 1. The specific statistical methods that can be used to summarize or to describe a collection of data is called: a. Descriptive statistics b. Inferential statistics c. Analytical statistics d. All of these 2. The need for inferential statistical methods derives from the need for ______________. a. Population b. Association c. Sampling d. Probability 3. A population, in statistical terms, is the totality of things under consideration. It is the collection of all values of the _________________ that is under study. a. Instance b. Variable 15 CU IDOL SELF LEARNING MATERIAL (SLM)

c. Amount d. Measure 4. Non-sampling errors are introduced due to technically faulty observations or during the______________________ of data. a. Processing b. Analysis c. Sequencing d. Collection 5. Sampling is simply a process of learning about the __________________ on the basis of a sample drawn from it. a. Census b. Population c. Group d. Area Answers 1. b, 2.c, 3.b, 4.a, 5.b 1.11 REFERENCES Reference Books: • Dr. J. Ravichandran, Probability & Statistics for Eng., Willey Publications. • Quantitative Methods for Business & Economics by Mouhammed, Publisher: PHI, 2007 Edition. • Quantitative Techniques for Managerial Decisions by A. Sharma, Publisher: Macmillan, 2008 Edition. Textbooks: 16 CU IDOL SELF LEARNING MATERIAL (SLM)

• S.C. Gupta, V.K. Kapoor, Fundamental of Mathematical Statistics, Sultan Chand and Company. • Quantitative Methods for Business & Economics by Mouhammed, Publisher: PHI, 2007 Edition • Quantitative Techniques for Managerial Decisions by A. Sharma, Publisher: Macmillan, 2008 Edition. 17 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 2: FREQUENCY DISTRIBUTION & MEAN Structure 2.0 Learning Objectives 2.1 Introduction 2.2 Graphical Representation of a Frequency Distribution 2.3 Types of Bar Charts 2.4 Frequency Polygon 2.5 Mean 2.6 Merits and Demerits of Arithmetic Mean 2.7 weighted Mean 2.8 Summary 2.9 Keywords 2.10 Learning activity 2.11 Unit End Questions 2.12 References 2.0 LEARNING OBJECTIVES After studying this unit students will be able to: • Explain a frequency table of quantitative data. • Differentiate a histogram or a frequency polygon of quantitative data. • Differentiate normal distribution, positively skewed distribution and negatively skewed distribution. • State that means median and mode is a measure of central location. • Analyse how to choose the appropriate measure of central location based on the shape of the distribution. • Outline that standard deviation is a measure of dispersion/variation. 2.1 INTRODUCTION A frequency distribution is a representation, either in a graphical or tabular format that displays the number of observations within a given interval. The interval size depends on the data being 18 CU IDOL SELF LEARNING MATERIAL (SLM)

analyzed and the goals of the analyst. The intervals must be mutually exclusive and exhaustive. Frequency distributions are typically used within a statistical context. Generally, frequency distribution can be associated with the charting of a normal distribution. • Frequency distribution in statistics is a representation that displays the number of observations within a given interval. • The representation of a frequency distribution can be graphical or tabular so that it is easier to understand. • Frequency distributions are particularly useful for normal distributions, which show the observations of probabilities divided among standard deviations. • In finance, traders use frequency distributions to take note of price action and identify trends. Example 1: Sam's team has scored the following numbers of goals in recent games 2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3 Sam put the numbers in order, and then added up: • how often 1 occurs (2 times), • How often 2 occurs (5 times), etc, and wrote them down as a Frequency Distribution table. From the table we can see interesting things such as • getting 2 goals happens most often • only once did they get 5 goals Score Frequency 12 25 34 42 51 Example 2: These are the numbers of newspapers sold at a local shop over the last 10 days: 22, 20, 18, 23, 20, 25, 22, 20, 18, 20 19 CU IDOL SELF LEARNING MATERIAL (SLM)

Let us count how many of each number there is: Papers Sold Frequency 18 2 19 0 20 4 21 0 22 2 23 1 24 0 25 1 It is also possible to group the values. Here they are grouped in 5s: Papers Sold Frequency 15-19 2 20-24 7 25-29 1 2.2 GRAPHICAL REPRESENTATION OF A FREQUENCY DISTRIBUTION After creating a Frequency Distribution table you might like to make a Bar Graph or a Pie Chart using the Data Graphs (Bar, Line and Pie) page. Histogram A histogram is a graphical representation that organizes a group of data points into user-specified ranges. Similar in appearance to a bar graph, the histogram condenses a data series into an easily interpreted visual by taking many data points and grouping them into logical ranges or bins. • A histogram is a bar graph-like representation of data that buckets a range of outcomes into columns along the x-axis. 20 CU IDOL SELF LEARNING MATERIAL (SLM)

• The y-axis represents the number count or percentage of occurrences in the data for each column and can be used to visualize data distributions. • In trading, the MACD histogram is used by technical analysts to indicate changes in momentum. For example, a census focused on the demography of a country may use a histogram to show how many people are between the ages of 0 - 10, 11 - 20, 21 - 30, 31 - 40, 41 - 50, etc. This histogram would look similar to the example below. Bar Graph The pictorial representations of a grouped data, in the form of vertical or horizontal rectangular bars, where the lengths of the bars are equivalent to the measure of data, are known as bar graphs or bar charts. The bars drawn are of uniform width, and the variable quantity is represented on one of the axes. Also, the measure of the variable is depicted on the other axes. The heights or the lengths of the bars denote the value of the variable, and these graphs are also used to compare certain quantities. The frequency distribution tables can be easily represented using bar charts which simplify the calculations and understanding of data. The three major attributes of bar graphs are: • The bar graph helps to compare the different sets of data among different groups easily. 21 CU IDOL SELF LEARNING MATERIAL (SLM)

• It shows the relationship using two axes, in which the categories on one axis and the discrete values on the other axis. • The graph shows the major changes in data over time. 2.3 TYPES OF BAR CHARTS The bar graphs can be vertical or horizontal. The primary feature of any bar graph is its length or height. If the length of the bar graph is more, then the values are greater than any given data. Bar graphs normally show categorical and numeric variables arranged in class intervals. They consist of an axis and a series of labelled horizontal or vertical bars. The bars represent frequencies of distinctive values of a variable or commonly the distinct values themselves. The number of values on the x-axis of a bar graph or the y-axis of a column graph is called the scale. The types of bar charts are as follows: 1. Vertical bar chart 2. Horizontal bar chart Vertical Bar Graphs When the grouped data are represented vertically in a graph or chart with the help of bars, where the bars denote the measure of data, such graphs are called vertical bar graphs. The data is represented along the y-axis of the graph, and the height of the bars shows the values. Horizontal Bar Graphs When the grouped data are represented horizontally in a chart with the help of bars, then such graphs are called horizontal bar graphs, where the bars show the measure of data. The data is depicted here along the x-axis of the graph, and the length of the bars denotes the values. Properties of Bar Graph Some of the important properties of a bar graph are as follows: • All the bars should have a common base. • Each column in the bar graph should have equal width. • The height of the bar should correspond to the data value. • The distance between each bar should be the same. Advantages and Disadvantages of Bar Chart Advantages: 22 CU IDOL SELF LEARNING MATERIAL (SLM)

• Bar graph summarises the large set of data in simple visual form. • It displays each category of data in the frequency distribution. • It clarifies the trend of data better than the table. • It helps in estimating the key values at a glance. Disadvantages: • Sometimes, the bar graph fails to reveal the patterns, cause, effects, etc. • It can be easily manipulated to yield fake information. Difference between Bar Graph and Histogram The bar graph and the histogram look similar. But it has an important difference. The major difference between them is that they plot different types of data. In the bar chart, discrete data is plotted, whereas, in the histogram, it plots the continuous data. For instance, if we have different categories of data like types of dog breeds, types of TV programs, the bar chart is best as it compares the things among different groups. For example, if we have continuous data like the weight of the people, the best choice is the histogram. Example 1: Sam's team has scored the following numbers of goals in recent games 2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3 Sam put the numbers in order, then added up: • how often 1 occurs (2 times), • how often 2 occurs (5 times), etc, and wrote them down as a Frequency Distribution table. 23 CU IDOL SELF LEARNING MATERIAL (SLM)

From the table we can see interesting things such as • getting 2 goals happens most often • only once did they get 5 goals Example 2: These are the numbers of newspapers sold at a local shop over the last 10 days: 22, 20, 18, 23, 20, 25, 22, 20, 18, 20 • Let us count how many of each number there is: Papers Sold Frequency 18 2 19 0 20 4 21 0 22 2 23 1 24 0 25 1 • It is also possible to group the values. Here they are grouped in 5s: Papers Sold Frequency 15-19 2 24 CU IDOL SELF LEARNING MATERIAL (SLM)

20-24 7 25-29 1 Example 3: In a firm of 400 employees, the percentage of monthly salary saved by each employee is given in the following table. Represent it through a bar graph. Savings (in Number of Employees (Frequency) percentage) 20 105 30 199 40 29 50 73 Total 400 Solution: The given data can be represented as 25 CU IDOL SELF LEARNING MATERIAL (SLM)

This can also be represented using a horizontal bar graph as follows: 26 CU IDOL SELF LEARNING MATERIAL (SLM)

2.4 FREQUENCY POLYGON A frequency polygon is almost identical to a histogram, which is used to compare sets of data or to display a cumulative frequency distribution. It uses a line graph to represent quantitative data. Statistics deals with the collection of data and information for a particular purpose. The tabulation of each run for each ball in cricket gives the statistics of the game. Tables, graphs, pie-charts, bar graphs, histograms, polygons etc. are used to represent statistical data pictorially. Frequency polygons are a visually substantial method of representing quantitative data and its frequencies. Let us discuss how to represent a frequency polygon. Steps to Draw Frequency Polygon To draw frequency polygons, first we need to draw histogram and then follow the below steps: Step 1- Choose the class interval and mark the values on the horizontal axes Step 2- Mark the mid value of each interval on the horizontal axes. Step 3- Mark the frequency of the class on the vertical axes. Step 4- Corresponding to the frequency of each class interval, mark a point at the height in the middle of the class interval Step 5- Connect these points using the line segment. Step 6- The obtained representation is a frequency polygon. Example: In a batch of 400 students, the height of students is given in the following table. Represent it through a frequency polygon. Solution: 27 Following steps are to be followed to construct a histogram from the given data: • The heights are represented on the horizontal axes on a suitable scale as shown. CU IDOL SELF LEARNING MATERIAL (SLM)

• The number of students is represented on the vertical axes on a suitable scale as shown. • Now rectangular bars of widths equal to the class- size and the length of the bars corresponding to a frequency of the class interval is drawn. ABCDEF represents the given data graphically in form of frequency polygon as: Frequency polygons can also be drawn independently without drawing histograms. For this, the midpoints of the class intervals known as class marks are used to plot the points. 2.5 MEAN For a data set, the arithmetic mean, also known as average or arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic means of a set of numbers x1, x2, ..., xn is typically denoted by x . If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted x ) to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted  or x ). In probability and statistics, the population mean, or expected value, is a measure of the central tendency either of a probability distribution or of a random variable characterized by that 28 CU IDOL SELF LEARNING MATERIAL (SLM)

distribution. In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving  = xp(x) An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean (see the Cauchy distribution for an example). Moreover, the mean can be infinite for some distributions. For a finite population, the population mean of a property is equal to the arithmetic mean of the given property, while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual—divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers states that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean. Types of Mean The following are the important types of Mean: (i) Arithmetic mean/simple Mean (ii) Weighted Mean. Besides these, there are less important averages like progressive average, etc. These averages have a very application and are, therefore, not so popular. (i) Arithmetic Mean Mean, often referred to as the arithmetic average or arithmetic mean, is calculated by adding all the numbers in a given set and then dividing by the total number of items within that set. CASE (I) Ungrouped Data The general formula to find the arithmetic mean of a given data is: Mean = X = sum of all observations number of observations =Mean= 1 = 1 n X n ( x1 + x2 + ..... + xn ) n i =1 xi It is denoted by x . 29 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 1: Compute the mean of the first 6 odd, natural numbers. Solution: The first 6 odd, natural numbers: 1, 3, 5, 7, 9, 11 ¯x= (1+3+5+7+9+11)/ 6=36/6=6 Thus, the arithmetic mean is 6 Example 2: If the arithmetic means of 14 observations 26, 12, 14, 15, x, 17, 9, 11, 18, 16, 28, 20, 22, 8 is 17. Find the missing observation. Solution: Given 14 observations are: 26, 12, 14, 15, x, 17, 9, 11, 18, 16, 28, 20, 22, 8 Arithmetic mean = 17 we know that, Arithmetic mean = Sum of observations/Total number of observations 17 = (216 + x)/14 17 x 14 = 216 + x 216 + x = 238 x = 238 – 216 x = 22 Therefore, the missing observation is 22. Example 3: Find the mean of the following set of integers. 8, 11, –6, 22, –3 Solution: Example 4: The set of scores 12, 5, 7, -8, x, 10 has a mean of 5. Find the value of x. Solution: Example 5: 30 CU IDOL SELF LEARNING MATERIAL (SLM)

The heights of five runners are 160 cm, 137 cm, 149 cm, 153 cm and 161 cm respectively. Find the mean height per runner. Solution: Mean height = Sum of the heights of the runners/number of runners = (160 + 137 + 149 + 153 + 161)/5 cm = 760/5 cm = 152 cm. Hence, the mean height is 152 cm. Example 6: Find the mean of the first five prime numbers. Solution: The first five prime numbers are 2, 3, 5, 7 and 11. Mean = Sum of the first five prime numbers/number of prime numbers = (2 + 3 + 5 + 7 + 11)/5 = 28/5 = 5.6 Hence, their mean is 5.6 Example 7: Find the mean of the first six multiples of 4. Solution: The first six multiples of 4 are 4, 8, 12, 16, 20 and 24. Mean = Sum of the first six multiples of 4/number of multiples = (4 + 8 + 12 + 16 + 20 + 24)/6 = 84/6 = 14. Hence, their mean is 14. Example 8: Find the arithmetic mean of the first 7 natural numbers. 31 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: The first 7 natural numbers are 1, 2, 3, 4, 5, 6 and 7. Let x denote their arithmetic mean. Then mean = Sum of the first 7 natural numbers/number of natural numbers x = (1 + 2 + 3 + 4 + 5 + 6 + 7)/7 = 28/7 =4 Hence, their mean is 4. Example 9: If the mean of 9, 8, 10, x, 12 is 15, find the value of x. Solution: Mean of the given numbers = (9 + 8 + 10 + x + 12)/5 = (39 + x)/5 According to the problem, mean = 15 (given). Therefore, (39 + x)/5 = 15 ⇒ 39 + x = 15 × 5 ⇒ 39 + x = 75 ⇒ 39 - 39 + x = 75 - 39 ⇒ x = 36 Hence, x = 36. CASE (II) _ Grouped Data 1. Direct Method Let x1, x2, x3 ……xn be the observations with the frequency f1, f2, f3 ……fn. Then, mean is calculated using the formula: Here, f1+ f2 + .... fn = ∑fi indicates the sum of all frequencies. Example 1: Find the mean of the following distribution: x 10 30 50 70 89 32 CU IDOL SELF LEARNING MATERIAL (SLM)

f7 8 10 15 10 Solution: Now, use the mean formula. Therefore, Mean = 55 Example 2: Finding the mean of the following distribution: Class-Interval 15-25 25-35 35-45 45-55 55-65 65-75 75-85 Frequency 6 11 7 44 21 Solution: When the data is presented in the form of class intervals, the mid-point of each class (also called as class mark) is considered for calculating the mean. The formula for mean remains the same as discussed above. Note: Class Mark = (Upper limit + Lower limit)/2 33 CU IDOL SELF LEARNING MATERIAL (SLM)

Therefore, Mean = 39.71 Example 3: The mean score of a group of 20 students is 65. Two other students whose scores are 89 and 85 were added to the group. What is the new mean of the group of students? Solution: The formula: can rewritten as: Total score = Mean × Number of students Total score of the original group = 65 × 20 = 1,300 Total score of the new group = Total score of the original group + scores of the 2 new students = 1,300 + 89 + 85 = 1,474 34 CU IDOL SELF LEARNING MATERIAL (SLM)

Number of students in the new group = Number of students in the original group + Number of new data = 20 + 2 = 22 Example 4: The mean of a list of 6 numbers is 20. If we remove one of the numbers, the mean of the remaining numbers is 15. What is the number that was removed? Solution: Using the formula: Sum = Mean × Number of numbers Sum of original 6 numbers = 20 × 6 = 120 Sum of remaining 5 numbers = 15 × 5 = 75 Number removed = sum of original 6 numbers – sum of remaining 5 numbers Number removed = 120 – 75 = 45 Alternative Method: Let x be the removed number The removed number is 45 Example 5: 10 students of a class had a mean score of 70. The remaining 15 students of the class had mean score of 80. What is the mean score of the entire class? Solution: Total score of first 10 students = 10 × 70 = 700 Total score of remaining 15 students = 15 × 80 = 1200 Mean score of whole class 2. Short-cut Method 35 This method is called as assumed mean method or change of origin method. The following steps describe this method. Step1: Calculate the class marks of each class (xi). Step2: Let A denote the assumed mean of the data. Step3: Find deviation (di) = xi – A Step4: Use the formula CU IDOL SELF LEARNING MATERIAL (SLM)

Example: Calculate the mean of the following using the short-cut method. Class-Intervals 45-50 50-55 55-60 60-65 65-70 70-75 75-80 12 6 Frequency 5 8 30 25 14 Solution: Let us make the calculation table. Let the assumed mean be A = 62.5 Note: A is chosen from the xi values. Usually, the value which is around the middle is taken. Now we use the formula, 36 =62.50-0.25 =62.25 Therefore, Mean = 62.25 CU IDOL SELF LEARNING MATERIAL (SLM)

2.6 MERITS AND DEMERITS OF ARITHMETIC MEAN Advantages of Arithmetic Mean The uses of arithmetic mean are not just limited to statistics and mathematics, but it is also used in experimental science, economics, sociology, and other diverse academic disciplines. Listed below are some of the major advantages of arithmetic mean. 1. As the formula to find the arithmetic mean is rigid, the result doesn’t change. 2. It takes into consideration each value of the data set. 3. Finding arithmetic mean is quite simple; even a common man having very less finance and math skills can calculate it. 4. It’s also a useful measure of central tendency, as it tends to provide useful results, even with large groupings of numbers. 5. It can be further subjected to many algebraic treatments unlike mode and median. 6. The arithmetic mean is widely used in geometry as well. Disadvantage of Arithmetic Mean 1. The strongest drawback of arithmetic mean is that it is affected by extreme values in the data set. 2. In a distribution containing open-end classes, the value of mean cannot be computed without making assumptions regarding the size of the class. 3. It’s practically impossible to locate the arithmetic mean by inspection or graphically. 4. It cannot be used for qualitative types of data such as honesty, favourite milkshake flavour, most popular product etc. 5. We can't find arithmetic mean, if a single observation is missing or lost. 2.7 WEIGHTED MEAN The weighted mean is a type of mean that is calculated by multiplying the weight (or probability) associated with a particular event or outcome with its associated quantitative outcome and then summing all the products together. It is very useful when calculating a theoretically expected 37 CU IDOL SELF LEARNING MATERIAL (SLM)

outcome where each outcome has a different probability of occurring, which is the key feature that distinguishes the weighted mean from the arithmetic mean. The Weighted mean for given set of non-negative data x1, x2, x3, …. xn with non-negative weights w1, w2, w3…. wn can be derived from the formula given below. Where, x is the repeating value w is the number of occurrences of x weight x is the weighted mean Example: Suppose that a marketing firm conducts a survey of 1,000 households to determine the average number of TVs each household owns. The data show a large number of households with two or three TVs and a smaller number with one or four. Every household in the sample has at least one TV and no household has more than four. Find the mean number of TVs per household. Number of TVs per Household Number of Households 1 73 2 378 3 459 4 90 Solution: As many of the values in this data set are repeated multiple times, you can easily compute the sample mean as a weighted mean. Follow these steps to calculate the weighted arithmetic mean: Step 1: Assign a weight to each value in the dataset: x1=1, w1=73 x2=2, w2=378 38 CU IDOL SELF LEARNING MATERIAL (SLM)

x3=3, w3=459 x4=4, w4=90 Step 2: Compute the numerator of the weighted mean formula. Multiply each sample by its weight and then add the products together: = (1)(73) +(2) (378) +(3) (459) +(4)(90) = 73 + 756 + 1377 +360 =2566 Step 3: Now, compute the denominator of the weighted mean formula by adding the weights together. = 73 + 378 + 459 + 90 =1000 Step 4: Divide the numerator by the denominator =2566/1000 =2.566 The mean number of TVs per household in this sample is 2.566. 2.8 SUMMARY • A frequency distribution is a representation, either in a graphical or tabular format, that displays the number of observations within a given interval. • The interval size depends on the data being analyzed and the goals of the analyst. • Frequency distributions are typically used within a statistical context. • Communicate the largest amount of information as simply as possible. 39 CU IDOL SELF LEARNING MATERIAL (SLM)

2.9 KEYWORDS • Frequency distribution: A representation that displays the number of observations within a given interval. • Representation of a frequency distribution: It can be graphical or tabular so that it is easier to understand. • Mean: The average of the variable. 2.10 LEARNING ACTIVITY 1. George’s scores on three math tests were 70, 90 and 75. What score does he need on the fourth test to have a final average of 80? _______________________________________________________________________________ ________________________________________________________________ 2. Pedro’s luncheonette is open six days a week. His income for the first five days was $1,200, $1,200, $2,000, $1,400 and $3,000. How much money must she make on the sixth day to average $2,000 for the six days? _______________________________________________________________________________ ________________________________________________________________ 3. Given below are the weekly pocket expenses (in $) of a group of 25 students selected at random. 37, 41, 39, 34, 41, 26, 46, 31, 48, 32, 44, 39, 35, 39, 37, 49, 27, 37, 33, 38, 49, 45, 44, 37, 36 Construct a grouped frequency table with class intervals of equal widths, such as 25 - 30, 30 - 35, and so on. Also, find the range of weekly pocket expenses. ________________________________________________________________________________ _______________________________________________________________ 2.11 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. How do you describe frequency distribution? 2. What are the 3 types of frequency distributions? 40 CU IDOL SELF LEARNING MATERIAL (SLM)

3. What are the essential parts of a frequency distribution? 4. What is another name for a frequency distribution? 5. What is the mean of 17? Long Questions 1. If the heights of 5 people are 142 cm, 150 cm, 149 cm, 156 cm and 153 cm. Find the mean height. 2. Find the mean for the following distribution: Classes 0-10 10-20 20-30 30-40 40-50 Frequency 2 12 22 8 6 3. Find the mean of the following distribution. (a) The age of 20 boys in a locality is given below. Age in Years 12 10 15 14 8 64 Number of Boys 532 20 24 (b) Marks obtained by 40 students in an exam are given below. 64 Marks 25 30 15 Number of Students 8 12 10 4. Find the mean of the following distribution. 4 5 xi 1 2 3 10 3 fi 4 5 8 5. The daily wages of 50 employees in an organization are given below: Daily wages (in $) 100 – 150 150 - 200 200 - 250 250 - 300 8 Number of Workers 12 13 17 Find the mean daily wages. 41 CU IDOL SELF LEARNING MATERIAL (SLM)

B. Multiple choice Questions 42 1. One dimensional diagram is: a. Line diagram b. Rectangles c. Cubes d. Squares 2. Type of bar diagram is: a. Pictogram b. Sub divided diagram c. Line diagrams d. Pie diagram 3. The most commonly used device of presenting business and economic data is: a. Pie diagrams b. Pictograms c. Bar diagrams d. Line diagrams 4. A pie diagram is also called: a. Pictogram b. Angular diagram c. Line diagram d. Bar diagram 5. In volume diagram the three dimensions which are taken into account are: a. Length, weight, breadth CU IDOL SELF LEARNING MATERIAL (SLM)

b. Height, weight, breadth c. Length, height, breadth d. Length, weight, height Answers 1.a, 2.b, 3.c, 4.b, 5.c 2.12 REFERENCES Reference Books: • Dr. B. Krishna Gandhi, Dr. T.K.V Iyengar, M.V.S.S.N. Prasad, Probability and Statistics, S. Chand Publishing Co. • Quantitative Techniques for Decision Making by A. Sharma, Publisher: HPH, 2007 Edition. • Statistical Methods by S.P Gupta, Publisher: Sultan Chand & Sons, 2008 Edition. • Research Methodology by C. R. Kothari, Publisher: Vikas Publishing House Textbooks: • S.C. Gupta, V.K. Kapoor, Fundamental of Mathematical Statistics, Sultan Chand and Company. • Research Methodology by V. P. Pandey, Publisher: Himalaya Publication • Research Methodology in Management by Arbind and Desai, Publisher: Ashish Publication House • Research Methodology and Statistical Methods by T. Subbi Reddy, Publisher: Reliance Publishing House 43 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 3: MEDIAN Structure 3.0 Learning Objectives 3.1 Introduction 3.2 Median 3.3 Ungrouped Data 3.4 Grouped Data 3.5 Merits and Limitations of Median 3.6 Limitations 3.7 Summary 3.8 Keywords 3.9 Learning activity 3.10 Unit End Questions 3.11 References 3.0 LEARNING OBJECTIVES After studying this unit students will be able to: • Explain why their different formulas are for calculating the median for an odd versus even number of scores for a variable. • Analyse how to develop a strategic approach to organizing data. • Outline the relationship between numbers in a data set through the calculation of mode. • Analyze data from tables and interpret double bar graphs. • State the definitions of mode and will be encouraged to work together to get a class average. 3.1 INTRODUCTION In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of 44 CU IDOL SELF LEARNING MATERIAL (SLM)

as \"the middle\" value. The basic feature of the median in describing data compared to the mean (often simply described as the \"average\") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a \"typical\" value. Median income, for example, may be a better way to suggest what a \"typical\" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. 3.2 MEDIAN The value of the middlemost observation, obtained after arranging the data in ascending order, is called the median of the data. For example, Consider the data: 4, 4, 6, 3, 2. Let's arrange this data in ascending order: 2, 3, 4, 4, 6. There are 5 observations. Thus, median = middle value i.e., 4 We can see here: 2, 3, 4, 4, 6 (Thus, 4 is the median) The median by definition refers to the middle value in a distribution. In case of median one-half of the items in the distribution have a value the size of the median value or smaller and one-half have a value the size of the median value or larger. The median is just the 50th percentile value below which 50 per cent of the values in the sample fall. It splits the observation into two halves. As distinct from the arithmetic mean which is calculated from the value of every item in the series, the median is what is called a positional average. The term 'position' refers to the place of a value in a series. The place of the median in a series is such that an equal number of items lie on either side of it. Example 1: If the income of five employees is Rs. 5,900, 6,950. 7,020, 7, 200 and 8, 280 the median would be 7, 020. 5, 900 6, 950 7, 020 « value at middle position of the array 7, 200 8, 280 45 CU IDOL SELF LEARNING MATERIAL (SLM)

For the above example the calculation of median was simple because of odd number of observations. When an even number of observations are listed, there is no single middle position value and the median is taken to be the arithmetic mean of two middlemost items. For example, if in the above case we are given the income of six employees as 5,900, 6,950, 7,020, 7,200, 8.280, 9,300, the median income would be: 5,900 6.950 7,020 7,200 8,280 9,300 There are two middle position values Median = 7,020 + 7,200 = 14,220 22 =Rs.7, 110 Hence, in case of even number of observations median may be found by averaging two middle position values. Thus, when, N is odd the median is an actual value, with the remainder of the series in two equal parts on either side of it. If N is even the median is a derived figure, i.e., half the sum of the middle values. Example 2: Let us consider the data: 56, 67, 54, 34, 78, 43, 23. What is the median? Solution: Arranging in ascending order, we get: 23, 34, 43, 54, 56, 67, 78. Here, n (no. of observations) = 7 So, (7+1)/2=4 ∴Median=4th observation (7+1)/2 = 4 ∴Median=4th observation Median = 54 Example 3: Let's consider the data: 50, 67, 24, 34, 78, 43. What is the median? Solution: Arranging in ascending order, we get: 24, 34, 43, 50, 67, 78. 46 CU IDOL SELF LEARNING MATERIAL (SLM)

Here, n (no. of observations) = 6 6/2=3 Using the median formula, Median =(3rdobs.+4thobs.)/2=(43+50)/2 Median = 46.5 3.2 UNGROUPED DATA Ungrouped Data Step 1: Arrange the data in ascending or descending order. Step 2: Let the total number of observations be n. To find the median, we need to consider if n is even or odd. If n is odd, then use the formula: If n is even, then use the formula: Example 1: 47 From the following data of the wages of 7 workers, compute the median wage: Wages (in Rs.) 4100 4150 6080 7120 5200 6160 CU IDOL SELF LEARNING MATERIAL (SLM)

7400 Solution: Calculation of Median S. No. Wages arranged in S. No. Wages arranged in ascending order ascending order 1 4100 5 6160 2 4150 6 7120 3 5200 7 7400 4 6080 Median = Size of N+1th item = 7+1 = 4th item = Rs. 6080. 22 Size of 4th item = 6080. Hence the median wage = Rs. 6080 We thus find that median is the middle most items: 3 persons get a wag less than Rs. 5200 and equal number, i.e., 3, get more than. Rs. 5200. The procedure for determining the median of .an even-numbered group of items is not as obvious as above. If there were for instance, different values in a group, the median is really not determinable since both the 5th and 6th values are in the centre. In practice the median value for a group composed of an even number of items is estimated by finding the arithmetic mean of the two middle values that is, adding the two values in the middle and dividing by two. Expressed in the form of formula, it amounts to: Median = Size of N+1th item 2 Thus, we find that it is both when N is odd as well as even that 1 (one) has to be added to determine median value. Example 2: Obtain the value of median from the following data of the monthly income of 10 employees of a company in Rs: 48 CU IDOL SELF LEARNING MATERIAL (SLM)

4,391 Income arranged in ascending order (in Rs.) 5,384 4,391 5,591 5,384 5,407 5,591 6,672 5,407 6,522 6,672 6,777 6,522 6,753 6,777 7,850 6,753 7,490 7,850 Solution: 7,490 Calculation of Median: S. No. 1 2 3 4 5 6 7 8 9 10 Median= size of N+1th item 2 49 CU IDOL SELF LEARNING MATERIAL (SLM)

=11th item = 5.5th Item 6,522 + 6,672 2 Size of 5.5th item = 5th item + 6th item = 22 = 6,597 Hence, median is Rs. 6,597. Example 3: From the following data, find the value of median: Income (Rs.) No. of persons 4,000 24 4,500 26 5,800 16 5,060 20 6,600 6 5,380 30 Solution: Calculation of Median Income Income arranged in arranged in No. of persons c.f. ascending No. of persons c.f. order ascending 24 30 100 50 5,380 16 116 order 70 5,800 6 122 6,600 4,000 24 4,500 26 5,060 20 50 CU IDOL SELF LEARNING MATERIAL (SLM)