FIRST EDITION: HASNIDA BINTI ISMAIL SITI NOR BINTI FAUZI STATISTICS: WORKBOOK & REVISIONS
Managing Editor Siti Hajar Binti Khazali Editor Hasnida Binti Ismail Siti Nor Binti Fauzi Faezah Binti Kamisan Writer Hasnida Binti Ismail Siti Nor Binti Fauzi Designer Hasnida Binti Ismail Siti Nor Binti Fauzi Application Publishers and Developers Hasnida Binti Ismail Siti Nor Binti Fauzi 1 st Edition 2022 e ISBN 978-967-2498-66-7 All rights reserved. No part of this work may be reproduced, stored in any form or by any means without the written permission of the publisher. Published by: Politeknik Muadzam Shah, Lebuhraya Tun Abdul Razak, 26700 Muadzam Shah, Pahang Darul Makmur
ACKNOWLEDGEMENT We would like to record our warm appreciation and thanks to the many parties who have provided encouragement and helpful comments towards the arrangement of this Statistic Workbook and Revisions. It is our hope that this Workbook would help students to gain better understanding of this course. HASNIDA BINTI ISMAIL SITI NOR BINTI FAUZI Lecturer Department of Commerce Politeknik Muadzam Shah
PREFACE All praise to Almighty Allah the Lord of the universal for his infinite mercy upon us. Greetings to our beloved Prophet. I am grateful for his nearness and consent upon completing this book. The Statistics Workbook and Revisions was written carefully and specifically to help and equip the students with knowledge of statistics. It is written for students pursuing their diploma level especially for polytechnic based on the syllabus by Department of Polytechnic and Community College. This book is designed to complement any workbook on Statistics and making it suitable reference for those who are new or never been exposed to statistics concepts. This book consists of seven chapters: Introduction to Statistics, Data Presentation, Measures of Central Tendency, Measures of Dispersion and Skewness, Correlation and Regression, Elementary of Probability and Estimation and Hypothesis Testing. Each of the chapter begins with Learning Objectives. Each chapter also contains self-revision. Model examinations papers are provided at the end of the book. I would like to thank everybody involved in the production of this book and it is my hope that this book would serve its purpose in helping students and lecturers gain better understanding of the subject. HASNIDA BINTI ISMAIL SITI NOR BINTI FAUZI Lecturer Department of Commerce Politeknik Muadzam Shah
Acknowledgements Preface Contents CHAPTER 1 : INTRODUCTION TO STATISTICS 1 Introduction 5 1.1 Definition 5 1.2 Who use Statistics 6 1.3 Types of Statistics 6 1.3.1 Descriptive Statistics 6 1.3.2 Inferential Statistics 6 1.4 Sources of Data 7 1.4.1 Primary Data 7 1.4.2 Secondary Data 7 1.5 Types of Data 8 1.5.1 Qualitative Data 8 1.5.2 Quantitative Data 8 1.6 Statistical Terms 8 1.6.1 Population 8 1.6.2 Sample 9 1.6.3 Sampling 9 1.6.4 Pilot Test 9 1.6.5 Variables 9 1.6.6 Respondents 9 1.7 Data Collection Method 9 1.7.1 Direct Observation 9 1.7.2 Experimental Method 10 1.7.3 Direct Questionnaire 10 1.7.4 Mail Questionnaire 10 1.7.5 Telephone Survey 11 1.7.6 Internet Survey 12 1.7.7 Face to Face Interview 12 Self Revision 14
CHAPTER 2 : DATA PRESENTATION 15 Introduction 16 2.1 Frequency Distribution Table 16 2.2 Histogram 21 2.3 Polygon 22 2.4 Ogive 23 Self Revision 26 CHAPTER 3 : MEASURES OF CENTRAL TENDENCY 29 Introduction 30 3.1 Measures of Central Tendency for Ungroup Data 30 3.1.1 Mean 31 3.1.2 Median 32 3.1.3 Mode 34 3.2 Measures of Central Tendency for Group Data 36 3.2.1 Mean 36 3.2.2 Median 39 3.2.3 Mode 43 3.2.4 Relationship Between Mean, Median and Mode 47 Self Revision 50 CHAPTER 4 : MEASURES OF DISPERSION AND SKEWNESS 53 Introduction 54 4.1 Measures of Dispersion for Ungroup Data 54 4.1.1 Range 54 4.1.2 Mean Deviation 54 4.1.3 Variance and Standard Deviation 55 4.2 Measures of Dispersion for Group Data 58 4.2.1 Range 58 4.2.2 Mean Deviation 58 4.2.3 Variance and Standard Deviation 59 4.3 Measures of Skewness 62 4.3.1 Pearson’s Coefficient of Skewness 1 (PCS 1) 63 4.3.2 Pearson’s Coefficient of Skewness 2 (PCS 2) 63 Self Revision 66
CHAPTER 5 : CORRELATION AND REGRESSION 68 Introduction 69 5.1 Scatter Diagram 69 5.2 Correlation 72 5.2.1 Pearson’s Product Moment Correlation Coefficient 72 5.2.2 Spearman’s Rank Correlation Coefficient 76 5.3 Regression 80 Self Revision 84 CHAPTER 6 : ELEMENTARY OF PROBABILITY CONCEPTS 86 Introduction 87 6.1 Definition of Probability 87 6.2 Use of Probability Theory 87 6.3 Basic Definition of Terms Used 88 6.4 Independent Events 89 6.5 Dependent Events 89 6.6 Venn Diagram 89 6.7 Two Way Table 91 6.8 Tree Diagram 93 6.9 Approaches to Probability 97 6.10 Calculating Probability 99 6.11 Probability Mutually Exclusive 100 6.12 Probability Mutually Inclusive 101 6.13 Probability of Independent Event 103 6.14 Probability of Dependent Event 104 Self Revision 107 CHAPTER 7 : ESTIMATION AND HYPOTHESIS TESTING 117 Introduction 118 7.1 What is Hypothesis 119 7.2 Hypothesis Testing 119 7.3 Five Steps Procedure For Testing a Hypothesis 119 7.4 Terms in Hypothesis 120 7.5 Writing an Hypothesis 120
7.6 Basic of Hypothesis Testing 122 7.7 Alternative Hypothesis 122 7.8 Test of Hypothesis for Large Sample 123 7.9 Test of Hypothesis for Small Sample 125 Self Revision 126 PAST SEMESTER QUESTION 128 Set 1 129 Set 2 133 Set 3 137 References 140
DPB 30063 / INTRODUCTION TO STATISTICS 1 CHAPTER 1 : INTRODUCTION TO STATISTICS
DPB 30063 / INTRODUCTION TO STATISTICS 2 CHAPTER 1 : INTRODUCTION TO STATISTICS Learning Objectives: End of chapter, student be able to: 1. Explain the meaning of statistics. 2. Compare types of statistics. 3. Identify sources of data. 4. Identify types of data. 5. Explain statistical terms. 6. Explain data collection method. Figure 1: The use of statistics (Malaysian Economics)
DPB 30063 / INTRODUCTION TO STATISTICS 3 Figure 2: The use of statistics (Malaysian Population)
DPB 30063 / INTRODUCTION TO STATISTICS 4 Figure 3: The use of statistics (Data Collection Method)
DPB 30063 / INTRODUCTION TO STATISTICS 5 Introduction • Everyday, thousands of data is generated. • There are data on demand, economic growth, company’s profit, population of a country, examination and etc. • Managers of organizations collect, summarize, analyse and transform data into useful information that they can use to compare with the current performance of their organizations and use it to guide and plan for the future needs of the organization 1.1 Definition Statistics is a scientific procedures and methods for collecting, organizing, summarising, analysing and presenting data, as well as obtaining useful information, drawing conclusion and making effective decisions based on the analysis. Figure 4: The use of statistics (Exam Result)
DPB 30063 / INTRODUCTION TO STATISTICS 6 1.2 Who Use Statistics • Management – to increase the sales or profit for the company. • Finance – to develop ways of forecasting values of these measures at future points of time. • Teachers – to improve the student result. • Football Coach – to improve the team performance. • Ourself – to make monthly budget / manage own income (salary) 1.3 Types of Statistics 1.3.1 Descriptive Statistics • Data are compiled, summarized and presented in suitable visual forms which are easy to understand and suitable for use. • Various tables, graphs, charts and diagrams are used to exhibit the information obtained from the data. • Raw data are transformed into meaningful forms, so the manager can make conclusions just by taking a quick look at the visual presentations. • Example: Sales manager presenting sales performance for the past five years using time series chart. 1.3.2 Inferential Statistics. • Generalisations were made about a population by analysing the sample. • If the sample is a good representation of a population, accurate conclusions about the population can be inferred from the analysis of this sample. • This is because the sample values are close representations of the actual values of the population of interest. STATISTICS
DPB 30063 / INTRODUCTION TO STATISTICS 7 • Example: The caterer wants to know what did student prefer for lunch either fish or chicken. Therefore 250 students were selected and from this result they can make a conclusion. 1.4 Sources of Data 1.4.1 Primary Data • Researcher collects data from primary sources or from samples. • Normally it is called as the first-hand information. • Example: Census Survey (Banci). Advantages: • It is more accurate and consistent with the objectives of the research. • The researcher is able to explain how the data are collected and the limitations of their use. • The data needed by decision makers are not available from the secondary. Disadvantages: • It requires more time, manpower and higher cost to collect. • Researchers have limited budget and time to complete the project. 1.4.2 Secondary Data • Secondary data are normally published data collected by other parties. • It is also called second-hand information. • Examples: Data from articles, journals, magazines, newspaper, annual report and others publications. Advantages: • It is accessible from the internet, annual report and newspaper. • It is inexperience because there is no field work required. • It is also less time to collect data. • Some data such as import and export data are only available from secondary sources.
DPB 30063 / INTRODUCTION TO STATISTICS 8 Disadvantages: • It may lack accuracy because the measurement and method of data collection are not explained by the previous researcher. • The data maybe biased because the original purpose of data collection is not known. • The data may not meet the specific needs and objectives of the current research. 1.5 Types of Data 1.5.1 Qualitative Data • It is measured with non-numerical scale. • Example: Gender, Citizens, Ethnicity, Colours etc. 1.5.2 Quantitative Data • It is measured on numerical scale. Example: Numbers of student in a class, How many cars are there in a campus. • The answer is numerical (can be count 1, 2, 3 etc.) i. Discrete • Numerical responses which arise from a counting process (round number). • Example: Number of children, number of residents, number of flight. ii. Continuous • Numerical responses which arise from a measuring process (decimal number). • Example: Height, temperature, distance etc. 1.6 Statistical Terms 1.6.1 Population • The complete set of items that are of interest in the research. • Example: If the study is about the average monthly income of workers in a factory, the group of all workers in the factory is the population
DPB 30063 / INTRODUCTION TO STATISTICS 9 1.6.2 Sample • A subset of items that are chosen from the populations. • Example: A survey for the population of Malaysia, therefore N.Sembilan state is the sample. 1.6.3 Sampling • Scientific method of selecting a sample from a population. • Various sampling techniques such as simple random sampling, convenience sampling, quota sampling etc. 1.6.4 Pilot Test / Study • A study done before the actual fieldwork is carried out. • This study is also used to test out questionnaires and to improve them in terms of flow, question design, language etc. 1.6.5 Variable • The characteristics of the population of interest. • Example: Monthly income, gender, level of education etc. 1.6.6 Respondents • Person who involve in survey. 1.7 Data Collection Method 1.7.1 Direct Observation • This method is the most commonly method used for collecting statistical data directly from the population. • It is used in work, studies and organizations. • The direct observation method enables the researcher to record the behaviours, events, activities while they are happening.
DPB 30063 / INTRODUCTION TO STATISTICS 10 Advantages Disadvantages 1.7.2 Experiment Method • An experiment is a controlled study in which the researcher attempts to understand cause and effect relationships. • The study will be supervised by researchers of which there are two groups - a control group and experimental group. • Example: Researchers will provide study notes to the one group of students to prepare for test while the other group, the researchers will ask them to find their own notes. The results for both groups will be assessed by researchers in finding cause and effect relationship. 1.7.3 Direct Questionnaires • The researcher explains briefly his intentions before giving the questionnaire to the respondents. • The questions should be simple, clear and easily understood and related to the purposed of the survey. 1.7.4 Mail Questionnaires • A questionnaire is sent to each with a stamped addressed envelope attached and it is usually enclosed with the questionnaire. • The respondents are requested to answer the question in the questionnaire and return it to the researcher within a certain period of time.
DPB 30063 / INTRODUCTION TO STATISTICS 11 Advantages Disadvantages 1.7.5 Telephone Survey • An interviewer asks only a few questions from prepared questionnaire. • The duration of this method is normally short and has some limitations because respondents are restricted only to individuals who can be reached by telephone. Advantages Disadvantages
DPB 30063 / INTRODUCTION TO STATISTICS 12 1.7.6 Internet survey • An online survey is a questionnaire that the target audience can complete over the internet. Online surveys are usually created as Web forms with a database to store the answer and statistical software to provide analytics. • People are encourage to complete online surveys by an incentive such as being entered to win a prize. Advantages Disadvantages 1.7.7 Face to Face / Personal Interview • This method is widely used in the research and is based on a direct meeting between interviewer and interviewee to encourage response. • Through personal communication, it is possible not only to obtain much more information but also to use visual materials. Advantages Disadvantages
DPB 30063 / INTRODUCTION TO STATISTICS 13
DPB 30063 / INTRODUCTION TO STATISTICS 14 SELF REVISIONS 1. State the types of data (qualitative or quantitative) for each following characteristic: i. Number of family ( ) ii. Blood glucose level ( ) iii. Eye color ( ) iv. Marital status ( ) v. Ethnicity ( ) 2. Differentiate primary and secondary data. Answer: 3. Define the following terms: i. Statistics ii. Population and sample Answer: 4. List five types of data collection method. Answer:
DPB 30063 / DATA PRESENTATION 15 CHAPTER 2 : DATA PRESENTATION
DPB 30063 / DATA PRESENTATION 16 CHAPTER 2 : DATA PRESENTATION Introduction • Data presentation is the method by which people summarize, organize and communicate information by using a variety tools such as charts, histograms, diagrams and graphs. • Some common data presentations are frequency distribution table, frequency polygon, ogive, bar chart and pie chart. 2.1 Frequency Distribution Table • The easiest way to present raw data is by constructing a frequency table. • Frequency table is a summary table which the data are arranged into numerically ordered class groupings. • Bacis elements of frequency distribution table is as follow: Class Interval Frequency Cumulative Frequency Class Boundaries Mid Point Relative Frequency • There are three steps involved in constructing frequency distribution table: Learning Objectives: End of chapter, student be able to: 1. Construct frequency distribution tables. 2. Organize quantitative data
DPB 30063 / DATA PRESENTATION 17 i. Determine the number of class (k): ii. Determine the data range: iii. Determine class width / class size ( C ) : Example 1: Consider the following data on hoursspent by 50 workers for period of a month in a factory. Construct a frequency distribution table consists of class intervals, tally/frequency, relative frequency and mid point, cumulative frequency and class boundaries. 165 110 130 103 140 175 133 156 204 144 161 195 167 162 187 157 151 124 149 184 155 71 164 79 197 108 94 146 113 87 164 87 116 69 40 128 41 149 121 114 30 104 93 203 178 62 141 143 148 122
DPB 30063 / DATA PRESENTATION 18 Solutions: Frequency Distribution Table
DPB 30063 / DATA PRESENTATION 19 Example 2 Below are the profit (RM Juta) made by 30 companies for the year 2007. Construct the frequency distribution table consists of class intervals, tally/frequency, relative frequency and mid point, cumulative frequency and class boundaries. 14.6 22.4 8.3 23.2 15.7 28.0 14.0 14.8 9.7 18.7 12.9 16.7 15.4 21.5 6.7 10.1 19.1 8.4 27.2 6.7 13.4 16.0 7.2 19.7 27.5 16.5 22.8 21.8 17.6 20.3 Solution: Frequency Distribution Table
DPB 30063 / DATA PRESENTATION 20 Example 3 Following are record of swimmers in a swimming competition. Construct a frequency distribution table consists of classintervals, tally/frequency, relative frequency and mid point, cumulative frequency and class boundaries. 6.01 4.85 5.71 3.19 5.84 2.96 4.75 5.44 4.52 3.00 3.45 2.65 2.86 3.67 4.99 4.82 6.42 3.05 2.37 4.80 5.05 3.26 3.32 3.98 5.96 5.02 4.30 4.12 5.67 2.06 1.78 2.56 4.40 5.55 2.58 6.14 4.65 5.71 5.37 3.22 1.88 1.67 3.90 1.67 6.20 6.25 1.38 2.05 4.27 6.41 Solution: Frequency Distribution Table
DPB 30063 / DATA PRESENTATION 21 2.2 Histogram • Histogram is a graphical representation of the frequency distribution in which bars represent frequencies. • Histogram is construct by using class boundaries and frequencies of the classes Element needed to draw a histogram: Step 1: Mark the class boundaries at the horizontal axis (x) Step 2: Insert the frequency at the vertical axis (y) Step 3: Draw a vertical bar to show the frequency for each class Example 4 The following data shows the age of student at Taman ABC. Class Interval Frequency Class Boundaries 2 – 6 3 7 – 11 5 12 – 16 10 17 – 21 12 22 – 26 8 27 – 31 2 Total 40 Frequency Class Boundaries Title : Histogram ………….
DPB 30063 / DATA PRESENTATION 22 2.3 Polygon • Polygon can be constructed in two way: a. By joining the mid point of the top of each bars in histogram in a straight line. b. By connecting the mid points of every class in one line. Example 5 The length of 30 pandan leaves were measured and shown below. Draw a polygon. Class Interval Frequency Mid point 10 – 14 3 15 – 19 8 20 – 24 12 25 - 29 7 Frequency Mid point Title : Polygon …….
DPB 30063 / DATA PRESENTATION 23 2.4 Ogive • An ogive is a graph or a line chart of a cumulative frequency distribution. There are two typesof ogive that is: i. Ogive Less Than ii. Ogive More Than • Drawing ogive ‘less than’ Step 1: Find class boundaries and create a ‘less than’ cumulative frequency Step 2: Mark the lower class boundaries on the x-axis Step 3: Mark the cumulative frequency on the y-axis Step 4: Draw an ogive • Drawing ogive ‘more than’ Step 1: Find class boundaries and create a ‘more than’ cumulative frequency Step 2: Mark the lower class boundaries on the x-axis. Step 3: Mark the cumulative frequency on the y-axis Step 4: Draw ogive Example 6 Construct an ogive for the following data using the ‘less than’. Class Interval Freq 21 – 25 10 26 – 30 25 31 – 35 16 36 – 40 5 41 – 45 8 46 – 50 6
DPB 30063 / DATA PRESENTATION 24 Cum Freq. Less Than Class Boundaries Less Than Example 7 Construct an ogive for the following data using the ‘more than’. Class Interval Freq 21 – 25 10 26 – 30 25 31 – 35 16 36 – 40 5 41 – 45 8 46 – 50 6 Title : Ogive Less Than ……………
DPB 30063 / DATA PRESENTATION 25 Cum Freq More Than Class Boundaries More Than Title : Ogive ‘more than ……………
DPB 30063 / DATA PRESENTATION 26 SELF REVISION QUESTION 1 The followings are the speed of 50 cars that passes through a highway done by traffic police. 113 119 125 123 116 128 118 125 128 111 111 127 124 123 120 125 124 116 121 128 122 118 117 128 125 122 113 119 112 126 110 125 120 110 129 123 129 114 122 119 121 113 128 124 116 129 112 127 117 115 You are required to :- a. Calculate and prepare the frequency distribution table: i. Range ii. Numbers of class iii. Class interval iv. Frequency v. Class boundaries vi. Mid point Draw the ‘less than’ ogive for the data above.
DPB 30063 / DATA PRESENTATION 27 QUESTION 2 a. List down THREE types of graphical method to present the data b. Define frequency distribution. c. State TWO category of quantitative data. d. The regional transport authority is concerned about the speed of motorbikes ridden by college students on a section of the road. The following data shows the speed of 45 riders (speed in km/hr) 15 38 69 69 32 52 18 61 45 49 52 47 38 56 57 18 29 52 55 61 56 46 58 55 58 42 62 64 44 62 39 29 42 49 48 55 31 48 39 58 48 47 68 48 49 i. Determine the number of classes ( k ). ii. Find the data range. iii. Determine the class width ( c ). iv. Fill in the missing item using the raw data given. e. Class Interval Class Boundaries Freq Relative Frequency Mid Point 15 – 24 25 – 34 35 – 44 45 – 54 55 – 64 65 – 74 Construct a histogram, polygon and a less than ogive for the given data.
DPB 30063 / DATA PRESENTATION 28 QUESTION 3 Data below shows the unemployment rate in Malaysia from January 2019 until February 2020. 3.8 4.1 4.0 3.8 3.6 3.2 3.5 3.6 3.4 3.5 3.4 3.6 3.6 3.2 3.3 3.7 3.3 3.1 2.9 3.1 3.2 3.1 3.6 3.5 a. Using your knowledge, find the: i. Range ii. Number of class (k) iii. Class size b. Construct distribution table that consists of: i. Class interval ii. Tally iii. Frequency iv. Class Boundaries v. Midpoint c. Based on the above answer in (b), draw a frequency polygon. d. State your conclusion based on the polygon graph. e. Construct another table that consists of upper boundaries and cumulative frequency. f. Draw a less than ogive.
DPB 30063 / MEASURES OF CENTRAL TENDENCY 29 CHAPTER 3 : MEASURES OF CENTRAL TENDENCY
DPB 30063 / MEASURES OF CENTRAL TENDENCY 30 CHAPTER 3 : MEASURES OF CENTRAL TENDENCY Learning Objectives: End of chapter, student be able to: 1. Explain the measure of central tendency. 2. Calculate the measure of central tendency for ungroup data. 3. Calculate the measure of central tendency for group data. 4. Explain the relationship among mean, median and mode. Introduction • The measures of central tendency is usually called the average. • Central tendency is a single value situated at the central of data and can be taken as a summary value for that data set. • There are a two group of measures of central tendency that are ungroup data and group data. • There are also three types of measures of central tendency that is mean, median and mode. 3.1 Measure of Central Tendency For Ungroup Data. There are two types of ungroup data. i. Ungroup data that are scattered and obtained in original form (raw data). Example: The marks obtained by 10 students for AKPK course are: 70 50 55 60 80 95 75 83 72 66 ii. Ungroup data in frequency distribution table. Example: The marks obtained by 10 students for AKPK Quiz are given below Marks Number of Student 50 1 60 3 65 3 75 2 Total 10
DPB 30063 / MEASURES OF CENTRAL TENDENCY 31 3.1.1 Mean • Mean is a measure of central tendency that is computed by taking the sum of all data values and then dividing it by the number of data. • It is also known as average. • Mean is the most commonly used in central tendency. Formula Scattered Frequency Distribution Table Example 1: Find the mean for the following data: a. 3, 5, 7, 5, 6, 10 b. 1, 4, 8, 15, 14, 20, 23 Solution:
DPB 30063 / MEASURES OF CENTRAL TENDENCY 32 Advantages of using mean as measurement of central tendency: • It is the most frequently used information • It is simple to calculate and is accurate • Considers the whole set of data Disadvantages of using mean as measurement of central tendency: • Mean value are not accurate and not appropriate for data with extreme (very large) values. 3.1.2 Median • Median is the middle of a set data when the data are arranged in ascending or descending order. • It more suitable to be used for representing the data when there is an extreme value (very large or very small) in the set of data. Steps to find median 1. Arrange the data in ascending (from small to large number) 2. Find the location for median = +1 2 3. Find the value for median Example 2 Find the median for the following data: 11, 14, 3, 21, 17, 16, 19, 5, 7, 19, 8, 9, 20, 4, 15
DPB 30063 / MEASURES OF CENTRAL TENDENCY 33 Solution Example 3 A real estate broker intends to determine the median selling price of 9 houses listed below. What is the median price? 67000 105000 148000 167000 91000 95000 122000 189000 116000
DPB 30063 / MEASURES OF CENTRAL TENDENCY 34 Solution Advantages of using median as a measurement of central tendency: • Straightforward and uncomplicated. • Easier than using mean. • The magnitude of extreme values does not significantly affect the median. • The calculations of median will not be affected if the largest or smallest data value is not stated. Disadvantages of using median as a measurement of central tendency: • The accuracy of the median value is affected by the number of data given. • Median cannot be determined for non-quantitative data. 3.1.3 Mode • Mode is the value that occurs most frequently (highest frequency) in a data set. • Mode is located by arranging the data in ascending or descending order.
DPB 30063 / MEASURES OF CENTRAL TENDENCY 35 Example 4 The quantities ordered (in million units) for the first 20 weeks of a certain product are given as follows: 14.25 25.00 27.00 24.00 43.20 27.00 19.00 28.00 34.00 15.00 19.00 11.00 7.00 23.00 14.00 19.00 15.00 21.00 15.50 22.00 Solution Therefore the value for mode is 19.00 because the quantities occurs 3 times compare to others. Advantages of using mode as a measurement of central tendency: • Easy to understand • Mode value is not affected by the extreme data. • To find the mode is much easier than to find the mean. Disadvantages of using mode as a measurement of central tendency: • The accuracy of the mode value is affected by the number of data given. • Mode value does not measure the whole data.
DPB 30063 / MEASURES OF CENTRAL TENDENCY 36 3.2 Measure of Central Tendency For Group Data. 3.2.1 Mean Steps to find mean: a. Find the mid-point ( x ) b. Find the total of the frequency c. Multiply f (frequency) and x (mid point) than get the total of fx Example 5 The frequency distribution table below shows the distribution of weight for 21 students. Find the mean weight. Mass (kg) Frequency (f) 40 – 44 2 45 – 49 4 50 – 54 6 55 – 59 5 60 – 64 4 Total 21
DPB 30063 / MEASURES OF CENTRAL TENDENCY 37 Solution Example 6 A survey was conducted by a group of students as a part of their environment awareness program, in which they collected the following data regarding the number of plants in 20 houses. Find the mean number of plants per house. Number of plants 0 - 2 2 - 4 4 - 6 6 - 8 8 – 10 10 - 12 12 - 14 Number of houses 1 2 1 5 6 2 3
DPB 30063 / MEASURES OF CENTRAL TENDENCY 38 Solution
DPB 30063 / MEASURES OF CENTRAL TENDENCY 39 3.2.2 Median Formula Median Steps to find median: i. Obtain the cumulative frequencies. ii. Determine the location of median class interval using cumulative frequency column. iii. Find the frequency and lower limit (from class boundaries) of the median class. iv. Find the class width / size (c) of the median class. v. Find the cumulative frequency for the median class vi. Calculate the value of median using the formula.
DPB 30063 / MEASURES OF CENTRAL TENDENCY 40 Example 7 Find the median for the following distribution Marks Frequency 30 – 39 20 40 – 49 18 50 – 59 22 60 – 69 24 70 – 79 0 80 – 89 14 90 - 99 20 ∑
DPB 30063 / MEASURES OF CENTRAL TENDENCY 41 Solution
DPB 30063 / MEASURES OF CENTRAL TENDENCY 42 Example 8 Below are the wages for workers in a factory Wages No. of Workers 200 – 300 3 300 – 400 5 400 – 500 20 500 – 600 10 600 - 700 6 ∑