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# Introduction to statistics and Measure of central tendency

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Introduction to Statistics: Meaning of Statistics, Introduction to data, Variable, order array, frequency distribution, Types of frequency distribution. Illustrative examples. Central Tendency: Mean: definition and calculation, General and mathematical properties of mean, Combined arithmetic mean, weighted arithmetic mean. Median: definition and calculation, Properties of median, Location of median, Partition values: Quartiles, Deciles, Percentiles, Mode: Definition and calculation, properties of mode, Empirical relation of mean, median and mode. Illustrative examples.

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### Introduction to statistics and Measure of central tendency

1. 1. ShakehandwithLife.in Quantitative Techniques Volume-2 (Revised) Introduction to Statistics and Measure of Central tendency E-Book Code : QTVOL2 by Narender Sharma “Save Paper, Save Trees, Save Environment”
2. 2. www.shakehandwithlife.in Page 1 Quantitative Techniques Telephonic classes + e-book+ login Id and password of your Online Study Room To Practice MCQ’s test paper, Online assignments, Discussion forum and many more things to do……. Visit: www.shakehandwithlife.in e-mail: shakehandwithlife@gmail.com, narender@shakehandwithlife.in
3. 3. www.shakehandwithlife.in Page 2 Click on Contents Introduction to Statistics .............................................................................................................................................................................. 4 Meaning of Statistics .................................................................................................................................................................................. 4 Statistics in plural sense ...................................................................................................................................................................... 4 Statistics in Singular Sense ................................................................................................................................................................. 5 Scope of Statistics ................................................................................................................................................................................... 5 Functions of Statistics ........................................................................................................................................................................... 5 Uses and Importance of Statistics .................................................................................................................................................... 5 Limitations of Statistics ........................................................................................................................................................................ 6 Distrust of Statistics ............................................................................................................................................................................... 6 Introduction to data ........................................................................................................................................................................................ 6 Variable ............................................................................................................................................................................................................ 6 Discrete Variable ..................................................................................................................................................................................... 6 Continuous Variable .............................................................................................................................................................................. 7 Data ................................................................................................................................................................................................................... 7 Ordered Array or Individual series ................................................................................................................................................. 7 Frequency Distribution ........................................................................................................................................................................ 8 1. Discrete Frequency distribution ............................................................................................................................................. 8 2. Grouped Frequency Distribution ........................................................................................................................................... 8 Types of Grouped Frequency distribution ................................................................................................................................ 10 1. Exclusive Series .......................................................................................................................................................................... 10 2. Inclusive Series ........................................................................................................................................................................... 10 3. Open Ended Series ..................................................................................................................................................................... 10 4. Mid value series .......................................................................................................................................................................... 11 5. Cumulative frequency series ................................................................................................................................................. 11 6. Relative Frequency Distribution .......................................................................................................................................... 11 Statistics of Love : A story of statistical decision making ............................................................................................................. 13 Central Tendency .......................................................................................................................................................................................... 16 Functions of good Average ............................................................................................................................................................ 16 Properties of a good Average .......................................................................................................................................................... 16 Mean ............................................................................................................................................................................................................... 16 General Properties of Mean ............................................................................................................................................................. 17 Mathematical Properties of Mean ................................................................................................................................................. 17 Mean for Individual Series ............................................................................................................................................................... 17 Mean for Discrete Series ................................................................................................................................................................... 18 Mean for Continuous series ............................................................................................................................................................. 20 Combined Arithmetic Mean: ........................................................................................................................................................... 24 Weighted Arithmetic Mean .............................................................................................................................................................. 24 Median........................................................................................................................................................................................................... 25
4. 4. www.shakehandwithlife.in Page 3 Properties of Median .......................................................................................................................................................................... 25 Location of Median .............................................................................................................................................................................. 25 Median for Individual Series ........................................................................................................................................................... 26 Median for Discrete Series ............................................................................................................................................................... 28 Median for Continuous Series ........................................................................................................................................................ 28 Partition Values ......................................................................................................................................................................................... 33 Quartiles .................................................................................................................................................................................................. 33 Deciles ...................................................................................................................................................................................................... 33 Percentiles .............................................................................................................................................................................................. 33 Mode .............................................................................................................................................................................................................. 35 Properties of Mode .............................................................................................................................................................................. 35 Mode for Individual Series ............................................................................................................................................................... 35 Mode for Discrete Series ................................................................................................................................................................... 35 Mode for Continuous Series ............................................................................................................................................................ 36 Empirical Relation of Mean, Median and Mode .......................................................................................................................... 38 References ........................................................................................................................................................................................................ 40 Feedback ........................................................................................................................................................................................................... 40
5. 5. www.shakehandwithlife.in Page 4 Introduction to Statistics  A Population (Universe) Is The Whole Collection Of Things Under Consideration  A Sample Is A Portion Of The Population Selected For Analysis  A Parameter Is A Summery Measure Computed To Describe A Characteristics Of The Population  A Statistic Is A Summary Measure Computed To Describe A Characteristics Of The Sample Some Facts About Statistics  In Late 1970’s, Dr. Mikel Harry , a senior staff engineer at Motorola’s Government Electronics Group (GEG), experimented with problem solving through statistical analysis. Using this approach , GEG’s products were being designed and produced at a faster rate and at a lower cost.  Bill Smith , who is now called Father of Six Sigma , an engineer , and Mikel Harry together devised a 6 step methodology called Six Sigma, with the focus on defect reduction and improvement in yield through statistics.  The Motorola Corporation saved \$ 16 billion in 10 years. Meaning of Statistics Statistics in plural sense  Aggregate of facts  Numerically Expressed  Affected by Multiplicity of Facts  Reasonable Accuracy  Placed in Relation to each other  Pre-determined Purpose  Enumerated or Estimated  Collected in Systematic Manner Meaning of Statistics Plural Sense Numerical Data or Statistical Data e.g. Production, Income, Expenditure, Population, Prices Singular Sense Techniques /Methods e.g. Collecting, Classifying, Presenting, Analysing And Interpreting The Data Graphics: Narender Sharma
6. 6. www.shakehandwithlife.in Page 5 Statistics in Singular Sense  Collection of Data  Organization of Data  Presentation of Data  Analysis of Data  Interpretation of Data Scope of Statistics Functions of Statistics  Simple Presentation  Enlarges Individual Knowledge and Experience  It compares facts  Expressions of facts in numbers  It Facilitates Policy Formulation  It helps other sciences in testing their laws  It Establishes Relationships between facts  It helps in Forecasting  It enables realization of magnitude of a problem  Presentation of data in condensed form Uses and Importance of Statistics  Importance for administration  In the field of business, industry or agriculture Scope of Statistics Nature Statistics is both a science and an art Subject Matter Descriptive Statistics: It Deals with method used in collections, organisation, presentaion and analysis of data in order to describe various features and charateristics of such data Inferential Statistics: It helps the decisions maker to draw conclusions about the charateristics of large population on the basis of sample result. Limitations Statistics must be regarded as an instrument of research of great value but barring severe limitations which are not possible to overcome. Graphics: Narender Sharma
7. 7. www.shakehandwithlife.in Page 6  Importance in Economics viz. Consumption, Production, Exchange, Distribution, Revenue, Economic planning, National Income  Importance in political fields  Importance in the social fields  Importance in the fields of science and research  Importance for banking  Importance for insurance companies  Importance in the field of education Limitations of Statistics  Study of numerical facts only  Study of aggregates only  Not the only method  Homogeneity of data  Results are true only on an average  Without reference results may prove wrong  Can be used only by experts  Misuse of statistics is possible  Only means not the solution Distrust of Statistics  Statistics is a rainbow of lies.  Statistics are tissues of falsehood.  Statistics can prove anything.  Statistics can not prove anything.  Statistics are like clay of which you can make god or devil, as you please. Introduction to data Variable Variable or the Variate is the ability or the characteristics through which a direct quantitative measurement is being done. Height Weight , Production, consumption , marks etc. are all variables. A variable may be either discrete or continuous. Discrete Variable A discrete variable is that one which takes only isolated or discontinuous values. It must be complete number . For examples  Number of goals scored in a football match  Number of children in a family  Wicket fall down in a cricket match  Number of students in class  No. of cars manufactured in a car manufacturing company Note that all the values in the examples mentioned above has a complete and an isolated number. We can’t assume the number of goals could have 3.5 or 3.2 or any other continuous variable because it’s not possible. No. of children in a family. Can it possible that no. of children be one and half i.e. 1.5 or 1.2 or 1.8 or can have value between 2 and 3 or 3 and 4? No it can’t be possible. Hence the variable which takes a complete, isolated and discontinuous value is a discrete variable.
8. 8. www.shakehandwithlife.in Page 7 Continuous Variable A continuous variable is one which can take any value in a specified interval. Continuous variable is expressed by some measurement units. e.g. 4.2Kg, 2.32meter , 172.67cm, 33.120Celsius etc. A continuous variable can take any value in a specified interval. Like the value between 22cm and 24cm may have 22.1, 22.2, 22.01, 22.12, 22.3, 22.31, 22.312………….and so on these values may infinitely large. Hence this kind of variable is called the continuous variable. Data The data means information or set of given facts. Data is in either attribute form or numerical form. If the data is in attribute form then first convert the data into numerical form. Numerical data can be classified into following categories; Ordered Array or Individual series An ordered array or individual series is an orderly arrangement of data according to the ascending or descending value of the given data. Hence an Individual series is nothing but arrangement of the data in ascending or descending order. Variable Direct Quantitative Measurement Discrete Variable e.g. No. of goals, Students in a class, Members in a family Isolated, discontinuous or Complete Number Continuous Variable Value in a specified interval, Expressed by some measurement units e.g. 4.2kg , 172.33cm, 330 celsius Numerical data Ordered Array or Individual series Frequency Distribution Discrete Frequency Distribution Grouped Frequency Distribution Graphics: Narender Sharma Graphics: Narender Sharma