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Quantitative Techniques Part II
 

Quantitative Techniques Part II

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• Construction of frequency distribution and their analysis in form of measures of central tendency and variations. ...

• Construction of frequency distribution and their analysis in form of measures of central tendency and variations.
• Measure of Dispersion , Types of measures , their relative merits, limitations and characteristics.
• Skewness, meaning and coefficients of skewness.

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    Quantitative Techniques Part II Quantitative Techniques Part II Presentation Transcript

    • Cutting Edge for Management and Engineering Professionals By Narender Sharma LSL USL Specification Width LCL UCL Higher the number in At Level Normal front of i.e. Z , the Distribution covers Process Width lower possibility of 99.99966 observations defect occurrence MeanShakehand with Life “Promoting quality culture in every sphere of human life.”www.shakehandwithlife.blogspot.com
    • Few words to all management and engineering professionals……………… Few Motivational wordsAll of you have studied about the Software and Hardware as these “ I know you are goingunderstand as a heavy source of earning but I believe in Humanware and in to make it ………..my view “Nothing costly than a human mind as it has unlimited capabilities.” It may take timeSo my Mission statement is and hard work “Shakehand with Life” You may become frustrated andAll Calculations and graphs have made in this booklet with the help of at times you will feelMS – Excel and Advanced Statistical Software Minitab and every precautionhas been taken while solving the questions. Subject matter is try to clear by like giving upstories, examples and pictures. Sometimes you may evenAs a quality professional It is my effort to induce the statistical wonder if it’s really worth ittemperament in the minds of all management and engineering professionals But I have confidence in youwho are coming in management programme from different corporate areas. and I know you’ll make it,Statistic is the basic tool to reach on a precise and specific decision, asdecision making is the key responsibility in the management. If you are not If you try.”aware about the role of Statistic in the management then definitelysomewhere you will lag behind in your area. ……………..Ananda PierceI always seeking feedback from your side so that, I can continuously makeimprovement in my work.With RegardsNarender Sharma Owner Manager and Blogger at Shakehand with Life web portal. Working As A Quality Professional In A Leading Container Glass Manufacturing Organization. Visiting Faculty in Leading Management Institutes in Delhi. Six –Sigma Green Belt. M.B.A. (Production And Operation Management). B.Sc. (Electronics, Physics, Mathematics).Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 2www.shakehandwithlife.blogspot.com
    • Train-ed To be a LeaderPeople are often like a Train, Some are like its EngineLeading the train forward,Some are like bogies chugging along, following the leaderWhile few others are likes the brakes, putting a stop to its motionTherefore the leader is like the Engine of this trainA man who will lead with trust and honesty,with speed and also ensure that there are no accidents. Narender Sharma Lead India MAILBOX, My Times, My Voice Times of India, New Delhi, Aug24, 2007, p. 2Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 3www.shakehandwithlife.blogspot.com
    • 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 AUnit II Characteristics Of The Population A Statistic Is A Summary Measure Computed To Describe A Characteristics Of The Sample Construction of frequency distribution and their analysis in form of measures of central tendency and variations. Some Facts About Statistics Page No. 3-26 In Late 1970’s, Dr. Mikel Harry , a senior staff engineer at Motorola’s Government Electronics Group (GEG), experimented with problem Measure of Dispersion , Types of measures , their relative merits, solving through statistical analysis. Using this approach , GEG’s products were being designed and produced at a faster rate and at a limitations and characteristics. lower cost. Page No. 26-36 Bill Smith , who is now called Father Of Six Sigma , an engineer , and Mikel Harry together devised a 6 step methodology called Six Sigma, Skewness, meaning and coefficients of skewness. with the focus on defect reduction and improvement in yield through statistics. Page No. 37-38 The Motorola Corporation saved $ 16 billion in 10 years.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 4www.shakehandwithlife.blogspot.com
    • Meaning of Statistics Nature Statistics is both a science and an art Plural Sense Singular Sense Descriptive Statistics: It Deals with method used in Numerical collections, organisation, Data or Techniques presentaion and analysis of Statistical /Methods data in order to describe Data various features and charateristics of such data e.g. Production, e.g. Collecting, Income, Expenditure, Classifying, Subject Population, Prices Presenting, Analysing And Interpreting The Scope of Matter Data Statistics Inferential Statistics: It helps the decisions maker to draw conclusions aboutFeature of Statistics in Plural Sense or in terms of Numerical Data the charateristics of large population on the basis of• Aggregate of facts sample result.• Numerically Expressed• Affected by Multiplicity of Facts• Reasonable Accuracy• Placed in Relation to each other• Pre-determined Purpose Statistics must be regarded• Enumerated or Estimated as an instrument of• Collected in Systematic Manner Limitations research of great value butFeature of Statistics in Singular Sense or Statistical Methods barring severe limitations which are not possible to• Collection of Data overcome.• Organization of Data• Presentation of Data• Analysis of Data• Interpretation of DataShakehand with Life “Promoting quality culture in every sphere of human life.” Page 5www.shakehandwithlife.blogspot.com
    • Functions of Statistics Distrust of Statistics• Simple Presentation • Statistics is a rainbow of lies.• Enlarges Individual Knowledge and Experience • Statistics are tissues of falsehood.• It compares facts • Statistics can prove anything.• Expressions of facts in numbers • Statistics can not prove anything.• It Facilitates Policy Formulation • Statistics are like clay of which you can make god or devil, as you please.• It helps other sciences in testing their laws• It Establishes Relationships between facts Construction Of Frequency Distribution: A) Discrete Frequency Distribution : It is statistical table which shows the• It helps in Forecasting value of variable individually and also the corresponding frequencies side• It enables realization of magnitude of a problem by side. We use tally bars to construct it.• Presentation of data in condensed form e.g. Twenty students of MBA 1st secured the following marks inUses and Importance of Statistics managerial economics out of 30 marks in assignments• Importance for administration 11, 12, 14, 11, 16, 11, 17, 16, 17, 14, 17, 18, 20, 14, 20, 17, 20, 17, 14, 20• In the field of business, industry or agriculture• Importance in Economics viz. Consumption, Production, Exchange, Now discrete frequency distribution of data Distribution, Revenue, Economic planning, National Income Marks Tally Bars Frequency• Importance in political fields 11 ||| 3• Importance in the social fields• Importance in the fields of science and research 12 | 1• Importance for banking 14 |||| 4• Importance for insurance companies 16 || 2• Importance in the field of education 17 |||| 5Limitations of Statistics 18 | 1• Study of numerical facts only• Study of aggregates only 20 |||| 4• Not the only method Total 20• Homogeneity of data• Results are true only on an average• Without reference results may prove wrong B) Grouped Frequency Distribution : It is statistical table which shows the• Can be used only by experts value of the variable in groups and also the corresponding frequencies• Misuse of statistics is possible side by side. e.g.• Only means not the solutionShakehand with Life “Promoting quality culture in every sphere of human life.” Page 6www.shakehandwithlife.blogspot.com
    • Marks of the students in Q.T. No. of Students Rules For constructing a grouped frequency distribution 40-50 10 I) Selection of number of classes ( ) : Prof. H.A. Struge have given a 50-60 8 formula by which the number of class interval can be ascertained. 60-70 7 70-80 5 The formula is 80-90 2 Here number of class intervas; Total number of observations Total 32Important Terms associated with Grouped Frequency Distribution: II) Size or width of the class- intervals ( ) : Size of the class interval can1) Class Interval or Class : It is a group of numbers in which items are placed be find out by the formula given here is such as 10-20, 20-30 etc.2) Class Frequency: The numbers of observations falling within a class is III) Precautions in finding of class interval called its class frequency denoted by ‘f’. a) Class interval as such the class limits are multiples of 5 e.g. 0-5, 5-10, 10-3) Class Limits: Each Class is Located between two numbers .These two 15, 15-20 etc. However , any number can be taken as class interval. numbers constitute class limits. The lowest value of a class is its lower b) The class interval should be uniform throughout the distribution. limit and the higher value is termed as higher limit. e.g. in the class 10-20, c) Every class interval should have convenient mid point. the lower limit is 10 and the upper limit is 20. IV) Selection of class limits4) Class Mark (or Mid-value): a) the mid values of the classes coincide or come very close to the point of concentration in the data. b) The overlapping of classes is avoided.Width or Magnitude of the class: where is the size of the class c) Class Limits must be stated precisely enough so that there will be nointerval. confusion as to what they include. Marks 10-15 15-20 20-25 25-30 30-35 Kinds of continuous seriesFrequency 4 5 8 5 4 Total=26 1. Exclusive series: The series in which every class interval excludes items corresponding to its upper limit. In this series , the upper limit of one class interval is the lower limit of the next class interval. e.g.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 7www.shakehandwithlife.blogspot.com
    • 2. Inclusive Series: 4. Frequency Series containing mid values of class intervals and their An inclusive series is that series which includes all items up to its corresponding frequencies upper limits . Mid- 5 15 25 35 45 In such series , the upper limit of class interval does not repeat itself Values as a lower limit of the next class interval. Frequency 4 5 8 5 4 Total=26 There is a gap between the upper limit of a class interval and the lower limit of the next class interval which ranges between 0.1 to 1.0 e.g. Mid Value series may be converted into simple frequency series using the Marks 10-14 15-19 20-24 25-29 30-34 following method Find the difference between mid values ( ) Frequency 4 5 8 5 4 Total=26 Mid- Values 5 15 25 35 45Conversion of Inclusive series into Exclusive series Frequency 4 5 8 5 4 Total=26 First , find the difference between the upper limit of a class interval and the lower limit of the next class interval. Class 0-10 10-20 20-30 30-40 40-50 In the case of first class interval half of the difference is subtracted from the lower class and half is added to the upper class. In case of subsequent class intervals half of the difference is added to the 5. Cumulative frequency series : Cumulative frequency series is that series in upper limit and remaining half to the lower limit of the next class which the frequencies are added corresponding to each class interval in interval. e.g. the distribution. Marks 9.5-14.5 14.5-19.5 19.5-24.5 24.5-29.5 29.5-34.5 Marks Frequency Cumulative Frequency 5-10 3 3 Frequency 4 5 8 5 4 Total=26 10-15 8 8+3=11 15-20 9 11+9=20 20-25 4 20+4=243. Open Ended Series : ‘Less than or below’ is specified in place of the lower 25-30 4 24+4=28 class limit of the first class interval and ‘more than or above’ is specified in place of the upper class limit of the last class interval. e.g. Marks Before going for further study here is a story to understand the basic concepts Less than 5 5-10 10-15 15-20 20 and above of the statisticsFrequency 4 5 8 5 4 Total=26Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 8www.shakehandwithlife.blogspot.com
    • Now the boys went to the professor and tell him that the girl allowed them Statistics of Love to see her for ten days .Three boys saw an extremely beautiful girl . After seen the girl the boys were Now professor advised them, “Note down your heart beat per minute dailynot able to stop themselves to propose the girl. So, the boys approached the for ten days while seeing the girl.”girl and after came to her they said,” Excuse us Miss, we would like to saysomething to you.” Now the data of heart beat/min of the boys after ten days was like thisThe girl looked the boys very gently and replied, “ Please, gentlemen if you Day 1 2 3 4 5 6 7 8 9 10 Boy 1 98 96 92 94 95 89 79 73 92 75have any problem?.” Boy 2 96 88 86 82 89 90 79 75 82 78One of them said in a very soft and calm voice, “ Miss , we want to say that Boy 3 78 76 74 75 77 79 76 78 79 80after having seen you , we have some sweet –sweet feeling in our heart foryou, we think all of us are in Love with you. Will u marry one of us?.” After collecting the data the boys went to the professor and show the data.Now the girl who belongs to statistical background said, “ok, gentlemen it Professor calculated the averages of data of heartbeats of all three boys andfeels very nice to heard all these, but one thing I would like to say that can find thatyou prove it statistically? I would like to choose the best among you.” Average Boy 1 Boy 2 Boy 3Now the Lips of all three boys were sealed and they thought that how could heartbeat/minit be possible to prove the love statistically. 88.30 84.50 77.20 of all three boysThey got back to their position and continuously thinking about this throughthe whole day but never forget the face of the girl. Now professor said to first boy that show the data to the girl and tell her that your average heart beat is highest among three boys so you are the perfectIn search of solution they went to a professor of statistics and told him about life partner for the girl as your heart is beating more than other two boystheir problem. Boys went to the girl and first boy said very confidently by showing the data ,The professor advised them, “Ask the girl to allow you to see her “Miss, here is the data of our heartbeats for ten days whenever we saw youcontinuously for ten days”. our hearts were beating more than the normal and also my averageNext day on the same place , They go to the girl and asked her the same thing. heartbeat is more than other two boys , So I am the best person to get marry with you.”The girl replied to the boys , “she has no problem on this but they will not tryto misbehave with her at any cost” Girl replied in the answer of the boy, “OK, Gentleman, but I would like to analyze all the data by my own and then I will answer you that who will beThe boys had given promise on this. best person to get marry with me. I will answer you in next three days at three different places individually.”Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 9www.shakehandwithlife.blogspot.com
    • Next three days would going to be very crucial to all three boys as they don’t difference of your minimum and maximum value of heartbeat is 6 which alsoknow who will get the opportunity to marry with the girl. lowest among three which means your love to me is quite static throughout the life even the intensity of love is low. Above all, the data analysis showsThe girl analysis the whole data by own and found the result below you could be better life partner for me. Std. Coeff. Then the Girl bent on her knees asked the boy.Variable Day Mean Min. Median Max. Skewness Dev. Var. Boy1 10 88.30 9.17 10.38 73.00 92.00 98.00 -0.86 Will you marry me? Boy2 10 84.50 6.43 7.61 75.00 84.00 96.00 0.26 Boy3 10 77.20 1.93 2.50 74.00 77.50 80.00 -0.24 To listen the answer of the girl the third boy filled with joy and picked up the girl in his arms.Now on next day the girl meet the first boy and said , “ No doubt that your This way the girl chose the right partner for her using statistics.average heart beat is more than other two boys but Std. Dev. and coff. of var.is very high which is 9.17, and 10.38 respectively which shows that your love From the above story we conclude that Statistics make us enough capable toto me has very high variability and very low consistency also value of take right decisions at right time and in right way not only in corporate lifeskewness -0.86 shows high negative asymmetry of data and the difference but also in our personal life.between your minimum and maximum value of heart beat is 25 this shows The story got international appreciations from the corporate and studentthat sometime you will love me with high emotions and sometime this shows community. Few comments are mentioned belowneutral behavior to me , So in all in all the data of your heartbeat is notreliable and so I cannot marry with you, I am very sorry gentleman.” “Very Innovative way of teaching statistics!” via e-mailNext day girl meet the 2nd boy and said, “ Your average heartbeat is also good Viswanathan Balasubramanian (Head,Corporate Quality at Chennai Centresbut the Std. Dev and Coff. of Var. are 6.43 and 7.61. No doubt both are less at, HCL Technologies Chennai Area, India)than in case of the first boy but still the variation of data is quite high. Itshows that your love to me also has high variability and low consistency. “Very True. Statistics can help us to draw conclusion to almost anything.” viaThe value of skewness 0.26 shows that low positive asymmetry of data. The Linkedindifference between your minimum and maximum value of heartbeat is 21 Vijesh kumar Devasy (Assistant Manager Quality at Genisys Informationwhich means high difference of intensity of love towards me. Above all , your Systems Pvt Ltd. Chennai Area, India)data shows that your love is also not static to me, so gentleman I cannot gowith you, Sorry.” “Good one! Makes a point!” via linkedinAt last girl meet the third boy, who was looking very depressed as he had Madhavi Shrivatava (Senior Quality Professional Greater Pittsburghlowest average heartbeat. Girl said to the boy, “ Gentleman your heartbeat is Area Currently in Pittsburgh, PA USA at Exploring jobs & sponsorship options77.20 lowest among three but Std. Dev. and Coff. of Var. are 1.93 and 2.50 for H1B)respectively which shows low variability and quite good consistency. Also thevalue of skewness is -0.24 is also the lowest value of asymmetry of data. The “Interesting way to sell Six Sigma concepts to young college crowd!!” via linkedinShakehand with Life “Promoting quality culture in every sphere of human life.” Page 10www.shakehandwithlife.blogspot.com
    • Central Tendency/AverageLt Cdr Shalini Agrawal (Lean Six Sigma Initiative Lead at TCS Mumbai Area,India ,Black Belt at TCS) It is a single fig. that represents the whole data. It represented by“WOW...just hope it could be as true and as easy as depicted by this Meanexample...nonetheless, stats always allow you to reach a point :)” Via linkedinPraveen Gupta (Head -Operations & Quality New Delhi Area, India HeadOperations & Quality at Macro Outsourcing Pvt. Ltd.) Median“Good one.... But may be we should give other two guys chance to reduce thevariation by carrying DMAIC and lets observe the improvement.... After all itsa matter of Love and Life :P well just kidding.....”Commented directly on theblog page Mode Value that occurs mostManish Rawat (Operations , Manufacturing, Pune, Maharashtra)“Dear Sir, I have spent more than two hour to understand mean, median , Purpose and Functions of Good Averagestandard deviation and skewness so that I can understand why the girl reject • Study of numerical facts onlyfirst two boys and propose the third boy.” Via E-mail • Brief Discription • Helpful in comparisonSatish, Student of Management Programme (MBA) (DITM Managementcollege Gannore, Sonepat) • Helpful in formulation of policies • Basis of statistics Analysis • Representation of the Universe or the mass statistical data.Now, How would have the girl done the statistical analysis? All this we willlearn in this unit. Characteristics / Properties of a good AverageTo read the complete story with pictures , follow the link • It should be easy to understand .http://shakehandwithlife.blogspot.com/2010/10/statistics-of-love.html • It should be easy to compute. • It should be uniquely defined. • It should be based on all observations. • It should not be unduly affected by extreme values. • It should be capable of further algebraic treatment.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 11www.shakehandwithlife.blogspot.com
    • Mean Shortcut methodDiscrete SeriesDirect Method Formula for calculation of mean Where The above example can be solved by this method as followWhereExample : Calculate the mean for following data The above example can be solved by this method as follows Wages 10 20 30 40 50 Wages (Rs) No. of workers No. of 4 5 3 2 5workers 10-30=-20 10 4 -80Solution : Denoting wages by and No. of workers by Wages( No. of workers( 20-30=-10 20 5 -50 10 4 40 20 5 100 30=A 3 30-30=0 0 30 3 90 40 2 40-30=+10 +20 40 2 80 50 5 50-30=+20 +100 50 5 250Calculation of meanShakehand with Life “Promoting quality culture in every sphere of human life.” Page 12www.shakehandwithlife.blogspot.com
    • In Continuous Series Mean Can be calculated by using following method Example : Calculate the arithmetic mean using all three methods of following data Marks 0-10 10-20 20-30 30-40 40-50 Direct No. of 20 24 40 36 20 Method Students Solution : Calculation of Arithmetic Mean using Direct Method Marks No. of Students Shortcut Method 0-10 20 100 10-20 24 360 20-30 40 1000Step Deviation 30-40 36 1200 Method 40-50 20 900Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 13www.shakehandwithlife.blogspot.com
    • Calculation of Arithmetic Mean using Shortcut Method Calculation of Mean in an Inclusive series Marks No. of Example : ) Students 0-10 20 -20 -400 Marks 20-29 30-39 40-49 50-59 60-69 No. of 10 8 6 4 2 10-20 24 -10 -240 Students 20-30 40 0 0 Solution: No. of 30-40 36 +10 +360 Marks Students ) 40-50 20 +20 +400 20-29 -20 10 -2 -20 30-39 8 -10 -1 -8Calculation of Arithmetic Mean using Step deviation Method No. of 40-49 6 0 0 0 Marks Students ) 0-10 -20 50-59 4 +10 +1 +4 20 -2 -40 10-20 24 -10 -1 -24 60-69 2 +20 +2 +4 20-30 40 0 0 0 30-40 36 +10 +1 +36 40-50 20 +20 +2 +40Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 14www.shakehandwithlife.blogspot.com
    • Calculation of Mean in Cumulative Frequency Series Calculate the mean for the following data:Example: Less than Less than Less than Less than More than Marks 10 20 30 40 50 Marks More than More than More than More than More than No. of 0 2 4 6 8 5 17 31 41 49 Students No. of 30 28 24 18 10 Students Solution: Calculation of Arithmetic MeanSolution : Since cumulative frequencies are given, first we find the simple Marks No. of Studentsfrequencies ) No. of 0-10 5 5 -20 -2 -10Marks Students ) 10-20 15 17-5=12 -10 -1 -12 1 30-28=2 -4 -2 -4 0-2 20-30 25A 31-17=14 0 0 0 3 28-24=4 -2 -1 -4 30-40 35 41-31=10 +10 +1 +10 2-4 40-50 45 49-41=8 +20 +2 +16 5A 24-18=6 0 0 0 4-6 7 18-10=8 +2 +1 +8 6-8 9 =10 +4 +2 +20 8-10 Combined Arithmetic Mean: If we have the arithmetic mean and the number of items of two or more than two related subgroups, we can calculate the combined arithmetic mean of the whole group by applying the following formula;Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 15www.shakehandwithlife.blogspot.com
    • Mathematical Properties of Arithmetic Mean1) The sum of the deviations of the items from arithmetic mean is always zero Symbolically,2) The sum of the squared deviations of the items from arithmetic mean is minimum i.e.3) If each item of a series is increased, decreased , multiplied or divided by some constant , then A.M. also increases, decreases , multiplied or is divided by the same constant.4) The product of the arithmetic mean and number of items on which mean is based is equal to the sum of all given items5) i.e.6) 5) If each item of the original series is replaced by actual mean , then the sum of these substitutions will be equal to the sum of the individual items .Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 16www.shakehandwithlife.blogspot.com
    • Weighted Arithmetic MeanSimple Arithmetic Mean , as discussed above gives equal importance(or weights) to each item of the series. But there can be some cases where all the items of aseries are not of equal importance. In case of Science, Economics or Technology each subject has its own importance. We have to provide different weightsaccording to their importance and weighted arithmetic mean is used as an average in such cases. The following formula is used to calculate the weightedarithmetic mean; ;Note : In weighted A.M. is taken instead ofPerformance of the students of the three universities given below using simple and weighted averages; University Mumbai Kolkata Chennai No. of Students No. of Students No. of Students Course of Pass% Pass% Pass% In (hundreds) In (hundreds) In (hundreds) Study X W WX X W WX X W WX M.A. 71 3 213 82 2 164 81 2 162 M.Com. 83 4 332 76 3 228 76 3.5 266 B.A. 73 5 365 73 6 438 74 4.5 333 B.Com. 74 2 148 76 7 532 58 2 116 B.Sc. 65 3 195 65 3 195 70 7 490 M.B.A. 66 3 198 60 7 420 73 2 146Simple and Weighted Arithmetic Means The Simple arithmetic mean is the same for all the three universities , i.e.,72 and hence it may be concluded that the performance of students is alike. But this will be a wrong conclusion because what we should compare here is the weighted arithmetic mean. On comparing the weighted arithmetic means , we find that for Mumbai the mean value is the highest and hence we can say that in Mumbai University the performance of students is best.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 17www.shakehandwithlife.blogspot.com
    • Median: “The median is that value of the variable which divides the Sr. No. 1 2 3 4 5 6 7 8 9group into two equal parts , one part comprising all values greater and the Items(X) 13 15 16 17 18 19 20 22 23other values less than the median”. Median is denoted by symbol ‘M’. Here N=9Note : A median is always determined by first arranging the series in anascending or descending manner. Calculation of Median (M) Individual Series (Even Number Series) Example : Calculate median for following data: Individual 200, 217, 316, 264, 296, 282, 317, 299 Series Where M=Median , Solution: N=8, arranged the data in ascending order N=Total no. of items in the series Sr. No. 1 2 3 4 5 6 7 8 Discrete Items(X) 200 217 264 282 296 299 316 317 Series Where M=Median , N=Total no. of items in the series Here N=8 Continuous Where Series ; Hence , M=289 Discrete Series Steps for calculation of median i) Arrange the data in ascending or descending order of size. ii) find the cumulative frequency column.Individual Series (Odd Number Series)Example : Calculate median for the following data: iii) Apply the formula22, 16, 18, 13, 15, 19, 17, 20, 23,Solution : Here N=9 Arranged the data in ascending orderShakehand with Life “Promoting quality culture in every sphere of human life.” Page 18www.shakehandwithlife.blogspot.com
    • iv) Now locate item in cumulative frequency column. It is Continuous Series, Steps for calculation: done by comparing with the cumulative frequency at Calculate cumulative frequency. Find out the median size using the formula each stage. The value of the variable is the value of the median. Determine the median class in which median lies. Example : Calculate the Median from the following data Substitute the values in the formulaSolution : ; Example : 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 Solution:The value of 30th item lies against 39 whose value is 16. Hence M=16Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 19www.shakehandwithlife.blogspot.com
    • Median Median . 50thitem lies in class 15-20. Hence median class is 15-20. Now 62.5th item lies in class 30-40. Hence median class is 30-40. NowCumulative frequency Series:Example Inclusive Series Marks: Less Less Less Less Less Less Less Less Example : than than than than than than than than 10 20 30 40 50 60 70 80 Marks: 1-10 11-20 21-30 31-40 41-50 No. of 4 16 40 76 96 112 120 125Students: No. of 4 12 20 9 5 Students: Solution: First we convert it into exclusive one by deducting 0.5 from theSolution : lower limit and adding 0.5 to the upper limits. Marks No. of Students (f) c.f. Marks No. of Students (f) c.f. 0-10 4 4 0.5-10.5 4 4 10-20 12 16 10.5-20.5 12 16 20-30 24 40 20.5-30.5 20 36 30-40 36 76 40-50 20 96 30.5-40.5 9 45 50-60 16 112 40.5-50.5 5 50 60-70 8 120 N=50 70-80 5 125 N=125Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 20www.shakehandwithlife.blogspot.com
    • Median25th item lies in class 20.5-30.5. Hence median class is 20.5-30.5 Now Descending Class Intervals Example: Marks: 30-35 25-30 20-25 15-20 10-15 5-10 0-5 No. of 4 8 12 16 10 6 4 Students:Unequal Class IntervalExample: Solution : 1st Method. 40and Converting the descending order series into ascending order series. Marks: 10-15 15-17.5 17.5-20 20-30 30-35 35-40 onwards Marks No. of Students (f) c.f. No. of 10 15 17 25 28 30 40 0-5 4 4Students:Solution: First convert it into a series with equal class intervals by adjusting 5-10 6 10the frequencies correspondingly. 10-15 10 20 Marks No. of Students (f) c.f. 15-20 16 36 10-20 10+15+17=42 42 20-25 12 48 20-30 25 67 25-30 8 56 30-40 28+30=58 125 40and onwards 40 165 30-35 4 60 N=165 N=60Median Median82.5th item lies in class 30-40.Hence median class is 20.5-30. Now 30th item lies in class 15-20. Hence median class is 15-20. NowShakehand with Life “Promoting quality culture in every sphere of human life.” Page 21www.shakehandwithlife.blogspot.com
    • Alternative Method:Median can also be calculated by keeping the series in descending order. The Merits of Median Demerits of Medianformula used is; 1. It is easy to understand . 1. It requires arranging of data in 2. It is easy to locate or compute. ascending or descending order 3. It is not affected by extreme but other averages do not need items. this.Where 4. Median can be located 2. It is not based on all Marks No. of Students (f) c.f. graphically . observations of the series, since 30-35 4 4 5. It is most suitable average in it is a positional average. dealing with qualitative facts 3. It is not capable of further 25-30 8 12 such as beauty, intelligence, algebraic treatment like 20-25 12 24 honesty etc. arithmetic mean. 15-20 16 40 6. It is most appropriate average 4. It cannot be computed exactly in case of open ended classes. where the number of items in a 10-15 10 50 7. It is rigidly defined. series is even. 5-10 6 56 5. It is very difficult to calculate if 0-5 4 60 the number of item is very small or large, N=60Median30th item lies in class 15-20. Hence median class is 15-20.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 22www.shakehandwithlife.blogspot.com
    • Partition Values-Quartiles, Deciles and Percentiles: Example: Calculate for the given dataQuartiles: Quartiles divides a series into 4 equal parts.For any series thereare three quartiles denoted by is known as first or lower 21, 15, 40, 30, 26, 45, 50, 54, 60, 65, 70quartile, covering 25% items . The second quartile or is the same as Solution : First Arrange the data in Ascending orderMedian of the series . is called third or upper quartile, covering 75%items. Sr.No. 1 2 3 4 5 6 7 8 9 10 11Deciles: Deciles divide a series into 10 equal parts. For any series , there are X 15 21 26 30 40 45 50 54 60 65 709 deciles denoted by These are called as first decile,second decile and so on.Percentiles: Percentiles divide a series into 100 equal parts. For any series ,there are 99 percentile denoted by . Boxplot of xFormula for calculation of Quartiles , Deciles, and Percentiles are given in 70below Table 60For Individual and Discrete Formula to be used 50 For Continuous Series Series in Continuous Series 40 x 30 20 10Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 23www.shakehandwithlife.blogspot.com
    • Mode If the modal values lies outside the modal class , the following formula used to calculate the mode : If the mode is ill defined , then we use the following formula: where Example: Calculate the mode from the following data 2 1 2 4 5 1 3 1 Marks 10 10 10 10 10 10 10 10 and and and and and and and and Mode Between 15 20 25 30 35 40 45 50 No. of 4 12 30 60 80 90 95 97 students The value of variable which occurs most frequently in a distribution is Solution : Since this is a cumulative frequency series, we first convert it into called the mode simple frequency series Not affected by Extreme Values. Marks 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 Between There May not be a Mode No. of 4 12-4 30-12 60-30 80-60 90-80 95-90 97-95 students =8 =18 =30 =20 =10 =5 =2 There may be several mode By inspection the modal class is 25-30 Used for either numerical or categorical data.Calculation of modeContinuous Series First determine the modal Class After determine the modal class Mode can be found by using the Empirical Relation Between Mean, Median and Mode formula The difference between and is 3 times the difference between , andWhere i.e. The equation can be expressed asNote :if the first class is the modal class, then is taken as zero. Similarly , if thelast class is the modal class, then is taken as zero.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 24www.shakehandwithlife.blogspot.com
    • Merits of Mode Demerits of Mode1. It is easy to understand and 1. It is not suited to algebraic 10-13 8 8 11.5 -12 -4 -32 simple to calculate. treatment. 13-16 15 23 14.5 -9 -3 -452. It is not affected by extreme 2. Therefore can be bimodal values. frequency series. 16-19 27 50 17.5 -6 -2 -543. It can be located graphically 3. It is not based on all the items 19-22 51= 101 20.5 -3 -1 -51 with the help of histogram. of the series.4. It can be easily calculated in 4. It is not rigidly defined. 22-25 75= 176 23.5=A 0 0 0 case of open ended classes. 5. It has no mathematical 25-28 54 230 26.5 +3 +1 +545. All the frequencies are not property. needed for its calculation. 28-31 36 266 29.5 +6 +2 +726. It is true representative of 31-34 18 284 32.5 +9 +3 +54 frequency distribution , since it is the value which occurs most 34-37 9 293 35.5 +12 +4 +36 frequently. 37-40 7 300 38.5 +15 +5 +35 N=300Combined Example on Mean, Median and ModeExample : Var. 10-13 13-16 16-19 19-22 22-25 25-28 28-31 31-34 34-37 37-40Freq. 8 15 27 51 75 54 36 18 9 7Solution: Calculation of Mean , Median and Mode Median lies in the class 22-25Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 25www.shakehandwithlife.blogspot.com
    • M=22+1.96=23.96 Measure Of DispersionBy inspection, mode lies in the class 22-25 Meaning of Dispersion 1) Dispersion is defined as scatterness or spreadness of the individual items in a given series. 2) If the value of all the items of a series is the same , there will be no variation among the various items and the dispersion will be zero.On the otherhand , the greater the variation among different items of a series, the more will be dispersion.Mode can also be found by the relation 3) Dispersion refers to the variation of the items around an average.If the difference between the value of items and the average is large , theThus, Mean ( ) = 24.19, Median ( ) =23.96, Mode ( )=23.6 dispersion will be high and on the otherhand if the difference between the value of the items and average is small, the dispersion will be low. 4) Dispersion is a measure of the extent to which the individual items vary 5) The degree to which numerical data tend to spread about average is called variation or dispersion of data. Objective of Measuring Dispersion 1) To determine the reliability of an average: for small dispersion or variation the average will be reliable and for large dispersion average will be quite unreliable. 2) To compare the variability of two or more series 3) For facilitating the use of other statistical measures 4) Basis of Statistical Quality Control: Properties of good measure of dispersion 1) It should be easy to understand. 2) It should be easy to calculate.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 26www.shakehandwithlife.blogspot.com
    • 3) It should be uniquely defined. Range4) It should be based on all observations. It is simplest measure of dispersion. It is defined as the difference between the largest and smallest value in the series. Its formula is;5) It should not be unduly affected by the extreme items. R= L - S6) It should be capable of further algebraic treatment. Where R= Range , L =Largest value in the series , S= Smallest value in theAbsolute and Relative measure of dispersion seriesAbsolute measure of dispersion: The relative measure of range , also called coefficient of range , is defined asAbsolute measure of dispersion are expressed in the same unit in which dataof the series are expressed.They are expressed in same statistical unit , e.g.,rupees , kilogram, tons, years, centimeters etc., e.g. for the given series 20, 35, 25, 30, 15Relative Measure of dispersion:Relative measure of dispersion refers to the variability stated in the form ofratio of percentage. Thus , relative measure of dispersion is independent ofunit of measurement. It is also called coefficient of dispersion. Thesemeasures are used to compare two series expressed in different units.Method of Measuring Dispersion Merits of Range Demerits of Range 1. It is simple to understand. 1. It cannot be calculated in case Range 2. It is easy to calculate. of open ended distribution 3. It is widely used in statistical 2. It is not based on all Interquartile Range quality control. Range charts observations of the series. Coefficient of and Quartile are useful in controlling the 3. It is affected by sampling Variation Deviation quality of the product. fluctuations. 4. It is affected by extreme values in the series. Standard Mean Deviation DeviationShakehand with Life “Promoting quality culture in every sphere of human life.” Page 27www.shakehandwithlife.blogspot.com
    • Inter Quartile Range and Quartile Deviation Mean Deviation or Average Deviation Mean deviation is defined as the arithmetic average of the deviation of theInter Quartile Range (IQR) The difference between upper Quartile (Q3) and various items of a series computed from some measure of central tendencylower Quartile (Q1) is called Inter Quartile Range say mean or median.Inter Quartile Range (IQR)=Q3-Q1 In taking deviation of the various items , algebraic signs + and – are not takenThe Inter Quartile ranges covers dispersion of middle 50% of the items of the into consideration.series. Mean deviation can be computed either from the mean or median , butQuartile Deviation : Quartile Deviation , also called Semi-Inter theoretically median is preferred because the sum of the deviations of theQuartile Range is half of the difference between the upper and lower quartile items taken from median is minimum when signs are ignored. The formulai.e. half of the Inter Quartile Range. Its formula is as for calculating mean deviation are:Quartile DeviationRelative Measure of Quartile deviation:Coefficient of quartile deviation The relative measure of mean deviation, also called the coefficient of meanMerits of Quartile Demerits of Quartile deviation is obtained by dividing mean deviation by the particular averageDeviation Deviation used in computing mean deviation1. It is easy to compute and simple 1. It gives 50% of the items i.e. the Thus to understand. first 25% and the last 25%.2. It is less affected by extreme 2. It is not capable for further values. algebraic treatment.3. It can be computed in open 3. It is affected by sampling ended classes. fluctuations. Example: Calculate the mean deviation from mean and its coefficient from the4. It is superior and more reliable 4. It is not a good measure of following data: than the range. dispersion particularly for Marks 0-10 10-20 20-30 30-40 40-505. It is useful when dispersion of series in which variation is Students 5 8 15 16 6 middle 50% items is to be considerable Solution: Calculation of Mean Deviation from Mean calculated.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 28www.shakehandwithlife.blogspot.com
    • 0-10 5 5 25 22 110 100-120 110 4 4 43 172 10-20 15 8 120 12 96 120-140 130 6 10 23 138 20-30 25 15 375 2 30 30-40 35 16 560 8 128 140-160 150 10 20 3 30 40-50 45 6 270 18 108 160-180 170 8 28 17 136 180-200 190 5 33 37 185Example: Calculate the mean deviation from median and its coefficient fromthe following data; Size 100-120 120-140 140-160 160-180 180-200Frequency 4 6 10 8 5Solution: Calculation of Mean Deviation from MedianShakehand with Life “Promoting quality culture in every sphere of human life.” Page 29www.shakehandwithlife.blogspot.com
    • Merits of Mean Demerits of Mean Standard Deviation( ) Deviation Deviation Standard deviation is defined as the square root of the arithmetic mean of1. It is easy to compute and simple 1. Ignoring signs is not correct the squares of the deviation of the values taken from the mean and is to understand. from mathematical point of2. It is less affected by extreme view. calculated as follows values. 2. It is not capable for further3. It is based on all the algebraic treatment. observations. 3. It is not accurate method when4. It is very useful in various fields it is calculated from mode. Also called Root Mean Square Deviation such as economics, commerce 4. It is difficult to compute when Denoted by small Greak Letter and social fields. the value of mean or median Standard deviation is the most important and widely used measure of5. Comparison about formation of comes in fractions . dispersion. different series can be easily 5. It is not used in statistical made as deviations are taken conclusion. from a central value. Difference between Mean Deviation and Standard Deviation Algebraic signs of deviations (+or -)are ignored while calculating mean deviation whereas in the calculation of standard deviation signs of deviations are not ignored i.e. they are taken into account. Mean deviation can be computed either from mean , median or mode .The standard deviation, on the other hand , always computed from the mean because the sum of the squares of the deviations taken from the mean is minimum.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 30www.shakehandwithlife.blogspot.com
    • Examples for calculation of standard deviation are given underMethods for calculating Standard Deviation Individual Series Actual Mean Method Example: Calculate the S.D. for the following data:Actual Mean Method 16, 20, 18, 19, 20, 20, 28, 17, 22, 20 Solution: calculation of standard deviation Assumed Mean 16 -4 16 Method 20 0 0 18 -2 4 19 -1 1 Step Deviation Method 20 0 0 20 0 0 28 8 64 17 -3 9 22 2 4 20 0 0Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 31www.shakehandwithlife.blogspot.com
    • Discrete Series Assumed Mean Method(Short Cut Method) Example: Calculate mean and standard deviation for the following data:Example: Calculate the S.D. from the following data given below: 3 4 5 6 7 8 9 Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 7 8 10 12 4 3 2 No. of 5 10 20 40 30 20 10 StudentsSolution : Solution: 3 7 -3 -21 63 0-10 5 5 -30 -3 9 -15 45 4 8 -2 -16 32 5 10 -1 -10 10 10-20 15 10 -20 -2 4 -20 40 6 12 0 0 0 20-30 25 20 -10 -1 1 -20 20 7 4 +1 +4 4 30-40 35 40 0 0 0 0 0 8 3 +2 +6 12 40-50 45 30 +10 +1 1 +30 30 9 2 +3 +6 18 50-60 55 20 +20 +2 4 +40 80 60-70 65 10 +30 +3 9 +30 90 70-80 75 4 +40 +4 16 +16 64 OrShakehand with Life “Promoting quality culture in every sphere of human life.” Page 32www.shakehandwithlife.blogspot.com
    • Variance Variance is the square of the standard deviation. Combined Standard Deviation The combined standard deviation of two groups is denoted by and is computed as follows: Example: Two samples of size 100 and 150 respectively have means 50 and 60 and standard deviations 5 and 6.Find the mean and standard deviations of the combined sample of size 250. Solutions: Given :Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 33www.shakehandwithlife.blogspot.com
    •  If a constant amount is added or subtracted from each item of a series, then S.D. remains unaffected i.e. S.D. is independent of the change of origin.  If each item of a series is multiplied or divided by a constant `a’ , then S.D. is affected by the same amount i.e. S.D. is not independent of the change of scale.  Standard deviation has the following relation to quartile deviation (Q.D.) and mean deviation (M.D.) in a symmetrical (or normal)distribution:Mathematical Properties of Standard Deviation  The standard deviation of first natural numbers can be found from the following formula:  The standard deviation has the following relation with arithmetic mean in a symmetrical distribution i.e. under normal curve 1) Area under the normal curve between is 0.6826 i.e. covers 68.26% area under the normal curve. 2) Area under the normal curve between is 0.9545 i.e. covers 95.45% area under the normal curve. 3) Area under the normal curve between is 0.9973 i.e.  The combined S.D. of two or more groups can be found by using the covers 99.73% area under the normal curve. following formula:  The sum of the squares of the deviations of the items taken from arithmetic mean is least, that is why standard deviation is computed from the A.M.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 34www.shakehandwithlife.blogspot.com
    • Below fig. Illustrates the Area property: Graph Showing Area Relationship of standard deviation with mean under normal curve. 68.26% 95.45% 99.73% x-scale z-scale  Standard Deviation Works as basic unit for Six-Sigma Methodology as shown below table World Class Industry Average Non- CompetitiveShakehand with Life “Promoting quality culture in every sphere of human life.” Page 35www.shakehandwithlife.blogspot.com
    • Merits and Demerits of Standard Deviation Solution : For finding out which share is more stable in value , we have to compare the coefficient of variation Merits of Standard Deviation Demerits of Standard Deviation1. It is based on all the 1. As compared to the quartile 41 -4 16 91 -4 16 observations. deviation and range, etc. it is2. It is rigidly defined. difficult to understand and 44 -1 1 93 -2 43. It is not very much affected by difficult to calculate. 43 -2 4 96 1 1 the fluctuations of sampling and , 2. It gives more importance to 48 3 9 92 -3 9 therefore is widely used in extreme observations. sampling theory and test of 3. Since it depends upon the units 45=A 0 0 90 -5 25 significance. of measurement of the 46 1 1 97 2 44. It is capable of being treated observations , it cannot be used 49 4 16 99 4 16 mathematically. For example , if to compare the dispersions of the standard deviation of a number distributions expressed in 50 5 25 94 -1 1 of groups are known , their different units. 42 -3 9 98 3 9 combined standard deviation can 40 -5 25 95=A 0 0 be computed.Coefficient of VariationCoefficient of variation is an important relative measure of dispersion. It wasdeveloped by Karl Pearson and is widely used in comparing the variability of Calculation of C.V.two or more series. Coefficient of variation is denoted by C.V. and is given by Share X Share YUses of Coefficient of VariationCoefficient of variation is used to compare the variability, homogeneity,stability , consistency and uniformity of two or more series.The series having less value of the coefficient of variation is considered moreconsistent in comparison to a series having a higher value of the coefficient ofvariation.Example: From the prices of shares of X and Y given below , state which shareis more stable in value: X 41 44 43 48 45 46 49 50 42 40 Y 91 93 96 92 90 97 99 94 98 95 Since the Coefficient of variation is less for share Y, hence share Y is more stable in price.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 36www.shakehandwithlife.blogspot.com
    • Skewness A) Positively Skewed distribution : If the longer tail of the frequency curve ofMeaning of Skewness: distribution lies to the right of the central point, it is called a positivelyThe term skewness means lack of symmetry in a frequency distribution. skewed distribution.Skewness denotes the degree of departure of a distribution from symmetry In Positively skewed distribution , the value of the mean will be greaterand reveals the direction of scatterness of the items. It gives us an idea about than median and median will be greater than mode i.e.the shape of the frequency curve. When a distribution is not symmetrical , It B) Negatively Skewed Distribution:is called a skewed distribution. Skewness tells us about the asymmetry of the If the longer tail of the frequency curve of the distribution lies to the leftfrequency distribution. of the central point , it is called a negatively skewed distribution. In theDefinition of Skewness: negatively skewed distribution , the value of the mean will be less then1. Skewness is the degree of asymmetry or departure from symmetry of a median be less than mode i.e. distribution -------- M. R. Speigal. Positively Skewed Negatively Skewed2. When a series is not symmetrical , it is said to be asymmetrical or Symmetric Distribution Distribution Distribution skewed-------------Croxten and Cowden.3. By skewness of a frequency distribution , we mean degree of its departure from symmetry----------Simpson and kafka.Skewness and frequency Distribution:1) Symmetrical Distribution: In a symmetrical distribution or symmetrical curve , skewness is not present. The values of mean , median and mode coincide i.e. . The spread of the frequencies is the same on both sides of the central Difference Between Dispersion and Skewness point curve. 1) Dispersion is concerned with measuring the amount of variation in a2) Skewed Distribution: series rather than with its direction. Skewness is concerned with A distribution which is not symmetrical is called skewed distribution or direction of variation or the departure from symmetry. asymmetrical distribution . A skewed distribution may be either 2) Dispersion tells us about the composition of the series whereas skewness positively skewed or negatively skewed. tells us about the shape of the series.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 37www.shakehandwithlife.blogspot.com
    • 3) Measure of dispersion are based on average of the first order such as etc. whereas measures of skewness are based on averages of Measures of Skewness first and second order such as Karl Pearson’s MethodTests of Skewness Absolute Measure of1) Relationship between Averages : If In a distribution , the values of mean , Coefficient of Skewness Skewness median and mode are equal i.e.Skewness is absent in it. On the otherhand, if the values of mean median andmode are not identical i.e. , then skewness is present in it. When mode( ) is ill defined, then When mode( ) is ill defined, then2) Distance of Quartiles from the median : If in a distribution, the quartiles( ) are equidistant from the median i.e. , then skewness is absent and if , then Bowley’s Methodskewness is present in the distribution. Absolute Measure of3) Graph of the data: When the data plotted on the graph paper gives us a Coefficient of Skewness Skewness bell shaped curve , skewness is absent. On the other hand , when the data plotted on the graph do not give the normal bell shaped curve i.e. the two values of the curve are not equal, then skewness is present in the Kelly’s Method (*Not so popular method) distribution. Absolute Measure of Coefficient of Skewness Skewness 1) 1) 2) 2)Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 38www.shakehandwithlife.blogspot.com
    • Ref. Books :JAIN T.R., DR. AGGARWAL S.C., Quantitative Methods for MBA, VK INDIAEnterprises.PARMANAND GUPTA , Comprehensive Mathematics , LAXMI PUBLICATIONS(P) LTD, NEW DELHI.SRIVASTAVA U K, SHENOY G V, SHARMA S C, Quantitative Techniques forManagerial Decisions , NEW AGE INTERNATIONAL PUBLISHERS.DALE H. BESTERFIELD, CAROL BESTERFIELD-MICHNA, GLEN H.BESTERFIELD, MARY BESTERFIELD-SCARE , Total Quality Management , “Success can never lower its standard, you have toPERSON EDUCATION raise your standard to achieve it.”M. S. MAHAJAN , Statistical Quality Control , DHANPAT RAI & CO. (P) LTD I wish for your dreams to be succeed.Ref. notes : All the Best !My Six – Sigma Green Belt training.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 39www.shakehandwithlife.blogspot.com
    • ContactMob:E-mail :shakehandwithlife@gmail.comsharma_ndr21@rediffmail.comWeb:www.shakehandwithlife.blogspot.comhttp://www.facebook.com/profile.php?id=100000837976546http://www.linkedin.com/profile/view?id=68376470&trk=tab_pro Call me for telephonic support at 7:00PM to 9:00PM. I request for your seriousness and honesty. Plz have patience if you hear “your call is on wait”. Organization , Institutions and Students are requested to first contact by mail or phone to organize the classes at their premises.Shakehand with Life “Promoting quality culture in every sphere of human life.” Page 40www.shakehandwithlife.blogspot.com