Math Exploration

TED ANKARA COLLEGE FOUNDATION HIGH SCHOOL

MATH EXPLORATION

Determining the Problem by Pareto Analysis and Finding Solution by

Normal Distribution Statistics

Instructor : Derya Çelik ErgevStudent : Umay AtayIB Diploma Number:

001129 - 0015

Umay Atay, 1129 - 0015

Introduction

I’ve dreamed about founding my own company or administer a well-known international company since the freshman year of high school, the year I started to think about my future career. As a first step through the path to this career, I decided to study IB Diploma Programme to gain an international vision, at the same time I do researches about most used methods and steps used by companies from innovation or design of a product to distribution and marketing.

As a Math Exploration subject, I chose to combine my interests on these subjects with the statistics, the subject I enjoyed the most in Math classes. I will use Pareto Analysis (ABC) method for determining the problem of a model company and compare the other companies the model one will work to minimize their problem by Normal Distribution Statistics.

What Is Pareto Analysis? 1

Pareto Analysis is a statistical technique in decision-making used for the selection of a limited number of tasks that produce significant overall effect. It uses the Pareto Principle (also known as the 80/20 rule) the idea that by doing 20% of the work you can generate 80% of the benefit of doing the entire job. In terms of quality improvement, a large majority of problems (80%) are produced by a few key causes (20%). This is also known as the vital few and the trivial many.

In the late 1940s quality management guru, Joseph M. Juran, suggested the principle and named it after Italian economist Vilfredo Pareto, who observed that 80% of income in Italy went to 20% of the population. Pareto later carried out surveys on a number of other countries and found to his surprise that a similar distribution applied.

The 80/20 rule can be applied to almost anything:

• 80% of customer complaints arise from 20% of your products and services.

• 80% of delays in the schedule result from 20% of the possible causes of the delays.

• 20% of your products and services account for 80% of your profit.

• 20% of your sales-force produces 80% of your company revenues.

• 20% of a systems defects cause 80% of its problems.

1 http://www.projectsmart.co.uk/pareto-analysis-step-by-step.php

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The 1st Problem : Determining the Prior Problem

I pretend as I am manager of a construction company which buys wooden blocks from other companies. To maintain the problems that can effect my business and benefit, I decided to use Pareto Analysis with the collected data of complains received and errors reported by the purchasing department of my model business about the wooden blocks bought in last three years.

781 complains and errors are collected and categorized under ten subheadings , their cumulative count and cumulative percentage in the total number of complains& errors is shown below.

Complains & Errors Reported in The Model CompanyError/Complain

(Cause)Count Cumulative

CountCumulative %

Difference btw. Lenghts

217 217 27,8

Storage 169 386 49,4Packaging 89 475 60,8Logistics 77 552 70,7

Availibility 65 617 79,0Deficient

Description57 674 86,3

Endurance of Sticks

46 720 92,2

Difference btw. Diameter

28 748 95,8

Natural Patterns of Wood

19 767 98,2

Security Warnings

14 781 100,0

Table 1.1

Then, a bar graph that shows the number of complains for each subheading and the cumulative line graph to determine the factors that forms 80% of the complains because according to the Pareto Analysis Rule ‘20% of a systems defects cause 80% of its problems ’ . According to this rule, by handling with the problems that forms that 80% at cumulative graph, I can increase my profit and decrease the rate of problems and complains efficiently.

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Graph 1.1: The bar graph shows the number of complains & errors reported under that subheading and the line graph represents the cumulative frequency of the complains & errors for ten subheadings.

As it can seen in the graph, difference between the length of bought wooden blocks is the major complain observed with the number of 217 in total, 781 complains & errors. The difference between length of wooden blocks is so effective because it cause further problems while cutting off these blocks by using automatic machines, that uses the standard measurements. Storage is the second major problem with the number of 169 errors with 49.4% cumulatively. Packaging, logistics and availability follows these two factors with decreasing number of complains and errors.

Difference between the lengths of wooden blocks, storage, packaging, logistics and availability are the subheadings that forms the 80% of the complains & errors data. Probably, these subheadings has a chain relation between themselves, means every problem at a stage enhance the other one. For example; the difference between the length of wooden blocks may cause storage and packaging error. These problems can be solved or decreased by working with another company as wooden block supplier, that supplies better logistics, availability and storage options and most importantly grater precision and accuracy of length of wooden blocks at the requested length 50.0000 m.

Three companies A, B and C applied as a supplier of wooden blocks to my model company with the equal prices and the length of 40 sample from their products are measured to chose the right one as the supplier for the next three years. Chosing the right company to work with is evaluated as second problem of this exploration.

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The 2nd Problem : Determining the Appropriate Company as Wooden Block Supplier

The table below shows the data that gathered from the samples of three different companies, the length of 40 samples of wooden blocks in meter with ten significant figure. Since the length of wooden blocks is an important factor that effect the other processes and the company’s benefit, measurements were made by sensitive measurement devices that are able to measure with ten digit numbers.

Company A Company B Company C49,8579924 49,87586743 50,04447664

50,05195817 50,18380579 49,6960852450,05875148 49,86370752 49,7878498649,72992004 50,1902737 50,2493027949,66896614 50,50988357 49,5486955250,00013981 50,35364809 49,4788586450,27859743 50,38873286 50,2698946449,88792635 50,36371878 50,1766777550,02340585 49,81216255 49,9301310349,72852971 49,89650929 50,2418320749,99287657 49,91714194 50,1956852850,36013746 49,65077528 50,2375363450,17229756 49,98998375 49,5491525449,98080032 49,91294515 49,9883381149,91876711 50,05331452 50,0539398650,05508837 49,59871096 49,678192150,14055704 50,32680127 49,7274253650,17660806 49,94165734 49,9265953749,65467522 49,57576118 50,6796446849,90736271 49,67663743 50,0502500849,83853459 49,51647037 49,841614950,06173578 50,23731205 49,9590252649,93978895 49,81754869 49,7370854950,10368726 49,4888342 50,096231749,72994489 50,22951902 50,40196349,98323915 49,97532471 50,5345825949,91719209 50,07505144 50,2362617549,97978274 49,6673007 50,0877462650,13685388 50,27796321 50,3094770949,9939808 49,89727811 50,03278902

50,32435799 49,96588648 50,709036449,92409655 50,07053257 49,9696083749,8277103 49,93772392 50,28204073

49,82646064 50,00346028 50,2370511450,1662845 50,17932795 50,16971369

50,03852202 49,96916068 49,7327641949,84193215 50,37848134 49,84722484

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49,94093325 49,96692737 50,2005406249,91799454 50,25581929 50,0968843849,78548374 50,12425165 50,57793986

The mean and standard deviation of the length of samples from each company is calculated by Excel with the same significant figure to conserve sensivity of the measurement devices.

Company A Company B Company C

Mean 49,97309684 50,00290531 50,06425363Standard Deviation

0,169247993 0,259372416 0,299024243

Table 2.2: This table shows mean length of 40 samples of wooden blocks in meter and standard deviation of 40 samples for the company A, B & C with

rounding to ten significant figures as the sample length values.

These results gives a clue about which company is more appropriate for model business, low standard deviation and high accuracy to the length 50.0000 m are the factors needed to determine the next supplier.

1. Since the requested length for wooden blocks is 50.0000 m, Company B is closer to that value with the mean length. 50,00290531 m

2. But low standard deviation is requested to minimize differences and errors and Company A has the lowest standard deviation with the value 0,169247993.

The decision cannot be made by only using standard deviation and mean calculations because different companies have the accurate mean length and lowest standard deviation. At that point, Normal Distribution graphs can be used for selection. For each company

Normal Distribution graphs are drawn according to the cumulative distribution function by the mean and standard deviation of the data set.

The normal distribution function gives the probability that a standard normal variate assumes a value in the interval ,

where erf is a function sometimes called the error function. Neither nor erf can

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be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically or otherwise approximated.

The normal distribution is the limiting case of a discrete binomial distribution

as the sample size becomes large, in which case is normal with mean and variance

with .

The distribution is properly normalized since

The cumulative distribution function, which gives the probability that a variate will assume a value , is then the integral of the normal distribution,

where erf is the so-called error function. 2

Normal Cumulative Distribution Function values of sample products according to their mean and standard deviation values, are calculated by Excel and normal distribution graphs are drawn separately for the company A, B & C. Tables of the calculated normal cumulative distribution functions of each company is shown separately with their normal distribution curves below, at the next six pages

2 http://mathworld.wolfram.com/NormalDistribution.html Retrieved Date : 25.12.2014

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Length of Samples (m) of Company A Normal Cumulative Distribution

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49,65467522 0,40157392949,66896614 0,46903962249,72852971 0,82978157249,72992004 0,8396618849,72994489 0,83983898749,78548374 1,27513838249,82646064 1,61951827949,8277103 1,629867403

49,83853459 1,71839716149,84193215 1,7456916549,8579924 1,87046744

49,88792635 2,07680804649,90736271 2,1859021449,91719209 2,23200115149,91799454 2,23547431349,91876711 2,23877567749,92409655 2,26039942649,93978895 2,31193947749,94093325 2,31496480849,97978274 2,35530783549,98080032 2,35470592749,98323915 2,35291773849,99287657 2,34110392249,9939808 2,339269751

50,00013981 2,32724775850,02340585 2,25527695350,03852202 2,18744843250,05195817 2,11466472750,05508837 2,09616102350,05875148 2,0738116450,06173578 2,05506832250,10368726 1,75027744950,13685388 1,47604826750,14055704 1,44478254650,1662845 1,228753786

50,17229756 1,17917596850,17660806 1,14398265450,27859743 0,46225148750,32435799 0,2735587650,36013746 0,1725041

Table 2.3: In this table, normal cumulative distribution function values calculated according to the mean 49,97309684 m and standard deviation 0,169247993 are shown for Company A.

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Company A

Graph 2.1: This is the normal distribution curve of the products of company A according to the Normal Cumulative Distribution Function(shown in table 2.2) by Excel.

The curve shows that the length of samples that Company A supplies forms a normal distribution. In the curve, the vertex is positioned at a point between 49.9 m and 50.0 m, at the mean length of Company A’s samples, 49.97. The normal cumulative distribution function of samples varies between 0,4 and 2,4 The end points of the curve samples limited are at 0,4 and 0,7 cumulative function calculations.

Length of Samples (m) of Company B Normal Cumulative Distribution49,4888342 0,215763649

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49,51647037 0,2649887649,57576118 0,39633938149,59871096 0,45672055449,65077528 0,61200270349,6673007 0,665943864

49,67663743 0,69724360749,81216255 1,17368039349,81754869 1,1914847449,86370752 1,33181631849,87586743 1,36425018249,89650929 1,41399284149,89727811 1,41570695949,91294515 1,44831901749,91714194 1,45627922649,93772392 1,49029621949,94165734 1,49581464249,96588648 1,52251968449,96692737 1,52337971649,96916068 1,52514373949,97532471 1,52943462849,98998375 1,53619850350,00346028 1,53810249850,05331452 1,50932978550,07053257 1,48670256950,07505144 1,47973980550,12425165 1,37865978250,17932795 1,22045320650,18380579 1,20602562550,1902737 1,184862984

50,22951902 1,05008995250,23731205 1,02242117650,25581929 0,9561332750,27796321 0,87655871850,32680127 0,70528103150,35364809 0,61645387350,36371878 0,58448129150,37848134 0,5391147350,38873286 0,50872895650,50988357 0,227695488

Table 2.3: This table shows the normal cumulative distribution function calculations of the length of products by Company B, according to the mean length 50,00290531 m and

standard deviation 0,259372416.

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Company B

Graph 2.2: The normal distribution curve is drawn by the calculated values of the normal cumulative distribution function (shown in table 2.3) of samples from Company B, by Excel.

According to the curve, the length of samples taken from Company B, has a normal distribution with vertex at the point 50.0 m. The cumulative function varies between the values 0,2 and 1,5. Both two sample limited end of the curve is at the point, 0.2 which shows that the curve is symmetrical.

Length of Samples (m) of Company C Normal Cumulative Distribution

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49,47885864 0,19632748949,54869552 0,30178869649,54915254 0,30258464149,6781921 0,579760647

49,69608524 0,62520566349,72742536 0,70742475449,73276419 0,72168103649,73708549 0,73325914149,78784986 0,87029637349,8416149 1,011176254

49,84722484 1,02521940649,92659537 1,20000691849,93013103 1,20647235249,95902526 1,25404369749,96960837 1,26896509549,98833811 1,29183700550,03278902 1,32678144350,04447664 1,33123218350,05025008 1,3326847950,05393986 1,3333536150,08774626 1,33003588650,0962317 1,326539756

50,09688438 1,32622699150,16971369 1,25370135150,17667775 1,2431088450,19568528 1,21130284150,20054062 1,20253019850,23626175 1,13071083950,23705114 1,1289911650,23753634 1,12793155750,24183207 1,1184651950,24930279 1,10164940150,26989464 1,05318793650,28204073 1,02333071550,30947709 0,953159843

50,401963 0,70507739950,53458259 0,38725097250,57793986 0,30505745850,67964468 0,16051236250,7090364 0,130484624

Table 2.4 : In this table, the calculated normal cumulative distribution function variables of the samples taken from Company C with the mean length 50,06425363 and standard deviation 0,299024243, is shown.

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Company C

Graph 2.3 : The graph shows the normal distribution curve of the calculated cumulative function of samples taken from Company C., according to the normal the mean length 50,06425363 and standard deviation 0,299024243

The graph shows a normal distribution curve, which means the length of wooden blocks produced by Company C has normal distribution. The vertex of the graph, is at a point between 50.0 and 50,2, which is equal to the mean length. The curve varies with the y-values 0.2 and 1.3. The sample limited ends of the curve at 0.2 and 0.1 function values.

Evaluation of Curve Graphs

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The graphs of the three companies supplied a normal distribution curve with mean 49.97 m, 50.00 m and 50.06 m (rounded to 4 significant figures) for company A, B & C respectively. The vertexes of the curves are positioned on the mean lengths of 40 samples for each company, as it should be in a normal distribution curve. Besides, the normal cumulative calculations, which determines the y-axis values of the curves, varies between different values for each company that are written in the explanation below the curves. Company B showed a symmetrical curve with equal normal cumulative values at two ends of the limited curve but Company A & C gave non- symmetrical curves according to the their arithmetical means.

Conclusion

In this Exploration, I assumed myself as a manager of Construction Company, which supplies its need of wooden block from other companies. First of all, I evaluated the complains and errors reported in last three years to determine the problem which has priority. Pareto Analysis is used at that stage with bar graph shows the number of complains & errors under ten subheadings and cumulative line graph is applied on the bar graph to determine the subheadings forms the cumulative 80% of the problems. It showed that difference between the lengths of wooden blocks, storage, packaging, logistics and availability are the main subheadings leads to the problems. I decided that these factors has a positive feed-back on each other so starting with the problem, difference between the length of wooden blocks would also decrease errors reported at other subheadings. Moreover, working with another company would also leads to the changes at each subheading.

My second problem was determining the most appropriate company to work with as a decision directed by the results of problem one. There were measurements of lengths of 40 wooden block samples from each company. I started with mean and standard deviation calculations, the appropriate company would have the lowest standard deviation and the most accurate mean length to my requested length of wooden block, 50.0000 m. However, results showed me that the company has the lowest standard deviation and the most accurate mean length, is different so these calculations weren’t enough to make a clear decision. I calculated the normal cumulative distribution function values of the data set of shows the length of 40 samples belongs to these three companies to draw their normal distribution curves by Excel. Since Company A has higher function valuees to produce wooden blocks with closer lengths to the requested lengths, 50.0000 m, working with Company A will be more efficient for my model company.

On the other hand, analyzing the logistics, packaging and storage facilities of Company A should be considered while making a certain decision, but evaluating all these factors according to the same variable with mathematics is extended with my knowledge. Observations and opportunities of the installations of Company A would also effect the decision.

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As a further study of this exploration, the comparison between Pareto Analysis done in this exploration and the prepared one according to the data collected after three years period working with Company A can be made.

Applications In Real Life

This exploration shows that how mathematics and most importantly statistics can lead economy and companies almost at every stage of business. It indicates that the numbers can be misleading even they were calculated accurately, so graphs and curves also should be considered during the evaluation process. There are many analyzing method and theories beside the ones used in this exploration that designed for different processes. While deciding which method will be followed for the calculations and simulations, the aim, the process and the field of the study should be stated clearly.

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