Design of Experiments | DOE | Ali Suleiman | Skillshare

# Design of Experiments | DOE

#### Ali Suleiman, Mechanical Design Engineer

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• ### 11. Sec04 Lec02 Analyze by ANOVA

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## About This Class

Design of Experiments ( DOE ) is a statistical tool which helps you to design any experiment properly toward right conclusions. In this beginner online course, you learn by examples and you will know first what is design of experiment and the aim behind it, then you will go deeper thus learning how to plan, execute and analyze any experiment properly using this powerful tool. You will also encounter both types of factorial designs here ( Full Factorial Design and Fractional Factorial Design). This tutorial will allow any newbie fully learn how to plan, execute and analyze any experiment properly, thus  making the right conclusions out it.

This tool is obligatory for any scientist or engineer, especially those working within the research and development sector and is also essential for those who practice and work with six sigma.

The course starts preparing you in PART I, thus making you familiar with the DOE definition, noise factors, relations and levels .

Part II then teaches you how to set all the relevant parameters so that you can then plan your experiment using the full or fractional factorial designs.

In Part III, you will learn how to execute your experiment afterwards in a way that you ensure highest accuracy in the results.

In part IV, you will then learn how to analyze the acquired results in your experiments using two different ways, either by charts, or by analysis of variance (Two-way ANOVA).

## Meet Your Teacher

#### Ali Suleiman

Mechanical Design Engineer

Teacher

I'm a mechanical design engineer working in research and development within the automotive industry. Through my career I have built a lot of knowledge in regard to product design and development. My passion toward all the tools and knowledge in my career has driven me to decide teaching. I will be providing a lot of beneficial and practical courses focusing on all skills related to design engineering.

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1. Sec01 Lec00 Introduction: will come from a course design of experiments. This course is brought to you by me, Alice Raman having a bachelor degree in mechanical engineering along with a master degree in mechatronics systems holding a six Sigma green belt certificates and a total of five years of experience in research and development within the automotive industry, all human innovations have become reality on Lee after they were proven through experiments . This course will help you to learn how to plan, execute and analyze any experiment properly towered, making right conclusions out of it. The course is ultimately for beginners. What I mean is that no prerequisites are needed from your side as part one in the scores would be taking care of that. In part two, you are going to learn how to set all the needed pedometers along with planning the experiment by either full or fractional factorial design. In the third part, we're going to know how experiments are then executed correctly to ensure highest accuracy possible. The last part will then introduced to professional ways for how to analyze your results and make the right conclusions out of them. 2. Sec01 Lec01 Definition: an experiment is a scientific procedure which is carried out to prove a hypothesis or discover something new. Engineers and scientists mainly perform experiments in order to bring more innovations. Do our life. Having that said design of experiments, also known as Do We is like a toolbox, which includes is set off techniques and instructions that will help us to plan, execute on, analyze any experiment correctly, thus making right conclusions out of it going deeper. In the definition, experiments are usually performed on a group of variables. One of these variables represents the point off interest for us. Thus, we call it output when we call other variables as factors in this case, when we perform any experiment, we are mainly trying to understand the effect off these factors on the output. I know that there are two types off effect which will exist in the experiment. The first is the effect off each factor on the output. This is called individual effect. The second type is the effect of combination between two or more factors on the same output . This is called effect off interaction. Now let's take a simple example in order to make the idea more clear for you. Assume that we have a farmer who would like to choose the best fertilizer for his garden off cucumbers. For that reason, the farmer has decided to run an experiment to fulfill his goal. This farmer is interested in having more produced cucumbers at the end of the season, so he chooses the amount off produced cucumbers as the output valuable. Let's say that he has three types off fertilizers to use in this experiment A, B and C. In this regard, these three fertilizers represent the factors in the experiment performing the experiment. The farmer has applied a single fertilizer along each of the rules cucumber plants. This will allow him to test the individual effect off each of these fertilizers over the specified output. In addition, he applied different combinations out of these fertilizers on other roles as well. This will allow him to test the effect off interaction among these fertilizers over the same up at the end of the season, the farmer accounts the amount of cucumbers which were produced by each of these rolls. Having this information, the former can't and analyze the results and conclude the best combination off fertilizers that he's going to use from now on in later seasons in a simple manner, the farmer has used design of experiments, toe plan and execute his experiment properly so he can grow right conclusions out of it, Thus helping came to make correct decisions for his business. 3. Sec01 Lec02 Relations: we have mentioned previously that engineers and scientists perform experiments in order to understand the effect off certain factors on the defined output. Looking deeper, this effect comes in a shape off mathematical relation that exists between each factor and the output. The relation is present in a way that when the factor variable changes its value, the output variable is expected to change its value as well according to this relation. Having that said, there can be three possible options for the way this output variable will behave according to the change in the fact that valuable first is that when the factor variable changes its value, the output variable does not change in value. This implies that there is no relation that exists between the factor and the output. Second is that when the factor valuable increases in value, the output variable increases in value as well. This is called proportional relation. Third is that when the factor variable increases in value, the output variable decreases in value and vice versa. This is called inversely proportional relation. As we were saying that these relations are actually mathematical, let's now represent each of them in both graphical and empirical way for that purpose. Let's designate the factor variable with X and the output valuable with Why emphatically? Because we mentioned that the output Why changes for the value off factor X. This means that why can be represented as a function of X when there is no relation, We would expect that the line in the graph will be horizontal. In this way, when the factor value changes, the output value is staying in the same level emphatically. The output being represented as function of X is equal to a constant. When the relation is proportional, it can be graphically represented as an inclined line with postive slope emphatically here . Why being the function off acts is a first order equation, another meaning align equation with postive slope when the relation is negatively proportional. It is also graphically represented as an inclined line, but with a negative slope emphatically Why Being the function off X is also a first order equation or line equation, but with a negative slope, as we have just presented the effect off a factor over the defined output by a leaner equation. Let's now show how the graphical and empirical illustrations will also show the amplitude of this effect, assume that we have three factors where factor one has no effect. Factor to has a small effect on the factor. Three has a higher effect over the boat. In this case, assuming a proportional relation factor, one will be represented as a horizontal line than factor to will be represented by an inclined line with a small slope than finally. Factor. Three will be represented by an inclined line with a higher slope. This can be applied as well, in the case of inversely proportional relations with just having negative slopes instead off postive ones. 4. Sec01 Lec03 Error: we mentioned previously that we perform experiments in order to find the relation between factors and the output. Assume having a factor. Acts and output y where the relation in reality between them is the one that you see here. But you don't know it when you do the experiment. You will not be able to get exactly this relation unless your experiment is extremely perfect. Well, practically, this is not possible. Having that said, When you perform the experiment, you will get a relationship at an officer distance from the real world. This officer distances called error. For that reason, when performing an experiment, we need to minimize the error as much as possible so that the relation we get is as close as possible to the real one. In other meaning, error need to be minimized so that our experiment can be more accurate for us to be able to minimize the error margin. We need to understand where this error is coming from. Mainly, the error comes from two sources in the experiment noise factors on the measurement device . Let's speak first about the noise factors has mentioned before. When we do an experiment, we are mainly changing the values off the X factors. Then we see and record how the output changes accordingly in other meaning. During the experiment, we actually have a control over the X factor's. However, at the same time there will be other factors which are changing their values. However, we have no control over them. This is because either it is hard to control them or at least they are out of the scope. In our experiment. These are called noise factors on. We designate them with Rosette to understand the idea better in this regard. Let's recall the same example of the farmer and his garden off cucumbers. The farmer wanted to understand the effect off each of the three fertilizers offer the amount of produced cucumbers. In this case, we have three X factors which the farmer can control during the experiment. However, there will be other factors during the experiment, which can also affect the amount of produced cucumbers. For example, the amount of water given in each line is a factor as well as this factor is not within the scope of the experiment, then it is considered as in noise factor. Another example is the genetics of these plants as if you apply exactly the same conditions onto cucumber plans, they may still produce different amounts of cucumbers. This is due to the fact that they are genetically different. To be able to minimize the error coming from noise factors, we simply need to control them. This can be done by fixing their value throughout experiment. Having that said, if we need to minimize the error coming from the amount off water, we need to keep exactly the same amount of falter given among all the lines during the experiment Coming to the second mentioned factor related to the genetic difference between the plans, we may choose the seeds initially to be from the same mother plant. In this way, we can be sure that the genetic difference among the plans in all the lines is as low as possible, fast minimizing the error coming from the side. Now we come for the second source off error related to the measurement devices. We mainly need these devices to measure the output and possibly the factors as well during the experiment. For example, if the output a new experiment is the temperature, then you need a 30 meter to mother This output. We all know that these measurement devices cannot be perfectly manufacturer another meaning that is always a marginal error accompanied with their measurement. Simply to minimize the error coming from these devices, you need to make sure to choose a measuring device with higher accuracy on the other side. When we used these devices in the lab, their accuracy will degrade over time. Accordingly. Before doing the experiment, scientists tend to make a check over these measuring devices using two methods. The first method, in case the device can be recalibrated, is to make sure that the periodic calibration off this device is being done on that will be by checking the last calibration report. The second method is to do what a so called measurement system analysis also called M s A. This can check the device for its accuracy for different attributes. No, that calibration methods and M. S A are not within the scope of this course, as these are big topics and they need separate course for addressed. Um 5. Sec01 Lec04 Levels: as we mentioned before when performing an experiment, we are mainly trying to find the relation between a factor acts and the output y. In order to find this relation, we need to change the value of factor acts from one level to another. Then we record the output value we got at each of these levels. Then we plugged the factor levels along with the output ones so that we can obtain this relation. Having that said, you are totally free to choose how many levels you would like to include for each X factor in the experiment. Just keep in mind that you must have at least two levels for each factor so that you can find its relation with the output, which in this case will be Alina relationship according to what has been mentioned. The number of factors and the number of the levels depend mainly on the goal behind your experiment. Know that it is not obligatory that you have the same number off levels for all factors in the experiment. Another meaning you may have a factor with two levels and other factor with three levels. It all depends on the specific case off your experiment and again on the gold behind it. Just keep in mind that when the number of factors or the number of the levels increase the coast and the testing time of the experiment will increase as well. Having that said, it is highly recommended that you choose the factors and their levels wisely so that you achieve your goal with the least coast and testing time possible for your experiment. 6. Sec02 Lec01 Set all Parameters: as we have learned about some basic and important topics in the previous section, we are now ready to begin learning design of experiments. We mentioned previously that any experiment should pass by three stages to be completed, landing the experiment than executing it, then, analyzing the results towered making right conclusions. In this section, we will focus on how to plan your experiment, where we will learn how to set all their living para meters than how to deploy the testing plan. Afterwards, we will first start by setting all the parameters, but before that, in order to enhance your learning experience throughout the course, let's introduce a real life problem on that will try to solve it. Using design of experiments. I assume there is an automotive tires company who is planning to release a new tire model which will help customers to save more fuel. The problem here is that the company need to make a correct choice for design, material and texture so that they ensure best performance of the tire. In regard to fuel economy of the car, the designers in this company have created two designs for the ultra body. Let's call them design A and design be as for the rubber material, the company to choose between two potential ones which are available in stock. Let's call the material A and material be. The designers in this company have also proposed to grooving textures for this tire texture a and tax, or be to solve the problem. The company has decided to conduct an experiment in order to figure out the best combination which serve their goal throughout the course. Step by step, we will help this company to conduct the experiment properly on may quite conclusions out of it. Using the techniques and procedures which will learn from design of experiments. Setting the pedometers will be done upon five main steps. The first step is to such a goal. I know that this is a straightforward but believe me, it is very important to write down your goal in a clear and specific manner from the very beginning, as all your activities later on will be serving the school in regard to the terrorist company. Their goal is to choose the right combination from design, material and texture in order to achieve the best tire model at saving fuel in the car after setting your goal, the next step will be to define a proper output which can always be extracted from the point off. Interest in your goal. No that are put valuable should be immeasurable centimeter in other meaning. The output variable that you choose should have unit off measurement. Each time you are choosing the output, make sure that these two criteria are fulfilled correctly. In our case, the point off interest is to save more fuel. Accordingly. The output perimeter can be the fuel consumption which can be measured in leaders. After we have defined the output, we can go to the next step where we need to define the equipment and procedure. For this experiment mainly in this step, we need to do a full description of the experiment in our case and ordered. That's the fuel consumption. We may use a car which is equipped with a tire model that we choose on. That car will be driven for 100 kilometers as a measuring device for the output. Assume having a built in digital fuel gauge the car panel on that. We will use it to know how much leaders are consumed, measuring the output will be simply by subtracting the final amount of fuel in the car from the initial. As we have set a full description for the experiment, it is time to go to the next step where we need to list all the X factors in this experiment. In our case, the company is mainly studying the effect off design, material and texture on the output being the fuel consumption. Having that said, we have three X factors which are as false. The first factor is the design off the Alfa body being with two levels designed A and design be. The second factor is the rubber material with two levels as well, which are material A and material be. The third factor is the texture design, with two levels as well. Dexter A and texture be The last step is to list all the possible noise factors which can impact the output results in our experiment. Yet they are not within the scope of the experiment. In our case, there are a lot off voice factors, so let's consider some of them only in order not to have an overwhelming list. The first nice factor can be the car model itself as there are car models that consume fuel differently. The second moist factor can be the driving person as the way you drive effects the fuel consumption as well. The third noise factor can be the ruled as driving on urban way or highway will also have an impact on the fuel consumption. As these noise factors will mostly affect our results. Yet they are not within the scope of the experiment. We need to minimize their effect as much as possible so that we ensure minimal error in the outcomes. We will speak more about this topic later on in the execution stage off the experiment. Fair enough. We have now listed all that living very meters for the experiment on we already to start deploying the testing plan. 7. Sec02 Lec02 Full Factorial Design: as we have tested all the living better meters, we can now move to the second part within the planning stage where we deploy the testing plant. The first step would be choosing the design of the plan. This will be by either choosing the full factorial design in which we test all the possible combinations or choosing fractional factorial design, in which we select certain combinations to be tested. Why we exclude others. We will come to speak more and compare between these two designs later on in the next lecture. As for now, let's go with the full factorial design. The next step will be to simplify our work by setting some designations. In this step, we will first designate letters to the Factors Capital de for Design Capital M for material on capital T for texture. Second, we will designate the number off levels for each factor with illiterate as well. For design. The number off levels is designated with a small D and is equal to two for material. The number off levels is designated with small M and is also equal to two for texture. The number off levels is designated with a small tea and is equal to two as well. Step number three will be to identify the type of experiment that you have. The type of experiment is mainly identified by the number off levels for its factors. For example, if we have an experiment where all factors have to levels on Lee, then it is simply called a two levels factorial experiment. Similarly, if we have an experiment where all factors have three levels on Lee, then we simply call this experiment a three levels factorial experiment. Ansel on the other side. If we have an experiment where factors have different number off levels among thumb, then this type is called general factorial experiment. Having that all said coming to our example as each of the factors have to levels, only then the type of our experiment here will be at two levels. Factorial experiment. In the next step, we need to calculate the size of the experiment. The size off the experiment will mainly inform you from the beginning how many possible combinations you will have in the experiment. The formula for calculating the size depends on the type of experiment. Let's start with the General factorial experiments assume having two factors factor A with three levels on factory. Be with two levels on Lee. Let's designate the number off levels with small A and small bees. Excessively, the size of the experiment will be simply equal to the multiplication off the number off levels. In this case, it will be equal to six. If you have a two levels factorial experiment, then the size will be equal to two to the Power K, where K is equal to the number of factors in the experiment, the formula goes similarly. If we have three levels factorial experiment or more as the type off. Our experiment is a two levels factorial experiment. Then we calculate the size. According in our example, we have three factors, which means that the size will be equal to eight. Having the size equal to eight means that we have a possible combinations which can be taken from these three factors. In a step five, we need to set the number off replicates. We mentioned in the previous step that we have eight possible combinations. In order to test each of these combinations, we need to prepare at least one sample at this combination to represent it on that. We performed the testing on this sample. This sample is called a replicate to ensure higher accuracy and confidence in the outcomes off the experiment, practitioners usually test each combination at multiple replicates in this way. Instead of taking one value from one replicate, they take multiple values from multiple replicates than they calculate the average value. In this way, the practitioners will have a higher confidence in the outcome of this combination. For our experiment, let's say that we would like Toa have three replicates to be tested at each combination. This will lead us to stop number six, where we calculate the total number of friends in the experiment. The total number off runs is simply equal to the size multiplied by the number off replicates. In our case, it is equal to 24. If we are testing one sample each time, the total number off runs being equal to 24 means that we need to run the experiment 24 times in order to be completed. Assuming that each one will take one hour, then the total number off runs will tell you how much time you need to complete all your experiment in this case, the total duration off the experiment will be 24 hours. Now we come to the final step where we need to set a table where this table represents the final plan off the experiment before setting the table. Let's just do a small thing to simplify our work further for the two levels factorial experiments. It is a good practice to designate the lower level with minus one on the upper level with plus one, you will see the benefit of having minus one on plus one later on when we come to analyze the results in section four. So let's just start now. Building our table planning by full factorial design means that we need to take all the possible combinations from these three factors using the size of the experiment. We have a possible combinations, which means that our people will have a trolls in the first column. We need to fill the combinations inside, not to consider the order. The possible combinations from the three factors in our example will be as false. One. The M T D M ditty empty Andi empty. The one here means that none off the factors is included on. We mainly use it as a reference. It's small note here is that practitioners make all combinations as configurations or treatments. Just keep in mind that these terms just mean the same thing as we have three factors in the experiment. Then we add three columns to the right. What each calm refers to. One of these factors. As we have said, the number off replicates to be equal to three. Then we also add three columns to the right. Let's now fill the columns off factors with the levels that we have being represented as minus one plus one for combination. One. All factors should be at the lower level, which is minus one for combination. The the design column goes to the upper level being at plus one, while others stay at minus one for combination M, The material column goes toe plus one, while others stay at minus one. For combination T, The texture column goes to plus one, while others stay at minus one. For combination. D m. Both design and material goto plus one, while texture stays at minus one for combination. DT design and texture goto plus one. While material stays at minus one for combination. Empty material and texture will be a plus one, while design stays at minus one for the last combination, the empty all factors will be a plus one. As for the last three columns, who will keep them empty for now as we will fill them after we record results within the execution stage of the experiment? By having this table, we have successfully deployed the testing plan using the full factorial design on that we are now ready to execute the experiment. 8. Sec02 Lec03 Fractional Factorial Design: as the full factorial design allows practitioners to test all possible combinations. The question here is that why should practitioners bother to use a different design? Well, the answer to this question is simply to reduce coast and testing time. If we recall the size formula for the two levels factorial experiment, we will see that as the number of factors increase, the size of the experiment will be increasing exponentially. This means that coasts and testing time off the experiment will be increasing exponentially as well as a bottom line. In situations, when you are limited with the budget or bounded in a tight time frame, you may need to deploy your testing plan using the fractional factorial design. So how the fractional factorial design would reduce the coast and testing time with our experiment, we know that in full factorial design, all combinations will be tested. This allows us to get full information from the experiment on build our conclusions accordingly. The fractional factorial design will allow us to exclude some of these combinations why we still maintain an acceptable percentage off the information that we need, thus making right conclusions as well. The executed combinations are believed to be the least important ones. The question that pops up here is that how can we identify which combinations are relatively less important? Historically by experience, practitioners came to the conclusion that combinations which include three factors or more , have a very low effect in the output when compared to other combinations. Having this conclusion, it is seen that in most cases it is not Worf peeper cost and time to test these combinations. Just an additional note here is that when we have a combination that has pull factors inside, we can call it as his second order combination. Similarly, if we have a combination which includes three factors, we call it 1/3 order combination on and so on. Recalling the size formula for the two levels factorial experiment. The key here will also reflect the maximum order which we have among the combinations in other meaning. If we come to the size of our experiment which is two to the power three, this three means that our experiment includes first order, second, older and third order combinations. So how much the fractional factorial design will reduce from the combinations for the two levels? Factorial experiment The size formula when using the fractional factorial design is two to the power K minus end and here is called the minimizing factor and it can be any positive integer in case off an equal one the size off. The experiment will decrease to 1/2 in cases unequal to to the size of the experiment will decrease the quarter on DSO. It is important to be mentioned that on maybe reflects the decrease by order in other meaning. Using our example where the sizes, too to the power three If we set end to be equal toe one, this means that we degrees the order in our experiment from third to second order. This eventually means that the third order combination D mt. Will be excluded in the experiment. Let's come now to deploy the final table using the fractional factorial design. The first step will be to define the minimizing factor and let's say that we choose and to be equal to one. This means that our experiment will decrease by half, first going from eight combinations before combinations on Lee. The second step is to set the Combinations column in the same way we did for the full factorial design next to it. We need to add the identity column designated with I and fill all its fields with plus one . After that, we need to add more columns to the right there, the combinations that we have. So if we exclude the difference combination one, then we have still seven combinations. Accordingly, we add seven columns for them. Know that the combinations should be listed from left to right in an increasing order. The next step will be to fill the empty fields for the first order combinations. We just fill them in the same way we have done for the full factorial design for the second order combinations. We will fill them by multiplying the individual factors inside. For example, for calm the M. We multiply the values in column D by column. Damn, then we fill the results accordingly. Similarly, for duty and empty for the third order combination, it is also filled by multiplying the individual factors inside another meaning for the combination called GMT, we won't apply the values in column B by those in column am by those in column T Then we feel the results accordingly. Now the table is filled. We mentioned previously that by choosing an equal toe one, we will decrease the combinations under testing from 8 to 4 combinations. But we still see all the combinations here in the table. That's because our work is not finished yet. If you look at that the anti column, you will see that it has four fields with a plus one on four fields with minus one. Let's rearrange the rows in the table in a way that we create to smaller tables. Table one has only plus one within the DNT column, while table toe has only minus one. There. In this way, you need to choose one off these two tables so that it's combinations are the ones to be tested in the experiment. You are totally free to choose which table you would like to have for the experiment. When choosing one off these two tables, you will see that the Empty will not be changing along all the roles, so it will be either all at minus one or all at plus one. As the empty will not be changing. This means that we are not studying its effect on the output anymore. In other meaning When choosing any off these two tables, the empty would be practically excluded from the experiment, mainly by choosing one off the too small tables, you are totally ready to execute the experiment. It is only left to show you how we will be studying the effects in this case. First of all, if you look at Table one, where the empty has only plus one in its fields, you will notice that identity column has also plus one in all its fields. This means that I is equal to the empty. If we look at table two in a similar manner, we will conclude that I hear is equal to minus the empty. These are called generators. Let's say that you have chosen table one for our experiment. In this case we have I is equal to the empty. You will notice that if we multiply any column by itself, it will give us only plus one. As a result, this means that I, which has only plus one in its fields, can be equal to double D or double M or double T as I is equal toe the empty as well, then double D or double M or double T are also equal to the empty. If we accept with the same factor from both sides of the equation will see that D will be equal to empty and would be equal to D T and T would be equal to GMM. What this means is that when we are studying the effect off the we will be actually studying the effect off the and Antico mind when studying the effect off em, who will be actually studying the combined effect off em and DT combined and so on. 9. Sec03 Lec01 Execute the Experiment: As we have said, the plan for this experiment, let's now go into the execution phase. Keep in mind that this stage is very crucial as improper execution can waste all your efforts and make you repeat the experiment all over again from the beginning. Having that said in this face, two important aspects should be preserved. First, the set up off the experiment should be 100% fulfilling the dusting plan. Second is that the experiment should ensure high accuracy of the results. So let's start with the first step in this phase where we recall the preliminary experiments set up which we have spoken about previously. Mainly, the set up will be a person who will drive cars which are equipped with a defined combination separately for a distance off 100 kilometers win done, we record the fuel consumption using the gauge, which is built in the car. Before we move to the next step, ask yourself this question. Is the set up and procedure of the experiment fulfilling the distinct plan? If no, then any work should be done before moving on. If yes, then we can go to the next step where we need to ensure high accuracy of the results. We mentioned before that in order to ensure higher accuracy, the total error in the experiment should be minimized as much as possible. We also mentioned that the total error in the experiment comes from the noise factors on the measuring device. As for the measuring device, we said previously that we can minimize the error here by pro chasing a device with small tolerance in its measurement. In case the devices already I used one. Then we need to make a check over its periodic calibration report on also the latest gauge arm our report as well. We said previously that these two topics are out of the scope and need a separate course to address them. So for now, let's assume that the measuring device is already fulfilling our needs in regard to accuracy. We come then toe the noise factors which we have listed previously in this face. Our job is to take the proper measures which ensure that the effect of these boys factors is minimized as much as possible for, said one were using Different car models can manipulate our results. As a measure we can use the same car model throughout the experiment. In this way, there will be no different car models in the experiment, thus eliminating this nice factor for that to were different drivers can lead to a different fuel consumption due to the individual way off driving, We can simply take as a measure that the same person will be driving all the cars throughout. The experiment in this way will have no different drivers. First. Eliminating this mice factor as well For that three were driving on different roles will also lead to difference in fuel consumption. We can choose the road to be a highway with no obstacles and that all cars will be driven on. Lee on this highway in this way will have no different roads in the experiment. First eliminating this noise factor too. So far, so good. We have now insured that our experiment will fulfill the plan and will ensure high accuracy of the results. We are now ready to execute the experiment and start recording the results in the empty fields assumed that we would like to execute the experiment using the full factorial design as a reminder, we have specified three replicates for each combination. This means that we need toe one each combination three times on record its results in the religion fields. So for the first combination will do the following. We will equip the first car with this combination, then let the driver go and cover the agreed distance, then record the fuel consumption. When he's done, we will also let the driver repeat that two more times on record a fuel consumption. Each time the second and third replicates excessively, we will per firm the same procedure for each of the left combinations in the table. As we have recorded all the results that we need now we can go forward to the next phase where we analyze these results. 10. Sec04 Lec01 Analyze by Charts: after we have successfully executed our experiment. It is time now to analyze the choir. It results in order to make the right. Conclusions afterward in this section will highlight two different ways. To analyze the results off any experiment, either by charts or by what is so called on over. We will come to explain each of these two way separately through this section, so let's start with the analysis by charts. Here we are going to analyze three different aspects. First, identify the optimal output response. Second is to study the main effects off factors over the output and finally to study the effect off interactions. Among the factors over this output, identifying the optimal output response means to screen the output, results off all combinations and then mark the combination, which gives you the output that fits your needs. By having this information, you may then set your system to be always with this combination so that you guarantee that you will have the fabric output response all the time. Having that said, the required up response can be the maximum minimum or in between. It all depends on your goal behind experiment. Let's reflect this on our example. Mainly, our goal was to ensure lowest fuel consumption by entire model made from a combination of design, material and texture in other meaning. We would like to identify the combination, which ensures the minimum output response in that Accordance as first step will calculate the average values off replicates and Mitro. After that, we will build a history graham out of these average values by looking on the history Graham , we simply identify the combination, which has the lowest response, which in our case, is the M. By performing this simple analysis, we mainly solved the problem of the company, thus choosing the combination the M as a best fit for the project. Sometimes your goal might not be limited to identifying the combination, which ensures an optimal output response. What I mean is that the goal off her experiment might be to study deeply the effect off all these factors over the defined output. Let's start with studying the main effect off each factor on the help to study the main effect off design over the output. Who will need to build a graph plot on the horizontal axis? Who will list the two levels off design being minus one and plus one. As for the vertical axes, it will be for the output values at each of these two levels for level minus one, we will take the average value off combination one for Level Plus one. We will take the average value of combination D. Then we connect these two coordinates with a line. We will do the same for material and texture. As the graph plots already. Let's do some interpretation for them. By looking at the three graphs at once, we can see that texture has the highest lines. Loeb. That's having the biggest effect over the output among the three factors. The slope here is positive, thus showing that when changing texture from level minus one to level plus one, this will dramatically increase. The fuel consumption design then has a relatively lower slow, thus having a lower effect on the output If compared to texture, the slope here is negative, which means that when we change design from minus one toe plus one, this will help decrease the fuel consumption. Finally, for material, we see that it has the lowest slope, which means that the material has the lowest effect over the output. Among all factors. The slope here is also negative, meaning that if we change material from minus one toe plus one in this way will be decreasing the fuel consumption. Let's come now to the final aspect in our analysis by charts, which is to study the effect off interactions. Among the factors over this output, studying an interaction is mainly to compare the main effect off factor one over the output with the effect off its interaction with another factor. For example, studying the interaction between design and material can be by comparing the effect off design alone over the output, with the effect off design and material combined over it. This can be done by a graph that includes two plots at once. Let's make the idea more clarified by studying the interaction between design and material over the output. In this case, we will first prepare the graph where the vertical axis will be for the output response. On that the horizontal axis will be for design first listing its two levels minus one and plus one. As we said, we need tohave to plots. The first plot, which is 40 alone, will be made out of the combinations one and d. This plot is made while I am stays at minus one. The second plot, which is for interaction, will be taken from combinations M and G m. This plot is made while M is at plus one. By interpreting the graph, we can see that design, when changed from minus one toe plus one, is decreasing the fuel consumption. However, this degrees is enhanced mawr when design interacts with material, thus confirming that moving material from minus one plus one, along with design enhances our saving off fuel further. 11. Sec04 Lec02 Analyze by ANOVA: Now let's come to the second way for analyzing the results on over which stands for analysis of variance. This analysis tool can provide you with a package of variable and useful information at once and regard the results off your experiment by another. You would be able to know the follow the level of effect by factors and that interactions over the output also the level off marginal error in your experiment and also the level of confidence in your experiment. Results on over is often called two way and over the two way means that on over will be able to analyze only two factors at a time if taking design and material as an example, the two way and over will be analyzing the main effect off each factor, along with the effect off interaction between them. If you repeat a two way and over, among other factors as well, then you gather all the information together for analysis. We can call it as invested two way and over. In this lecture, we will perform a two way and over on design and material so that you can automatically know how performance on other factors as well as we are, analyzing design and material as a first step will grab the roles which are only related to them, thus taking the combinations one D M and G m. So that we form a sub table. As you can see here out of this table, we will build another one in the way that you see. Then we copy the average values to it in accordance the combination off levels between design and material. After that, we start to calculate the average is vertically and horizontally. For the first column, the average will be 4.85 For the 2nd 1 the average will be 4.7. For the first roll, the average will be 4.95 for the 2nd 1 it will be 4.6. As for the field in the corner, this will be the total average off all four values being 4.775 We call this table as the table off averages. Let's also recall the designations off levels which we have set previously in the scores where a small D is for the number off levels and design and is equal to two small M is for the number off levels and material on also equal to two. Then we have small end, which reflects the number off replicates and is equal to three by the bill table on those perimeters. We can now calculate the sum off squares for design designated with SD. The calculation will be done by simply following the shown formula were as as the will be equal to 0.3675 for the sum of squares for material being designated with S M. The calculation will be done by is similar formula as shown here, where the result will be 0.675 After we have done that, we will need to calculate the sum off squares for the total error being designated with S S E. This calculation will take a different form. First, we will recall the table off averages. Then we form a second table as show will also recall the main table, which includes all the combinations and replicates. By matching the combinations and levels, we will move their values in the replicates fields into the newly formed one. Now, I need your focus on what is going to be done next, we will mainly subtract the averages from those replicates. Take care that the subtraction is only happening between averages and replicates which are related to the same combination off levels. We will then square the outcomes, thus having the results which are listed here now simply the sum of squares for total letter is the sum off all these results which in this case is 0.8 Now we will come to calculate the total sum off scores which is designated by SST. The calculation here is done by first subtracting the total average from all the replicates individually. Then we square them as well. That's having the following results. The total sum off squares is then calculated by having the some off all these shown results , thus being 0.52 to 5. So far, we have calculated as as the S m as s e. And as S T. What is left to be calculated is the sum of squares for both design and material combined being designated with SDM. I know that the total sum of squares should be equal to all other mention some of scores. By using this equation, we can simply calculate as as the M by subtracting others from the total sum of squares. In this case, as as the M will be equal to 0.75 now, we're ready to build the final table off. Unova, as seen here, will simply fill the fields in the table in accordance to the shown formulas inside. For the some off Squares column, we will directly fill it with the results, which we have calculated previously. As for the second column related to the degree of freedom, we will fill the fields according to the shown formulas, and using the values off small D small M and small and the Mean Square column, as shown, is filled by dividing the some off squares over the degree off freedom individually in vitro. The F Score column is then calculated per the shown formulas, thus dividing the mean square off the M and D M separately, over the mean square off error coming to the coordinates column, we simply fill it, using the degrees off freedom as advised in each field. As for the last column related to the critical F value, it needs a bit more effort and explanation before we can feel it. So find the critical F values we need to look into certain standard of value distribution tables. You can find such tables easily on the Web or within the last pages off any book for statistics. These tables are available at different confidence levels. When looking for these tables, you will see that somewhere around the table they mentioned the value off Alfa. If Alfa is equal to 0.5 which is equal to 5% this means that the data in this table contributes to a confidence level off 95%. Similarly, if Alfa is equal to 0.1, which is equal to 10% this means that the data in this table contributes to a confidence level off 90%. Let's say it is required for us by the management to set a confidence level off 95% as our criteria. Accordingly, we grabbed a value distribution table from the Web which contributes to this confidence level. Then we use the values in the co ordinates column in order to identify the critical of value. For example, for design, we have coordinates off one and aid successively, This means that the critical of value is the one which intersects column one in the distribution table, along with Row eight in it. In our case, the critical value is 5.32 We prefer the same procedure for other Rosa's well and list the critical values for them accordingly. Having that said, we can have three possible scenarios. If the F score value is bigger than the critical of value, then we can say that we are more than 95% confident about our results in the experiment. If the F score value is equal to the critical value, then we can say that we are 95% confident about our results in the experiment. If the F score value is smaller than the critical of value, then we should say that we are less than 95% confident about our results in the experiment . Now the another table is ready on all. What we need is to do an interpretation for it, thus growing our conclusions first. By looking into the S s column, we can mainly check the effect. By each of these listed items, we can see that design has the has effect over the output on it is far more than the effect by other listed items. We can also see that the effect by material is much lower than that off design, but still relatively more than the effect by the interaction. The M, which is seen toe, have the lowest effect among old. If we look at the effect by error, we can see that it is not relatively small where the effect by error is seen to be more than that off material and interaction, d m. A. By the way, this is not a good sign regarding the accuracy of our results. As if the experiment is done properly, the effect by error should be relatively much smaller than that off any off. The other items now coming to analyze the F score values, we can say the following as the F score for design is far greater than the critical value. We can say that we are more than 95% confident about the results and conclusion in regard to the design factor in regard to material. We also see the same thus saying that we are more than 95% confident about the results and conclusion related to the material factor. As for the interaction de m, we see that the F score is lower than the critical value, thus saying that we are less than 95% confident about our results and conclusions and regard to this interaction.