ASQ Six Sigma Green Belt – Objectives – Hyperledger Part 5

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This we will be talking about number of distributions which are listed here. Normal distribution, binomial, poison, chi square, students, T and F distribution. A variable can be of two types, continuous or discrete. Continuous variable is a variable which can take any value. For example, if I talk about the height of students, height of student can be anything. Let’s say height of student could be 150 CM or 150. 1 CM or 150. 135 centimeters. This is a continuous variable.

Continuous variables include things such as measurement, the length, the height, the width, the volume, the time. These are continuous variables. And then there are discrete variables. Discrete variables are counting. When I say counting, the example of that is if I flip a coin ten times, how many times do I have head? I can have head three times, four times, five times, but I cannot have head 3. 52 times.

So this will be the whole number. This is a discrete variable. Depending upon the type of variable, you have different distributions. So the six distributions which we talked earlier fall into these two categories. In continuous we have normal chi square, students, D distribution and F distribution. And in discrete we have binomial and poison distribution. Let’s start our discussion with binomial distribution. To understand binomial distribution, let’s take an example that if I flip a coin twice, a coin has a head and a tail. So if I flip it twice, what is the probability of getting one head? And as we know that if I flip a coin once, the probability of getting head or tail is . 5.

This is something which I know. But now the question is if I flip it two times, what is the chance of getting one head? So for that, what I can do is I can list down all the possibilities. And my possibilities are that it could be two heads. The first one is the head and the second flip also gives me a head. So this is possibility number one. Possibility number two is I get head in the first toss and then in the second toss I get tail. So this is the second probability, or the second way things could happen. Third way is tail first and head 2nd. Fourth way is two times I get tail. So these are four possibilities.

And now my question is what is the probability of getting one head? So I have one head here in this case. And I have one head here in this case. In this case I have two heads in this case I have zero heads. So out of four possibilities, there are two possibilities in which I can get one head. So the probability here in this example would be two divided by four is equal to one by two or 0. 5.

This was the simple thing which we learned in probability. So here is the summary of the discussion which we had. The probability of getting zero heads is 00:25, which is here zero point 25. The probability of getting one head is zero five, which is here. And probability of getting two heads is 00:25, which is here. If you are scared of binomial distribution, then this is the binomial distribution you have already drawn. This is the simplest binomial distribution I could draw.

Now let’s make things a little bit more complicated. Instead of two flips. Now I flip this coin four times. And now I am asking you what is the probability of getting one head? Let’s do the same way what we did earlier. List down all the probabilities. So first time if I flip, it becomes head, second time head, third time head, fourth time head. So this is one possibility. The second possibility could be the first three are head and the tail. Next one is two heads first two heads, second to tail and so on. And now after this, this becomes complicated. I really cannot list down all the possibilities because it will take me a lot of time to make the list of all the possible ways things could happen. That’s where binomial distribution or the equations or the formulas which we use come handy. Here I have the binomial distribution which I have drawn using minitab. We will talk about minitab later. But this is what I have drawn. This gives me the probability of each of these successes.

So the probability of getting zero success. And when I say success here, success is the head I’m interested in. I’m interested in getting heads. So the probability of getting zero head is this much, which is something zero five more than that plus. And then the probability of getting one head is something around 00:25. And then the probability of getting two heads, probability of getting three. Four I have here I cannot have more than four heads if I flip the coin four times. So this is the binomial distribution which I drawn using minitap software. Now question comes how did I draw that? Can we do it manually? What’s the background of that?

To do that, let’s understand binomial distribution first. So when we say binomial distribution, binomial distribution has following properties. One is that experiment consists of N repeated trials and which was the case. In the case which we are studying here, we are talking about four trials. So we flipped the coin four times. So we have four repeated trials. The second point is each trial can result in two possible outcomes. One of the outcome we call as success and the second outcome we call as failure. In this particular example of flipping the coin we are interested in head. So we call head as the success and tail as the failure. And there are only two possible outcomes and that is where the name binomial comes from. By means two. So here you just have two types of outcomes, head or tail.

If you’re doing inspection, the piece might pass or fail. In case of treating a patient, the treatment might work or treatment might not work, something like that, where we just have two possibilities. We are not looking at more than two possibilities. So whatever experiment, whatever trial we do, should have two results pass, fail, head or tail coming to the third bullet here, the probability of success is denoted by P, which is same on all trials. This is another important thing. The probability of that particular success should be same in all the trials. So whatever number of trials we do in case of flipping the coin, the probability of that will be 0. 5.

So you should have one fixed probability, which should be for all the cases, all the trials. And trials are independent. That means the result of one trial does not affect the outcome of other trial. We talked about independent events earlier also when we talked about probability. So these are the four requirements for binomial distribution. And these work very well in our example where we are flipping four coins and we are looking at the probability of getting one head. Now, here is the formula which you use in case of binomial distribution. PX is equal to NCX p to the power x one minus P to the power N minus x.

The second one is basically the expansion of NCR so here we have NCR. Now, if I expand NCR, which is number of combinations which we talked earlier in permutation combination, this is the same thing. Somewhere you might see a little bit deviation from this formula where one minus p could be written as Q. Q is the chance or the probability of failure. Q is equal to one minus p. But it’s all the same thing. Now in this what are these numbers? X n?

Let’s talk about that x is the number of successes that result from binomial experiment in our example which we are talking, we are looking at one success we are looking at the probability of getting one head when we flip the coin four times so x in our example is one, what is N? N is the total number of trials so what are the total number of trials? In our case we are doing four trials because we are flipping the coin four times what is the probability of success on an individual trial? In our case this is zero five because the chance of getting head or tail in each trial is zero five and what is PX? PX is something which we are interested in we are interested in finding out the probability of successes so probability of exactly x successes and remember the word exactly we are interested in probability of exactly one success which is PX. So PX we can find out using this formula.

Let’s do that so here we have done that all these numbers we have put in this formula so this was the formula at the top n was four so we put four factorial x was one, because we are looking at one head or one success, and p is zero five. So we have put zero five here, one minus zero five here, and we have put all the values of n and X. If I solve this, this gives me a value of zero point 25. That means the probability of getting one head when I flip four coins is 00:25, which you can see from this graph itself, which is zero point 25. This is for one head, exactly one head. Not one or less than one. For this, if you want to use Microsoft Excel, rather than going through this whole formula, you can use this command, this function which is binomist in binom dist, you need to input these variables here.

One is the number of successes you are interested in, four is the total number of trials, p is the probability which is zero five. And false means that you are not looking at the cumulative. And when I say cumulative, that means you are not looking at one or less than one, because that will give you the addition of the probability of getting zero heads and the probability of getting one heads. So that’s the reason you put false here and this gives you this value, which is 00:25.

Same thing if I do for finding out the probability of getting two heads. When I flip four coins, I can put two here instead of one, and that gives me 00:37 five. Now, what I expect from you is stop the video here, take a piece of paper and pencil or pen and do this calculation for two successes. When you flip coin four times, use this formula. Instead of one success, which we did here, one head, which we did here. You use the same formula for finding out what is the probability of getting two heads.

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That you did that and found the right answer there. Now, coming to the next thing in each of these distributions which we will be talking is finding out the mean and the standard deviation or variants of that distribution. In case of binomial distribution, the mean of distribution is given as N multiplied by P. And let’s take the same example of flipping four coins. If you flip four coins and the probability of getting head in each of these is zero five, then the mean of this will be four multiplied by zero five is equal to two. That means in a long run you will get on the average two heads. If you keep on flipping four coins. Number of times if you flip four coins, sometimes you might get one head. If you flip four coins, another time you might get two. Sometimes you might even get four heads or you might get four tails. But on the long run, the average of all these will be two, which is given by N, multiple by P. Similar thing is for variance and standard deviation.

Once we go further, we will talk about descriptive statistics. Then we will go into more details of finding out the mean and the standard deviation. But at this stage understand that the square of the standard division is called as the variance. So if you have the variance, take square root of that, that will give you the standard deviation. The variance or the standard deviation for binomial distribution is given by these two formulas. So in case of variance, it is N multiplied by P, multiplied by one minus P, in our case, the variance of this case will be let’s say if you are flipping four coins will be four multiplied by 0. 5 multiplied by one -. 5 which is also point five. Point five and point five multiplication will give me 0. 25 multiplied by four will give me one. So one is the variance and square root of one is also one.

So one will be the standard division. In this particular example, the calculation which I showed you earlier, this is what I did here. You can find out the mean and the variance of a binomial distribution using these formulas. Before we conclude the binomial distribution, let’s quickly look at one example here. In this example, a manufacturer has 12% defect rate. This manufacturer knows that I am creating 12% defective in my production. Now, this manufacturer has a buyer.

Now, buyer comes and buyer says that from your production I will pick 20 random pieces and I will accept you as a supplier. If there are two defectives or less than two defectives. If there are more defectives, then I’m not going to accept you. Now, this supplier or the manufacturer wants to know that what is the chance of them succeeding or getting this order. This is where you can use binomial distribution. The simplest and the fastest thing is using Excel, where you can put this command binom disk. Two are the number of defectives you are interested in. Here we are talking of two or less.

So two is the number of defective you are interested in. 20 is the total number of experiments. 20 pieces will be selected. Each has the probability of 00:12 as the success. And in this case, success is having defect. And that means defect. Here is success. The probability of that is zero twelveteen. And here I’m putting true, because I’m looking at the cumulative value here, two or less. So instead of finding the probability of zero defectives, one defective, two defectives, I can put this two and I can get this value as 00:56. This is a quick way to do that. Here in the bottom, I have used Minitab to plot a charge for all the possibilities. What is the possibility of getting one defective, two defectives, three defectives and so on. We will talk about that slightly later.

But now let’s use the manual formula, because in the exam you will not be using Excel, you will not be using Minitab. You need to do manual calculation. Let’s do that. So here is the calculation here p is zero twelveteen. The probability of success or the probability of getting a defective piece is 00:12. There are 20 pieces which we are selecting, there are 20 trials. And here we are interested in probability of zero defectives, one defective and two defectives. Once we add them all, we will get the probability of getting two or less defective, which we are interested in here, which is done by using this formula which we have seen earlier. First thing which we do is find out the probability of zero defectives. N is 20 in all the cases, x is zero one and two in different cases. And then zero twelveteen is the probability of success, one minus zero twelveteen. And here you put the value of N, which is same for all. And the value of x changes from zero one and two.

And you do all these three calculations separately that will give you these three numbers. You can do this calculation using calculator, and the total of this comes out to be 00:56, three, one, which is exactly the same thing which we calculated using Microsoft Excel. So after doing this calculation of finding out the probability of two or less defective, if we want to find out the mean of the distribution and the variance of this distribution, we can use these formulas which are NP for finding out the mean. And is 20 p is zero twelveteen.

This gives me on the average on the average you should be expecting 2. 4 defective items in that once you select 20 PCs. So in every time you pick 20 PC is on the average, you will see that you have 2. 4 defectives. Sometimes you might get one defective, sometimes you might get four defectives. But on the average this will turn out to be 2. 4. And if you want to calculate the variance, you can use this formula to calculate the variance and the standard deviation.