 So, now finally something mathematical is coming up let's talk about that first please write down the random errors the random errors can lie random errors random errors can lie both sides by both sides I mean that it could be it could be added to the actual value or subtracted from the actual you don't know you can lie both sides okay the only way the only way to minimize the error in the measurement the only way to minimize the error in the measurement is to take multiple readings what will happen then sometimes the random error will get added up sometime it gets subtracted so when you add up all the readings whatever the random it will add up to 0 getting my point so all of you listen here suppose you are measuring a length which is 5 meter and random error is let us say 1 meter for the sake of simplicity error is 1 meter okay so if I just take one reading I will get 4 it is a 4 so the error is 1 okay but if I take multiple readings so sometime I will get 4 sometime I will get 6 yes or no so when I take multiple readings when the average will be closer to the true value yes or no the random error sometime will be plus sometime will be minus so net net when you add up these random errors will add up to 0 IDT you should take infinite readings when you get exact reading exact true value the good part is it is random the bad part is you can't remove it and because it is random it can be both sides and since it can be both sides if you take multiple readings and add up random error will add up to 0 random then it can only be on one side and you take the mean if it is only on the one side it is not random no random error suppose you are getting 2.1 and then what you said 2.2 then then you will also get 2 and 1.9 if you take lot of readings the random the reason why it is random is because it can lie on both sides when you take average the average will close to be 2 if 2 is a true value okay if it is always on the one side suppose you know exactly the random error is let's say 1 meter and it is positive if you know it the reading whatever you get subtract it you get the exact value yes or no if you know that random error is plus 1 meters and your reading is plus 5 subtract 1 from the 5 that is a true value you don't have to do anything but if you're getting reading as 5 and the random error is 1 then it could be 4 it could be 6 it could be 5 itself you don't know now you understand why you take multiple readings in the experiments in the lab okay to manage the random error these are managed by step by step manner in conduct experiment systematically okay let me tell you a procedure these procedure you have already followed in conducting experiment but let's learn it systematically also write down absolute error relative error absolute relative percentage error so going forward if I'm using a word error I mean random error systematic errors gone by properly following that experimental procedures systematic error is taken care of understood okay now I will give you one example of error from which you will understand why there should be different definitions of error okay your home from the center let us say it is 10 kilometers okay the if you have measured 10 kilometer and the error is 1 kilometer but let's say error is 5 kilometer it's a huge error or not it's a huge error okay suppose you're measuring the distance of earth from the moon which is few lakh more than that kilometers okay the error there also is 5 kilometers is that huge error negligible right so it depends on exactly what is a value you are dealing with you can't look at the error in absolute terms if you're measuring 1 lakh and the error is 5 it is less but if you're measuring 10 and error is 5 it is huge okay so it is you know when we talk about errors we always tend to talk about the absolute value error is 2 mm is 10 mm error is 1 meter okay but in reality it doesn't mean much when you compare that two errors two readings if you compare you have to compare you don't look at the absolute errors fine so we are trying to quantify errors now how to talk about errors all right so there is something called absolute error which is what our understanding is absolute error is how far is your reading from the true value simple okay but the problem is I don't know what is a true value understood when you conduct the experiment do you know the true value you don't know so how will you know how far is your value from the true value so what you do is that you take multiple reading take the average and then you say that that average is my true value but the way someone and that there was someone the way he was doing is that he used to ask his fellow students to conduct the experiment and then they get the readings they get the reading let's say 2 3 6 late like that so that guy used to add little bit to every reading and then showcase as if it is a phone reading okay so there will not be much of a deviation so at times you add one here then you subtract one there more or less you get a different set of reading who knows it is a random error right who knows this what you have conducted the experiment and got it or these are your own creation okay but don't do that okay so this is we are talking about practically how to define the errors okay the errors are in the measurements you make in the experiments okay so step number one we will take multiple readings of anything suppose I am using the time period of pendulum or I am using the length of anything okay let's say my readings are a1 a2 a3 a4 like that okay let's say that I have taken n readings write down these readings are a1 a2 a3 up to let's say a n these are my readings okay fine now these readings will have I have removed the systematic errors already take it care of these readings have random error okay so using these readings I can arrive at the true values how by taking as many readings as possible ideally you should take how many readings okay but you take at least 10 readings okay whatever it is let's say you have taken n readings so the average of the readings average of the readings is what you add up all the readings you have seen this summation this is i equal to 1 to n divided by n this is your mean value you can also write it like this a bar this is your mean value and in the process of taking mean what will happen is that random error will get eliminated by itself because it is plus or minus okay so I will treat this mean value as my true value this is my true value understood now I can talk about let's say a bar is looks like a vector so let's say it is a n or a mean it is very easy for me to rub and write it down I can understand cut it and write down a mean a bar looks like a vector but nothing wrong with it as long as you know what it is okay now if I have to find error in let's say in the reading a1 how will you find how far a1 is from the true value let's call it as delta a1 what it will be equal to mod of a mean minus a1 that's an a mean minus a1 it could be plus it could be minus depending on whether a1 is less or more it could be anything not plus minus it could be plus it could be minus okay this is error in the measurement of a1 okay what is the error in the measurement of a2 simple a mean minus a2 and like this you can keep on writing and then at the end error in measurement of the nth reading is a mean minus an okay these are the errors okay on each of the readings now I need to come up with expression for this is my true value right I want to find out what is the error in the true value are you getting my point at the end I want to say that my length is this much with this much error you understood okay this is the length I am representing but right now I'm trying to calculate the error okay now whenever we talk about the errors whenever we talk about the errors we try to find maximum possible error why is that suppose I have to design a building and I know that there's an error in measurement of how much load a pillar can take okay right so I will try to find out what is the maximum possible error which I have calculated for the load a pillar can take and I'll design for that particular thing okay I'll not take a chance are you getting my point so whenever we are calculating the error we will be trying to find out the maximum possible errors understood all right so these are the errors in a1 a2 a3 these can be plus or minus suppose I have to put an error quantity for the mean value okay then this is how I should be writing the error in the mean value should be equal to write down mod of data a1 mod of data a2 and so on mod of data an divided by n make sense okay so this is this can be written as summation of mod of delta ai by n all right so how you write your readings as in suppose if I ask you okay fine you have calculated using some experiment methods can you write how much is the length now how you write a1 a2 a3 are my reading of the length this is the true value and that is the error that is still a random error only so a mean plus or minus delta a mean this is the value why mod because if you don't take mod that the total error may become zero but then the problem is this you might have taken n readings you might have taken n reading what if you take n plus 1 reading it will not become zero okay just because you've taken n reading at least equal to zero doesn't mean that it is zero fine so we are not taking any chance whatever it is you have taken out of that we're trying to find the maximum possible error we're taking mod of the errors we are not taking negative value of the error even if it's negative we're taking mod of that okay I'm not saying I'm not saying this is the error are you getting it if I know exactly how much is the error then what is the point of having the that much okay this is the maximum possible error so but how is this maximum like what if delta a1 is more than delta a2 understood this is the maximum possible error close to that max you can say that you can claim that the total error I'll not divide by n I'll say this then you are even more conservative you can say I'll multiply it by thousand the error I'll be very very conservative all right but this is in a way it represents the maximum possible error just to correct myself this is in a way a representation of maximum possible okay not exactly the maximum possible so it's like super the average error on either positive or negative side no so except like if I am correct this is the this is the mod of the mod of the whatever errors you have in all the readings so like if you want to calculate maximum possible to do this anyway you need to calculate every error so just find the biggest error and it's done but either way it could happen see it could happen that one of the readings you have taken there is some random fluctuation the probability of that is very less but suppose that kind of thing happens earthquake happens and you're conducting experiment you're measured with huge experiment huge error ideally that doesn't happen but suppose there are eccentric data points it could unnecessarily just increase your error by huge margin okay the probability of that happening is extremely less so you know it you know it always for safety point of view that is good suppose you found that there is a chance that the pillar has to withstand let's say one lakh Newton you had one reading okay but chance of that happening is once in 200 years okay to make sure that it withstands two lakh Newton you have to make it thousand times thicker than what it normally is so your amount of investment money which matters the most will be much more so you have to optimize between money and your life that is that's how it is i mean why do you think the pillars are of that shape pillars are thicker at the top why don't we have a big thick pillar like that to optimize the amount of material you're using whatever is the amount of less less material you have not used here and this kind while constructing like that you're making another pillar out of it okay so there's always an optimal point but yeah factor of safety is there don't worry much about it okay so what what essentially we are saying that any reading you know any reading let's say ith reading will be lying between what and what readings a maximum plus delta a mean and should be less than a max a mean minus delta a mean so if this is a representation of maximum possible error then any reading should lie between these two yes or no there can still be couple of points which could be beyond this range but i'm talking about max most of the readings will be between this and that sorry this is minus minus understood any doubts any doubts no doubts so we can solve a numerical on this coming coming one by one no doubts right have you done this this kind of thing in your labs come on no we didn't talk about blindly you copy no it's like we really don't need this right plus minus twice whatever you have written plus minus like this like once or twice