 Let's begin. So as you know, we are going to talk about statistics today. So we'll again go by our usual discussion on the weightage. So how much is the weightage for statistics? We have how much? 11. Statistics and probability together has a weightage of 11. So the proportion of statistics vis-a-vis probability would be around 6 is to 5 or 7 is to 4, mostly like that, 6 is to 5, 7 is to 4. So in total 11 marks question from statistics and probability. So what is removed? So this is removed. Step deviation method for finding the mean has been taken away. So you don't need to go by, go for that method. So there will not be any questions as such on step deviation method and cumulative frequency graph has been removed. So hence more than, less than frequency cumulative, cumulative frequency graphs have been removed. So you don't need to find out the, earlier the questions typically one question of four marks that is toward the end, one question will be on finding the cumulative frequency graph and then from there finding out the median. That was one of the, you know, very typical question every year, every year you'll see one question on that. So that has been taken away, never mind. So now, after that, what's next. So again, this has been now so many times that we don't need to repeat it, but every slide as you will see, I hope everybody is downloading or at least keeping it with you those slides which you're discussing. These slides will be helpful, even if no rush, if you have not done it, no worries, you can do it later as well. But these slides might help you, you know, flip through the entire chapter in a short span of time later when you are revising again for the boards. Okay, my dear, so let's go to the. What is our, you know, topic which we are going to discuss. Okay, so what is this measure of central tendencies, central tendencies measure our three ways we are measuring mean, medium and more, you know the definitions so we will first go through the formula for all of that and then we will solve previous year board papers. Okay, so what is central tendency so you know in statistics we do what we have a lot of data collected. Now once data is collected we want to analyze that data what message or information those data points are giving to you. So for that we have few, few techniques or few methods to analyze the data so there are two very famous or very popular or very useful, and one is to find out the central tendency, where in we are trying to trying to get let's say if I have to replace the entire set of data with one point what point, so one one data point so that is what is we call a central tendency, and the other is the expense the range so from what is the minimum to what is the maximum so how widely the data is varying so these two information are vital for us to analyze the situation and then, you know, this particular analysis and decision making later on. So for example right now, when you would have seen you know last year unfortunately there were lots of covert data which was coming in right so age wise, geography wise city wise, and all of that data is there so we usually tend to analyze that and basis that you know, we would do or we would take any call precautionary measure whether there has to be some lockdown, or no lockdown or let's say some you know, some other measures have to be taken so everything is driven by data. So hence, in your 10th grade we are discussing three primary mechanisms of finding the central tendency. And that's mean median and mode so you would have already gone through in your school and our classes will again you know touch base with all of these one by one so what is this mean median and mode okay so let's go to mean. So how do we calculate mean. So what is mean for that matter so in your childhood times you'd have anyways done this where just a minute let me change the curve. Right so where what do you do used to do you used to add all of the data points which are there and then find out the sum, some and divide by the number of data points you had so that was mean. So remember, we studied mean in arithmetic progression as well as well. So arithmetic mean of two numbers a and b is nothing but a plus b divided by two. So if there are more than two numbers then how do I find out mean or average average is nothing but some all, and then divide by the number of it so that you get some figure, which represents all the figures is it so that's what was the purpose of me so three, three, seven. So, whatever saying is. Yeah, so 379 and whatever so let's say 11 and 13. And let's say there are five numbers so used to some all and divide by five right so this is this will give you the mean part mean. Now, what we are going to do this you already did, I think in grade nine. Now, here, we are not doing mean of discrete data so you don't have data something is something like, you know, 592123 no so for example discrete data example would be let's say on a daily basis you are logging their temperature or rainfall let's say Bangalore on first Jan first Jan you are having let's say 20 centimeter of rainfall and second Jan let's say you have five centimeter of rainfall like that. Okay, so all this there are multiple such data you have data points your third Jan you are you are getting let's say zero centimeter of rainfall so the average will be nothing but 20 plus five plus zero by three this is one type of you know so this is not grouped data this is discrete data every, every day there is one data point you add all of them and submit to get zero summit and then divide by number of data points like that. Okay. Okay, press old. Yeah, so now let us say, what are we going to do in this great, we are going to take grouped data so for example, the same data point same data can be arranged like this for example how many days Bangalore received rainfall from zero to five centimeter let's say you will get five days like that. And how many days you got five to 10 centimeter, so you will say let's say three days. And now how many days you got from 10 to 15 you will say let's say 14 days like that. So, with this small example you can clearly say that Bangalore receives fairly reasonable amount of rainfall so 10 to 15 most number of 10 to 15 centimeter. Most number of days right so Bangalore receives good amount of rainfall. Now this is called grouped data where you know this is a class zero to five is a class this is called lower class this limit lower limit, lower limit and this is upper limit you know all that lower class limit upper class limit so this is class and the class width or class size is nothing but five minus zero so when you do this part this is called class size. Okay, so in this case our class size is uniform right everywhere 055101015 only thing to be taken care of here is the upper limit is not inclusive. So it's not included so let's say if Bangalore receives five centimeter of rainfall someday, then that will be counted in the second item second line. Right so you know all of this everyone is aware of this particular thing where how to find out class how to let's say what is the class size lower limit upper limit exclusive inclusive right there could be exclusive. There could be inclusive class also inclusive class is somewhere like one to five if I say one to five. And then I say six to 10, then I say let's say 11 to 15 this is inclusive inclusive meaning this five is in this class only. Okay, and then six is in this class 10 is in this class 11 and 15 both both limits are included in that class only that's called inclusive right so there could be data on of both types we have to be a little careful about that. Now how do we calculate mean of group to data so you understand how to group the data so we have club them together so instead of why do we group them for that matter so let's say one is we don't need that much precision. Every single data rainfall data is not let's say important, I just want to see you know, it's like grading of rainfall so for example it's like grading of a particular fruit, let's say you have pomegranate or. We don't need that much info that you know information so if let's say if you tell me that you know, I don't really want the exact gram value of a fruit let's say a banana or mango. Even if it's between 150 grams it's fine with me so hence that is called grading so they're grading of those those mango so less than 150 gram is one grade 150 to 200 gram maybe another grade and things like that. So I'm not interested in an individual value I'm interested only on the range where do they lie that's it okay. Okay folks let's move ahead so there are three methods which have been discussed in your grade direct method assumed mean method and step deviation method but as you know step deviation method has been removed from 2021 board syllabus so this part. We are not going to discuss we are going to discuss these two. Okay, let's begin. Now very easy the. What is direct method direct method is nothing but the formula is this I hope everybody understands this formula summation fi Xi divide by summation fi anyone has any issues in this in this formula then please do let me know. Okay. Never mind so now yeah so let's carry on so now tell me this formula is okay with everyone so what do you need to do what is X bar so you know the mean is given by this X bar. What is X bar F1 X1 plus F2 X2 plus FN X in like that you have to add all and divide by some of all the frequency so we'll take an example and understand but tell me is this formula clear to everyone anyone has any difficulty in this formula. Tell me quick. Yep, please formula. Any any issues in understanding the formula so we'll take an example that will make it much easier. So, you know, so basically every class. What is X1 X2 X3 and all that so first of all there has been questions like find the class mark. Now in a group data first thing is you have to find out the class mark. Okay, so what is class mark for example in our previous case if you see 025 and 5 to 10 this is inclusive data. This is exclusive data so hence this is exclusive grouping for that matter and 15 to 20 so what are the class marks class marks will be 2.5 the mid value so add these two and divide by two. Similarly, here this will be 7.5 this will be this is nothing but my X I okay so this is nothing but my X I this is 12.5 and this is 17.5 and like that. Okay, so what does this mean this means that two and half represents the entire class whether it is one whether it is four whether it is three whether it is 2.7 it is 2.9 whatever number between 025 every days. It will be represented by one number called 7.5 okay or 2.5 rather and same here. Yes, so this is class mark 2.5 7.5 12.5 every class will have different different class mark. Okay, and that's how you will find out your X eyes any doubt in that. Okay, so once you are done once X eyes are known then there in the table there will be the frequency that means how many times did you find rainfall between 025 centimeter in Bangalore and let's say the number is 25. 25, right 25 is the frequency so we write another table if I will we will do this in the next question that we make it much easier. So remember the formula what is it summation Fi Xi divided by summation Fi this is how we will pronounce it okay next. Yeah, so let's let's do this. So I've taken this question from your book. Yeah, so let's do this question. Right, so how to first of all, I can show you this is the table. This is the table which is there so what is this if you see, you know, class interval 0 to 10 1020 2030 3040 4050 this is the classes. These are the classes and the frequency is 812 1011 9. So what do we do. So first of all find out the class mark in this case what is the class mark 5. So 10 and 2015 so 20 plus 30 by 2 is 25 then 35 and then 45 okay and these are the frequencies, individual frequencies that means there were eight cases where the value was between 0 to 10. There were 12 cases where the values will were between 10 to 20 so it so hence 15 is the representative of all the values between 10 and 20. And why do we do that because since, you know, one we do not need that much of it, you know, deeper detail that whether it was, you know, all the numbers were around 10 or all the numbers were around 20. So I don't require that much. So sense of, you know, the spread the number is good enough so 15 is representing entire 10 to 20 class. Okay, so you have to choose one value, which will represent all the values between 10 and 20, and that number is 15. So in a way mean of mean we are doing. Okay. So 15 is there to 2535 then what do we do you multiply each class mark with the frequency respective frequency. So hence, here is where lots of mistakes will happen calculation errors will take place. And you have to just be careful about that. So this is there, then some all. So that is what is called summation fi xi. And some all the frequencies also divided you get the mean. Right. So this is the process. Okay. Now, next process is assumed mean method. So, let me just explain through the example that will make much more sense. So what will you do what is assumed mean and why do we use it. So assumed mean as you can see use when the numerical values of xi the numbers which you are, you know, the variables which are there, and their frequency, if you product them, it's too high. The product calculation is very big multiplication thing for example. So today, you are doing our 10 year day wise, you know, 10 year device rainfall of Bangalore. So let's say day one is, you know, or like like that let's say if you have zero to five centimeter rainfall, and you have 400 days for this, and five to 10, I'm giving a very small example there could be many bigger examples like that also so 10 to 15, let's say you have 2,142 days in the last, let's say 10 years or last 20 years something like that you're doing, and then you know it's a huge numbers will be there so when you create the table what will happen the class marks will be 0. Then 7.5 and then 12.5 and then multiplied by all these frequencies so 407 46 and 2142. Now multiplication will become a big challenge. Okay, so hence what do we do we assume some mean so you know either you assume one of them as the mean and do the calculations and how are the calculations done with very easy we'll explain that. Yeah, you're now multiplication. This thing load becomes less now today is the world of computers and calculators earlier it was not there so hence it was it used to make more sense when there was no calculation techniques, you know automatic error free calculation techniques today we have these machines available so we actually if you see we do not have that relevance of these methods, but at least you know for mathematical purposes we definitely require them. Okay, now, so let's take this example. And what is this example. Again, why is this chat not visible. Just a minute. I'm stopping the share and then resharing because the chat is somewhat not visible. Yeah, now it makes sense. Yes. Okay, now. So we were doing this yes so let's see how the assumed mean method is done. So what are this class interval is given. Now class interval is given and frequencies are given 1821 these are the frequencies so what you need to do is in any of these variable class intervals first find out the class marks. So let's say 51 53 so you can you can see upper limit, last lower limit divided by two everything remains the same till this point. Then what do we do we any of the x you assumed as you take as a doomed mean. It's not necessary that you have to take one of them only you can take any other value also which you think is more beneficial. So you can take but taking one of them reduces one calculation why because it will be reduced to zero and you don't need to multiply further. So but it's not necessary that you mean has to be one of the class marks, you can take anything but yes it has to be reasonable, which reduces your load. Just do not take any as you mean let's say in this case there was, there is no point taking assumed mean to be 80 because obviously 80 is nowhere closer to the mean in this range of data and anyway if you take 80. The calculations are not going to be reduced much when because if you do one of the steps where you have to subtract so for example when you are finding the deviations Xi minus a, and let's say you you're assuming is 100 for your calculation sake, but Xi is 51 so 51 minus 100 is minus 49. So it's as good or as bad as the number previous numbers so hence you're not going to get any benefit. So what do we do we take somewhere middle value and assume it to be mean and then then do the next find out the deviation so what is the deviation deviation is Xi minus a. So here 51 minus 57 so obviously this number is much reduced and multiplying with minus six is much better than multiplying by 51. So hence if you have to do manual calculations, it made so much of sense that you know instead of multiplying 51 to 18 if I multiply minus 6 to 18 it's much easier calculation and obviously if it is easier relatively lesser intensity so errors will also be lesser so hence imagine some 50 years back there was no calculator computer or even if it was there it was not that ubiquitous not everyone could afford it. Then it was very difficult you know in in let's say if you're in an insurance company or anywhere where there is a calculations required where you are calculating the mean it would be so tedious job and here. Can you see 1234567 only seven entries are there. Imagine if there are 1000 entries and you have to calculate the mean it would be our nightmare for someone who's doing it so hence people have found out some kind of smart ways of reducing the calculation load. So that's the di's are calculated okay so di's are minus six minus four minus two zero two four six here please take care of the signs because we usually tend to subtract the other way around so that should not happen. And now you can go for di times fi so every time frequency has to be multiplied so it will get minus 108 minus 84 so on so forth. All of them and the best part is when you add all these fi di's the number is going to be very very small because of this minus plus effect something will be cancelled. Right so this is the total sum summation fi di and this is summation fi so the only differentiation in the so what was the assumed or standard way direct method x this is summation fi xi divided by summation. Sorry fi. And it is assumed that it is going from one to n i is going from one to n. This was direct method what is assumed main method so here the only difference will be this a will be added to the same setup but xi will be removed or replaced by di that's it the only differences. The this so first write the same setup formula replacing xi by di and then simply add a so we are not going into the derivation of it. Because I have not seen but if anyone wants it I can do that for you but otherwise there is no question as such related to establish this result how do you know how is this how is this derived it's very simple actually. Derivation anyone needs help. Otherwise so far so good guys yep where are you all so silent on some grantee day we usually I don't know if you fly kites and all. Aditya you don't know so wait I will I will show you how to derive this. So how to derive never mind I will show you how to derive it also so how to derive this what is the formula formula is x bar is equal to a plus. X plus what is that summation. X sorry fi di divided by summation fi no worries I will explain no worries now. So my dear friend what is di di is xi minus a. Is it this is my di. Okay now summation. Sorry for that matter fi di will be how much fi xi minus a. Yes. Right now summation fi di so if you add all of them it will be summation fi into xi minus a. Is it simple both sides adding I hope this is clear so far so now and I'm just not putting the limit so it is assumed that it is going from one to n one to n. Okay so what will happen so if you see this this is summation fi di I'm just not writing the one to n part this is summation fi xi minus summation a times fi. Okay. Is it okay and I'm just not writing the limits about okay now constant inside a summation in that case you have you can take the constant out what do I mean. So if this is summation fi what does this mean this means nothing but a f one plus a f two plus a f three and so on and so forth a f n. So a can be taken out and f one plus f two plus so on and so forth f n so it is simply a summation fi. Is it so a can be brought out. Okay, so hence your summation fi di is equal to summation fi xi minus a summation a times summation fi. Now my dear friends divide the entire equation by summation fi di divide by summation fi. Divide the entire equation by this so it will be summation fi xi divided by summation fi minus a and that's that completes the proof. That completes the proof why because now you can write summation fi xi divided by summation fi is equal to a plus summation fi di divided by summation fi. Okay, so this is what by direct method this was our x bar right mean and this is a plus this is simply summation fi di. So instead of xi is now I have the idea divide by summation fi this is the formula for assumed mean method so whatever is assumed mean. And you can find out the mean of the entire group data. Oh, it is clear. Now, let's move forward. Okay, so when what happens if there is an inclusive series we have to be a little careful so this is inclusive series where you can see 29 is inclusive right and the upper limits are included in the. In the in the class itself so hence what do you do. You have to find the midpoint of 29 and 30 so this and this find the midpoint so 29.5 so hence yes so how to make it exclusive so hence half your half there. You add half to 29 reduce half from 30, and you start getting all these series so 30.5 30 like that 35.539.540.544.5 like that so you can see how. Yes, so point five to be added or not will depend on what kind of. You know you have these things you have the classes you have so how much you have to add or subtract. Okay, so depends on that yes to point five and plus point five will help in most wherever there is a gap of one. Okay, so you can see there is a gap of one, so simply add half of half both sides. Okay, my dear friends clear to everyone any doubt so far how to convert an inclusive series into an exclusive series and then the rest of the process stays the same. Okay, okay, clear. Right now let's come to median so mean will give you one one one observation but median gives you another centrality of the data so median what is median. Median is the mid value so you would have done in ninth grade ninth grade where you had discrete data you arrange them in ascending order or descending order. Yes, I can go back for a second with someone is asking to go back wait a minute. Yeah. Yes. Clear. Can we proceed. Yeah. Now, so medium of group data what is median you have studied in the previous grade. What is median median is the mid value right mid value. Something for example, in case of age ages, it works, you know, very effectively let's say let's change it for only three classes three class intervals then it will become much easier. Then you will be able to you know understand this properly. Let's say let's take a random thing. So let's say we have 10 to 20. Wait, let me let me show you. So let's say this is five. 20 to 30. Let's say this is seven and 30 to 40. Let's say we have three. So 12 plus three 15. Is that fine. There are 15 15 items are there. Okay. Now, so where do you think the medium will be so there are 15 items one, two, three 15 items are there where will be the median. Which number is the median eighth number is the median yes. Yes or no. Eighth number is median so where will the eighth number be five numbers top five are here gone so eighth number is not there certainly eighth number is here yes or no. Eighth number will be between 20 and 30 do you agree why because this is five and this is 12 five plus 12 12 so this is less than so 12 8 is less than 12 so certainly eighth number is in this class. Where is that and how do I calculate that so don't you think eighth number is certainly greater than 20 yes or no. It's more than 20. Right. And less than 30 also. So then 20 less than 30 how do I estimate where will it like so what I do is I take this gap of 20 minus 30 how much is the gap 20 30 minus 20 how much is the gap. 10 so what I do is I equally divide the eighth number so eight eighth number is greater than or equal to 20. Yes, greater than or equal to 20 yes. So now we will assume that between 20 to 30 the numbers are evenly distributed okay eight numbers are in so now you will apply that aim while a rule in arithmetic progression what we did so the eight numbers are evenly distributed between 20 sorry 10 numbers are evenly distributed between 20 and 30 what all would be 21 so first now so this is let's say sixth number. Right or sorry sixth number is 20 seventh number is 21 eighth number is 22 923 1024 1125 1226 like that do you understand. So now we are assuming the table to be like this 13th 27 14th. So I distributed between what 20 and 30 like that so yes six seven eight nine 10th like that 10 numbers or no I have to. Sorry there are only seven sorry there are seven seven numbers have to be distributed between 20 and 30. Right so this is not the now I hope you got this what I'm saying is once again. So what my one thing is clear what is that clear clarity clarity is it has to be more than 20 less than 30 now between 20 and 30 there are seven numbers. There are seven numbers yes so gap is 10 by seven very good. Yep or or each number is corresponding to seven by 10 is that okay so let's say there are seven seven numbers are there and I need to go to a third number do you guys get it. I need to go to third number in this gap no wife if I were already taken so I have to go to third number to this I'll get the median third number. Third in this gap where seven it is it has been divided into seven. Okay, so 10 has to be divided by seven and into three will give me the will give me the required number and that you have to add it to 20 yes or no. So 10 is the gap divided by seven and multiply by three you will get the median. So what is all this 20 what is 10 H what is seven F what is three n by two minus CF yes or no three is eight minus five eight was what n by two right minus five. So I'm just taking and here it will not be you know 7.5 it will come 7.5 minus CF but it's okay we are trying to get on you know almost there so I don't need exact values but is that okay. Is this formula what's now. So that is what I was trying to explain so hence lower limit of that median class plus divide the entire edge divide the entire edge by number of times the numbers were there so seven here in this case. So H by F I divided and then I took the difference between n by two minus the cumulative frequency of the previous class. So that is five here. So eight minus five is this. Now is it clear why this formula comes. So this is nothing but you know, linearization. So I've divided the entire class interval into assumed values and then I took the third value third assumed value from there. Right which was required here in this case understood clear any doubt. Forget median formula what median class do a lower value plus H the class size divided by the frequency of the median class into n by two. The median, if it would have been a discrete data set minus cumulative frequency of the previous class. Okay, this is what is the explanation to this formula. I know all this we spent a lot of time explaining this. And again if it is a inclusive series so please be a little careful while doing the frequency table. Now, these are the questions which will be asked. We'll solve this in terms of missing frequency so basically nothing but linear equation used in statistics. Find missing frequency like that will solve some issues on some problems later here. Now finally the more. So the mode is again, as Aaron was mentioning how many, you know, the number or the variable, which is appearing the most number of time is called the mode what is more value of variate which occurs most often. So example of that vaccine will be now implied or applicable over your value of the observation having the maximum frequency that value of the variable at which the concentration of data is maximum for these are some definitions of more model classes that class, the class having maximum frequencies called the model class, right, and how to calculate more mode is given by xk which is lower limit of the model class interval. And again I will not, I will give only hint how to get get this, you know, formula so you know this also formula can be again used, or again explained by discretization so if you convert into a discrete data set. Like what we did in the previous case, you will be able to find this out. So mode is equal to xk lower limit again, can you see the lower limit of the model class interval so mode is going to be between that. xk plus again h plus size into what is this fk minus fk minus one divided by I typically write this as the the you know the denominator I like I write like this fk minus fk minus one same as numerator. So why am I writing this and then minus fk plus one minus fk. Okay, so there are you can see why am I writing like this. So to fk you can see to fk is coming from here, and the other ones are negative. I am writing this for purpose. So again, if you if you look carefully, I am dividing what all are these fk minus fk minus one is frequency of the model class minus frequency of the previous class. And I am subtracting something out from it, what is that it is or rather you can write the other way around also so let me rewrite this here. So what I'm saying is, let me explain. So mode is xk again in this case L you can write L also xk lower limit of the model class plus again h that h appears over there that means the class size now I am distributing this class size. And then multiplying with something what is that in the previous case I was multiplying with n by two minus CF. Here I am multiplying with frequency of the model class minus frequency of the previous class. So I will get one value. Okay, divided by so fk is going to be highest anyways, right. Hence it is model plus and then plus fk minus fk plus one. So as if fk is in the middle, and you are subtracting the previous frequency and the succeeding frequencies and then adding both of them together in the denominator and multiplying this part only. To, to locate again, so if you if you are able to relate it to the previous this thing so this is my n by two in the previous case, this was cumulative frequency. And here, entire f the in the previous case I had only f here, if you recall, see, I have written this formula here can you see, it remains the same, it remains the same. And by two minus CF becomes fk minus fk minus one, and then lower f is nothing but the spread on both sides, right. So fk minus fk minus one on the left side, and fk minus fk plus one is on the right side so what do what is left and right. What does it mean, we'll show you an example. So this is the formula for mode. So the question will be direct, you will be just given some data set and you have to find out the mode. So here is the case, 0, 20, 20, 40, 40, 60 all that. So xk, how do I find out the, tell me one thing, one basic question in this mode calculation, can the denominator be zero? What happens if the denominator becomes zero? What happens if two fk invalid, what does it mean? So two fk minus fk minus one minus fk plus one becomes zero. Possible? Will that happen? Could be, right? Yes or no? What do you think? Can two fk, can the denominator be zero? What if you, let's say you're calculating, you're getting the denominator is equal to zero. Do you understand this? Let's say in the exam you're calculating and twice fk minus fk minus one minus fk plus one will become zero. Possible or not possible? Just say yes or no. Possible? Can it be possible? Yes or no? Can you get zero while calculating if the numbers are such? Yes? No. How many of you think yes? You can get zero. That's if fk is the, whatever it is, I'm just asking, can it be zero? Yes or no? Can the denominator be zero? Not possible. Why? Why can't it be zero? If the numbers are such, then fk is the largest frequency value. Okay. Okay. Yes. So fk is the, so fk minus fk minus one and fk minus fk plus one both are greater than zero. In the denominator what we discussed, hence if you see this part, where have I done another? This is greater than zero. Where are you? This is also greater than zero. Where have I done another? Are you clear? Both are greater than zero. If fk is less, no fk cannot be less. How can fk be less? fk is the modal class. He is saying, okay, okay, no problem. If fk cannot be less, fk will be the highest value. fk cannot be. If fk is the first term, then is fk minus one considered as zero? So fk minus one is nothing there. Yes? Zero? Yes. Now you got it right? So hence two fk is greater than zero. It will not be zero. Right? fk minus fk minus one. Okay. What if, okay, another question, another question. What if, let's say, let's take any, let's take this example. What if if this was also 12? Then which is the modal class? This will not be asked in the exam, but just asking. If you see, let's say both are 12. Possible? Possible or not possible? So which one is take, which one will you take as the modal class? 40, 60. Why are in reason being? Why will be that taken as? So because you then, if there are equal values, then you also consider the values before and after the modal class and find which one has greater values. I didn't understand. Come again. So because there are two classes with both qualifiers mode, you consider the values around that class also. So like 20 to 40 has eight. And I mean, so around 40 to 60, you have eight and 12. And around 60 to 80, you have 12 and six. So 40 to 60 has the greater like general overall value. Okay, but you know, there is data can have more than one mode. Okay, so that's called a Y model. It can have multiple modes also. Okay, so in this case, there can be two modes. That's okay. Multi-model. Yes. So there could be, you know, so unlike mean and median, we can have more modes. Okay, so something like that. Right, multiple, let's say most frequencies are, if you're plotting frequency here, your excise here. So hence see, there are multiple modes of this thing. Okay, so possible. So hence in that, it will not be there in your exam for sure. But just in case if that is the case, that means there are two medium to two values where the number of cases are more. Okay, so both are equally important for you from the analysis perspective. Understood. You can't ignore one. You can't ignore just one. No, no, no, this is yes, what are any saying since the neighborhood here is more than the neighborhood here. So hence the mode will be somewhat aligned towards in this direction skewed towards that direction. So I can understand what he's trying to say. But here we will, we cannot neglect both and you will have to take care of both of them. Okay. Anyway, so you know, you are now familiar with this. There is one empirical tendency. This is not always true, but you can remember this two times median minus three times two times mean minus three times median plus one times mode is zero. And I have written in this fashion because many times people get confused while remembering this code. So always remember mean median mode and two minus three plus one. So, right, and two plus one minus three is anyway zero. So you remember like this twice mean minus twice median plus more is zero. So many times this could be written as three median is equal to two mean plus one more for the same data. So it's not always true, but they will, you know, for your purpose, there was a question last year where they have given me in median mode, sorry, median and more. And you have to find out me the other way around of the same data. Okay, so this was the theory part. So this is the question now sample paper. So this is the data given 100 meter race top watch was used to find the time that it took a group of students any questions so far guys please tell me anything which you want to again revisit. It's not always used. It's not always in certain cases are in in certain cases you can use it in you know in in these where you know the way data doesn't vary that much. There is not not low so many outliers and all that then you can use this. Okay, so in but before applying this data in your case you can apply it, but they're in, you know, there will be many cases where the variation in data is so high that it will not always be true. Okay, you can just you tweak with let's say you put an extra additional outlier data outlier meaning total out of the way. Right, it was it is not supposed to be there in the first place but it is there. Let's say there is one data which will totally do it or change what let's say mode or mean. Let's say you're trying to find out some values which should be around 10 and all of a sudden you get a value 1.7 and all other values are 10.1 10.2 9.9 9.99 like that you're getting data and one value is totally taking it off. Always you know, and there is another value which is 21.3. Let's say total off. So in this case so both are both will balance the mean but let's say there are a couple of such outliers in a set of 1000 data and it's pulling the mean totally away from it then this equation might not be true, but there will be closer to zero for sure. Let me make myself clear. Is it clear to you. Hello. Am I am I clear. See me median mode for any data is is some is art is a what do you say is a manifestation of centrality of the data in reality they do not exist as for example let's say when I am counting the average number of children in a class in let's say your school nps will it be an integer average number of children in all the classes put together so what is the average class size if I ask for nps will will you come up with a number like exact 40 or 30 or 25 it cannot be right it will be somewhere around 27.56 right and or you know something like that so those these numbers are not you know they do not actually actually exist for 27.56 is not the number of people in the classroom but it gives you a sense of where it is so it is very clear that it is not beyond 30 it is also very clear that it is not beyond 25 so it gives you a sense of measurement but it's not exact measurement so hence in a in a case where we are dealing with now though you might have heard of something called big data and all that so much of data is getting generated. So, you know, so we are we are minds and we also don't know needed no need to go to each one of the data and you know be very particular about it so we are just a sense of it is good enough. That's where we talk in terms of mean, mediums and more. Okay, so what is the first answer 100 meter race stopwatch was used to find the time that it took group of students to run 100 meters estimate the meantime taken by a student to finish the race. Meantime taken what is the meantime taken so you have to fast class marks and. So eight into 10. First of all, try to find out summation fi so 810 1818 and 1331 and 637 and 340 so total is 40. Okay, 40 students have run so hence eight into 10 so you calculated this eight into 10 is 80 then 10 into 30. Is it it then 13 into 50 and then six into 70 and then three into 90 divided by 40. Could you do this or did you have some other shortcut method of finding it out how quickly you have to do this only right. Plus 300 1100 1100 plus 650 so 1750 so 1750 plus 420 plus 270. So this is a total 077 1412 to 141440 by 40. So it is you got the dd second one one what was the first answer a cup 43. But I messed up with the calculus or two it is sorry. 43 1700 5277 141784 121412244 I messed up then 80 300 1100. 650 1750. 420 and. Second one is 40 or a first one only I got stuck you guys are so fast they go and do like that. So hence eight into 10 is 80 here 10 into 3300 here 13 into 50 650 here. Six into 70 420 here and three into 90 270 here correct and divide by 40 is it am I doing correctly or not. Oh I did not add a okay so that was a mistake so this is. 380 and 380 430 1030 1050 1450 1450 1520 1720 so 1720 upon 40. So not this one. So this is. 43 first one is 43 what will be the upper limit of the model class model class is this so 60. You get the 60 part second is 60 right then third one construction of cumulative frequency table is useful in determining the median. We'll just discuss this they should not have asked this question but it is mean the sum of lower limits of median class and model class what is the median class here. Same as same as the model class is it how do I define the median class. N by two so 40 by 220 so 20 is. Here so you have to make the cumulative frequency chart so in this case eight 1831 37 and 40 will be. I'm just doing roughly okay so you don't do it you know because it's one marker so inside I'm not drawing a table you should have ideally drawn the table and then right so hence median class will be 20 why. Because. So 18 is done here so 20 is less than 31 here so hence this is the median class and the median class will have what same lower limit as this thing so 40 plus 40 80 is the right answer how many students finish the race within one minute this is but very easy. So this is again a less than all I think they should not have asked this but then they are asking no worries you must be ready you can see this is this is studied in but anyways you can you know. Find out from the data how many students finish the race within one minute so within one minute is 13 plus 10 plus eight these two people took more than 60 seconds so hence this some will be the final so 31. Very good so you have to answer four out of five, not all. So if at all you have to leave one which one would you would have left you have to leave one ABCDE which one will you leave. First one. Okay, everyone will leave first one. Yes, because. Yes, obviously it is not worthy enough for one minute and then there is a possibility of making errors also. So anyways, this is the second question which is again three marks in the same sample paper. Now here is where we will create a table and see this question is of missing frequencies. Right, so frequency missing so first of all class so you have to do so let me give you how will you solve this median is given as 16. So you must remember the formula of median. Usually I immediately write the formula of median so median is equal to L plus H by F into N by two minus CF. This is my median. Now, we have to create a table because a and B has to be found out. All of you also do I will also do parallelly so I will immediately make zero five. Then 12 510. I will not draw lines and all because I do not have the features here. So you can draw the lines if you wish in the exam definitely make a proper table and be very very careful while you are taking because if you miss any number and then gone. It's also good practice to match this revisit the numbers. Okay 35 to 40. Revisit the numbers so 0 512 correct 510 810 1512 15 B 664. Now you will find out class mark X I this is F I. So what is X I 2.5 5 7.5 now keep adding 5 12 12.5 17.5 22.5 27.5 32.5 and 37.5 so X eyes are done. Okay now what to calculate median I have to first find out the cumulative frequency so be very very careful. So cumulative frequency so 12 so keep adding 12 plus a then 12 plus a and then again 12 so 24 plus a. Okay then what 15 39 plus a then what 39 plus a plus B. Then 45 plus a plus B. Then 51 plus a plus B. And then 55 plus a plus B. Now we very very careful while again repeat 12 plus a. And then 12 again so 24 plus a 24 plus 39 plus a 39 plus a plus B 39 645 45 plus 651. This is the thing right now my dear friend how will you find out. And even done median is 16. Okay so where is the median class. What is the median class my dear friend. What is the median class in this. Can you can even tell me where is the median class which one is the median class. 15 to 20. Median is 16. What is the median 16. Why stress why it is 15 to 20 stress can you unmute and say or adipya. Since median is 16 so. Hence. How do you find out where will the 16 lie. 15 to 20 since 16 between 15 and 15. 15. Oh no my dear. You have to check cumulative frequency no. We're taking this. So since the median lies in the 15 to 20 class can't we say the median class is 15 to 20. Given median of the following data is 16. Yes. Yes. Okay. Yes. So what is the median class very good. Yes. I was thinking of. Okay. Okay. Yes. Yes. In that case. Okay. Yes. Very true. True. True. Sorry. Yeah. 15 to 20 correct. So say hence what is L. 15. What is H. 15 15. Is it. What is N by two. That is not known. That will be N by two is. 55 plus a plus B by two. What is CF 24 plus a. Yes or no. Are these data okay. Is this is this the data you are using. Now. So now next. How to do. yes so hence median is 16 so 16 is equal to l what is l 15 plus h h is 5 by f 15 into n by 2 minus cf so 55 plus a plus b by 2 and oh total is anyways given so we can directly write 70 so 55 plus a plus b is total of frequency is 70 so this is 70 so you can directly write 35 here so instead of this 70 by 235 minus cf minus cf cf is how much 24 plus a so minus 24 minus a okay so 16 minus 15 is 1 is equal to 1 upon 3 into 35 minus 24 is 11 minus a okay so 3 is equal to 11 minus a a is equal to 8 if a is equal to 8 then 55 plus a plus b is equal to 70 so 55 plus 8 plus b is equal to 70 so this implies b is equal to 70 okay so time consuming hence most of them will be four marks and things like that chalo good next easy one one marker find the mean of first n natural number so mix of apn this find n so you know first n natural number what is the sum sn n into this is the mean of first n sorry this is n sum of n natural number so in this case s15 will be 15 into 16 by 2 right oh it's a mean is given sorry my bad so mean is given as 15 so 15 is equal to n into n plus 1 by 2 by n am i right sum of n natural number divided by n will give you 15 so n and n will go so 30 is equal to n plus 1 so n is 29 you didn't understand the question is find the mean of first n natural number so 1 2 3 4 till n you have to find out the average of it mean how to find out average so 1 plus 2 plus 3 plus 4 plus dot dot dot plus n divided by n sum of first n natural number is this divided by n what is sum of first n natural number you know this so n by 2 use the formula of ap what is the formula of ap ap a formula sum sum of a n terms of ap is n by 2 into first term plus last term is it so n is a number of terms into first term is 1 and last term is n so this is the formula for sum of n terms of an ap so n plus 1 this is the sum and then divide by n and it is given that this is equal to 15 so find out n n comes out to be 29 okay next find the mean of the following distribution so first check these are exclusive so no problem so draw the frequency table so first of all x i so i'm just drawing here vertically i'm sorry horizontally you can do vertically as well so this is four this is six eight ten and twelve now you find out the mean mean two marks question mean now fi so you can use any of the methods because it's not mentioned go for direct method fi is equal to this is 5 10 10 7 8 you can go for assuming mean method also so x i fi or even take any of the it's going to be it's going to go so let us say this is a right so you find out di di is four minus eight minus four minus two four yeah zero two four okay and then multiply fi di will be minus 20 minus 20 zero 14 32 so summation fi di is equal to minus 40 and 32 and 14 46 6 only 6 right so x bar mean will be a plus summation fi di by cumulative frequency so summation fi so a is 8 plus summation fi di is 6 divided by total frequency parts thus pander of 25 25 plus 15 40 this one so 8 plus 6 by 40 so 1.5 0.15 two marks for this but don't do like this please draw a vertical table separately i'm just for the positive time i'm doing like that okay next find the mode of the following data so hence usually my practice is see i was very poor at remembering formula i don't know how you are so hence if you don't understand if i don't connect with the formula i will totally forget it so thankfully here i could derive that so hence my mode is x k plus h into two things were there f k minus f k minus one so difference of the frequencies divided by either you write in the typical way or i write like this f k minus f this is f k minus one plus f k minus f k plus one so once i write the formula then only i'll approach okay so this was my style you guys if you are confident you can just yeah so first i i'm very very clear what is being asked and what formula i have to apply because if i use the wrong formula gone calculate all of you so class frequency is again given you and so you have to find out the modal class first of all so this happens to be a modal class right modal class so what is x k 60 so i will write all the values also separately so that if i go wrong somewhere still i can get some some marks here and there if at all it is a quest of marks so h is the class size 20 f k frequency of the modal class 12 f k minus one frequency of the previous class 10 f k plus one six write all done now formula apply m is equal to careful calculation very important 60 plus h 20 into f k 12 minus f k minus 10 divided by 12 minus 10 plus 12 minus 6 that's it so this is 60 plus 20 into 240 divide by 12 minus 10 is 2 and 12 minus 6 is 6 so 8 so this is 65 2 marks okay this is how we'll solve so meet all questions of mean median mode done next median of the following data is again similar question missing frequency question this is four marks so this was asked along with another optional question which was related to oji but now that's gone so hence maybe you will get our optional between find the mean median or find this missing frequency so in missing frequency a lot of work has to be done so hence it becomes so anyways so summation f y is given total frequency is 100 so c you can predict only the values have changed everything the question remains the same totally median is 525 so what is the median class first of all where is the median class this is the median class right so you now know l but anyways you have to create a table so let's not be lazy and do it so 0 to 100 and I will then write the class marks just next to it instead of frequencies I'll write 50 here 100 to 150 sorry 100 to 200 it is so it is 150 this is my x i then 200 to 300 this is my 250 then 300 to 400 350 just check if all the data points are there many times something might be missing so keep a look keep an eye on that also so this is 450 then 400 to 500 then 500 to 600 this is 550 then 600 to 700 650 then 750 then 850 and then 950 right so you fill the fill the rest of this okay now be very very careful so two and then five then x then 12 17 then 20 then y then nine then seven then four okay enough so what is the median median is l plus h by f into n by 2 minus c f so let's write all of these values l is 500 h is 100 okay and then h is 100 f f is how much 20 then n how much 100 and c f how much 17 done so this is seven seven plus x 19 plus x 36 plus x 56 plus x 56 plus x plus y in these questions half of the time is wasted wasted on writing all these steps but then you can't avoid it 72 plus x plus y 76 plus x plus y okay yeah fair enough now median is given what is the value 525 so 525 is equal to l so 500 plus h 100 h 100 by f 20 into n by 250 minus c f 17 okay i hope i'm doing the right calculations so 25 is equal to 100 by 20 is 5 into 50 minus 17 is 33 okay what what something is missing what did i miss n by 2 minus f what was f frequency will not be 17 oh 17 plus x so thanks thanks thanks for 17 minus okay are we happy so sorry what is that not 17 this will be 36 plus x so there will be other errors these are all errors i'm old person will make more errors unlike you so 14 minus six correct so be very very careful guys so so 14 5 by 33 and what no this will not be 33 33 is gone 14 minus x yeah so 5 5 is equal to 14 minus 6 so x is equal to 9 correct and y so what is y so you can calculate y 17 plus x plus y is 100 so clearly x is 9 so 85 y 9 y is 15 yeah correct okay yes so that's done actually so this is all find the class marks in simple simple class marks so do it 10 plus 25 by 2 35 by 2 that is 17 point 5 am i right yeah and this one 35 plus 55 by 2 so 45 right am i right class marks of 10 to 25 is 17 half and 35 to 55 is 45 clear guys this was asked in 19 20 a mark okay hello all good any difficulty clear so far so good guys next do it again same you know you have to do same thing again and again you'll get bored but can't help maintain your calmness of your mind so size of items is given in centimeters compute the mode mode so again as i told you i will for the first i'll immediately write the formula and make sure that's correct so mode is equal to x k plus h into f k minus f k minus one divided by in your book it is 2 f k but i write i prefer to write because i can connect with it so hence i write like this two positive quantity f k plus sorry minus f k minus one that's it so now value so what is the modal class so write down modal class 12 to 16 correct now so now let's find out all the values x k 12 then h 4 then f k is equal to 17 f k minus 1 9 f k plus 12 right then calculation so m is equal to 12 plus 4 into 17 minus 9 that is 8 divided by 8 plus 5 right so this is 12 plus 32 by 13 right did you get this value guys 13 so 12 plus 2 point chubbis and 6 point 4 5 to 80 13 5 6 7 2.46 so this is 14.46 yep how many if you're getting 14.46 14.46 correct so this is again two mark mode so again we are going in the same thing mean median mode nothing great so far so i'm just solving in front of you so that even if you are not you're taking time don't worry again see every other board paper will have this missing frequency thingy so only need to do cumulative frequency chart so 3 it will so predictable 3 plus 6 9 9 plus 6 15 oh here the only one only one frequency is missing okay cool so 3 9 15 oh so total is not given hence only one f is given okay very good so 3 plus 9 plus 15 see here in the previous question they had given total frequency also summation f i was given here it is not even it's not even hence they have reduced to only one variable so you don't need to solve two variable equations so 3 9 9 plus 6 15 15 plus 13 28 28 plus f and then 33 plus f and then 37 plus f i am guessing f could be 3 looks like no maybe sir in one or two questions i don't know but i have seen more than one pair of correct answer it comes to finding two missing frequency can you share that question yeah what is the question one two questions i have seen more than one pair of correct answers when it comes to finding two missing frequency so if you yourself solving linear equation in two variables then there cannot be so basically what are we solving linear equations at the end of the day so if there are two variables two equations only one solution will be there if you come across anything please share let me have a look okay so okay so what to do next the mean of the oh it's a mean yeah look here you have that problem what is the problem this question is not about median it is about level hence the moment you see this thing people think about that it is median but thankfully i read the question once again before attempting anything so good the frequency f is the class interval 9 20 21 is missing so some what is mean so you have to do you this was useless actually see that that happens when you miss on the world so this was not needed but anyways so but we do we saw it in the time so hence 11 13 so 12 so let again make a table so quickly make a table 12 14 16 18 20 22 24 okay this is my x i then f i is 3 6 9 13 f and what else 5 4 then would you go for assume mean i would go definitely for assume mean so 12 14 16 18 20 24 let us say this is assume mean so let's find out di minus 6 minus 4 minus 2 0 2 you know 2 4 6 now summation fi di i know the values guys let's compare our notes 6 4 the 24 what is the value of this minus 18 did you do it or not 0 and 2 f and 20 24 okay so this 24 and this 24 is going to go so let me not do it so summation fi di summation fi di is how much 24 and 24 will go so 0 20 and minus 18 is minus 2 and minus 18 minus 20 is left once again so 24 24 goes 20 and minus 18 is minus 2 minus 2 and minus 18 is minus 20 plus 2f right now what is mean mean is given to be equal to 18 so 18 is equal to a so our a was also 18 minus 20 plus 2f divided by so divided by summation fi so oh anyways frequency was 37 plus f so that was anyways needed yeah so hence oh the 18 18 cancel 37 plus f is not needed so f is 20 by 210 did you get this f is equal to 10 guys how many if you got f equals to 10 anyone got f equals to 10 are you are you solving hello guys boda markas and grantee mod did you anyone solve f equals to 10 are you getting f equals to 10 guys are you there or i'm disconnected hello folks you're there you are still calculating others only aditya is calculating are in you got f equals to 10 hello still doing five oh regime what happened calculation so what is the observation type of questions one what are the typical questions you are going to get see i will i will what is the arithmetic mean of first n natural numbers again this is the same question you're using the empirical formula one question was there in 1920 yes we're using the empirical formula find the mode of a distribution whose mean is 8.32 it's called empirical it's not theoretically derived it's an observed value empirical means experimental so experiments observed or whatever observations we have could you go back yes where here aditya any question done so what i'm saying is hence is you know to answer what are in asked so it is one for few specific experiments we have seen this happening but we don't have any proof and and this empirical meaning derived from observations of experiments okay so hence there is no theoretical background of that formula if you ask me how to derive it it will be difficult but i if i see the data it makes sense are in yes like empirical probability where we perform the experiments and basis the data we come up with some conclusion but we do not have any theoretical basis for that why it is happening happening like that yes so maybe there might be theoretical proof which right now we do not know okay so hence we say that we observed it and hence we are saying the and again it is true for a few this thing so it is not you know maybe there could be several cases where it is not true so now apply it and find the value using the empirical formula find the mode of a distribution so you know the whose mean is eight point three two in the median is eight point zero five so let's write the formula so two times median mean minus three times median plus one time more is equal to zero so we have to find out the mode so what is more guys mode will be simply three times median minus two times mean so three times median median is eight point zero times zero five three to eight point zero five minus eight point three two into two so twenty four point one five minus sixteen point six four are in no i'm saying yes there is no true no proof for the entire set of things but for that particular say you can't prove it theoretically for the type of data you are getting also so let's say this particular cannot be used only in one case right this equation there will be certain cases where it is applicable certain cases then only you are using it isn't it but you cannot prove for those cases also how it is happening did you get the point so let's say in one particular specific case case a here you can apply let's say in in such cases these are the data hence you can apply this rule but for that set of data also can you prove this thing that's what i'm saying so hence it is called empirical where we don't have a theoretical mechanism to prove it but we have observed that happens okay so is this the calculation so what did you get no it's not only few observations stress see for example see everything evolves with time so till Einstein did not come no one was ready to challenge Newton's Newtonian mechanics but today we know that Newtonian mechanics works but it works on under certain circumstances only right classical mechanics works but it works under certain this thing but the moment you get into a subatomic level it doesn't work so hence as you as science progresses we come to know oh this is this has a limitation but this works well within that particular set of constraints so hence right but you understand what i'm saying so that's where so we do not generalize for everything so Newton's laws cannot explain the motion of an electron but it can certainly explain the motion of a rocket if it is not moving at a very high speed near to the speed of light so hence okay so this is the calculation finally so um what is the value 24 minus 16 oh so one four and seven uh am i right akshita is getting getting 7.51 it's not four or did i miss something five minus four is one and so it was not 24.15 it is 24.15 is it what i'm tired anyways thanks yeah so five one so totally tired should not do late night work anyways that's not for you okay so fair enough guys um there are lots of lots of questions which are similar typical typical typical typical typical typical typical typical all are like that see all are collated ahead you know so now this is something see they have vertical tables and all that table is very important by make table this is ruled out these this um this is not there in the syllabus anymore standard deviation so this was standard division but again you can see how tables have been put and a assumed mean has been highlighted and then all the calculations are done so you don't need to do this but for your own benefit if you think that okay um you're comfortable doing that so please do right like that and then uh can you please share the attachment for yes i will definitely so class size is again this assumed mean written down here mean is this mean is 21 plus minus 46 into into six so what happened yeah this one 21 minus 23 by 20 into 63 so like that every step has been mentioned see uh are we allowed to use step division because in many cases it is here i would not recommend are in to be honest you never know yeah because that's not there how can you use that kind of a thing um but you know so but i think they will not give you something which is so difficult to calculate so why take chances so because the number of volume of papers is so high so many question papers are there it becomes difficult to you know because you know you understand how many papers will be corrected by the faculty members there every day so many so if they start using their discretion it becomes difficult to manage the show so ideally yes very much any method you want you can do and hence in the exams like olympiad so now there is no restriction on methods