 Hello friends. So in this session, we are going to discuss the direct method of finding arithmetic mean of a given data So let us take an example. Let us say we had a sports meet where Two people got five medals four people got three medals and six people got two medals So let us say the distribution was something like this. So two people got five medal each Then four people got three medals each Three three three three and there were six people who got two medals each in the inter sports meet In the sports meet, right? Now the idea is you have to find out what question is what was the Average number of Medals One by the athletes, right? So what was the average number of medals one? So if you see two people got five four people got three each and six people got two each Then the idea is I have to we have to figure out what was the average number of medal one, right? So how do I do it if I have to do it then? We have to sum them up and then divide by the number of such Such values, so hence how many people actually participated if you see how many people actually won the medal So if you count them you can count from let us say this is one. This is two three four five six seven eight nine ten eleven and Twelve so there are twelve people who won medals. So hence to the average average number of medals One would be how much would be simply five plus five plus three plus three four times and Then two plus two plus two plus two six times Correct and then whole divide by total number of Sportsmen who won the medals, right? Now if you see this can be Done like this or in other way if you see this is nothing but five times two plus three times four and this is two times six Divide by twelve can be written as two plus four Plus six isn't it? Same thing in other. It's just a mathematical manipulation. So if you notice clearly If if you notice closely what you'll see this is nothing but the product of number of Medals one by each and how many people had one Okay, similarly, this is the first case then this one is the second case three number of medals one by four people and Two number of medals one by six people and this is nothing but two people plus four plus six Correct, which is if you see is nothing, but if I have if I coin five as x1 So this is x1 into f1 Plus x2 into f2 plus x3 into f3 and whole divided by F1 plus f2 plus f3. So what are the meanings of x and f so x1 is the Number of so x I now I when I write I I is a generalized Index so I is the index here. So it is ranging from what x it x is are nothing but x1 x2 and x3 and what are these x1 x2 and x3? number of medals and we had three types or three Cases where number of medals was five and number of medal one was three and number of medal one was two So XI values are five three and two and what is fi fi is nothing but Frequency it stands for frequency. How many times so how many times five medals were one, right? So how many times five medals were one two times? How many times three medals were one four times and how many times two medals were one six times? If you see notice here, this is the frequency table Frequency is nothing but how many times and XI is nothing but the values values of we say that these are values of the variables, right? now this particular Expression here is this particular expression here is easy to write when there are only three Values, but if there are many values, let's say hundred two hundred five hundred like that or many more Let's say if you are talking about a full Population or let's say a full Olympic game then it will be you know more than hundred countries participate in the Olympic game So it will be very difficult to find out average like that So hence we have derived a note notation to simplify this or shorten it up. What is that notation? So X1 F1 plus X2 F2 plus X3 F3 plus dot dot dot. Let's say if it was Xn Fn these many Values were there and then divide by F1 plus F2 plus F3 plus dot dot dot plus Fn Okay, so these were this was my X bar so bar means the moment there is a variable and there is a bar over it. It means Average or arithmetic mean right average slash Arithmetic mean, okay Now, so this is this particular expression can be written as This sign summation Then I is equal to 1 to n I is going from 1 to n and we write X I F I divided by Summation I equals 1 and it goes to n Fi What does this mean now? Let us interpret this once again So the moment I say submission X I Fi from I equals 1 To n it will mean you start from the value of 1 and go up to n Incrementing 1 each time. So what does it mean? So I'll have to stop. What is I equals to 1? I is 1 so we'll start with X1 F1, okay now check is I 1 equal to n No, then you add 1 to the previous number. That means now you write X to F2 now check whether 2 is equal to n If not, then again add one more. So X 3 F3 Then X 4 f4 and keep on adding till you hit Xn fn and then stop you can't go n plus 1 This is the value. This is the meaning of summation X I fi right similarly summation I equals to 1 to n fi means The value of I will keep on iterating till it reaches n. So F fi starts with F1 then increment it by 1 so F2 Then go to F3 and every time you add you keep checking whether you have reached the value n or not So if you reached n then stop Okay, let us take an example and understand so summation. Let us say I Is equal to 1 to let's say 5 X I fi will be nothing but X1 f1 so 1 is less than 5 so I can add one more X 2 f2 2 is again less than 5 so add more X 3 f3 3 is also less than 5 So add one more X 4 f4 again. I can add one more one more X 5 F5 now, I can't write X6 5 f6 why because the limit is put as 5, right? So this is the meaning of summation X I fi similarly summation I is equal to 1 to 5 fi will be simply F1 plus F2 plus F3 plus F4 Till F5 and I can't go beyond 5 right because the upper limit is 5 so this is what is the direct method. So direct method what you need to do How to find out mean by direct method? So if you have a table you have to first get a table So in the table you have to arrange the data You have to find the variables here X I and how many times they are appearing fi, right? And then you write X 1 f1 x2 f2 x3 f3 and so on and so forth Okay, so direct method says X bar will be nothing but summation X I fi from I is equal to 1 to n divided by Summation I is equal to 1 to n fi, right? So here you'll have to make another column another another column and Now put X I Fi so this will be nothing but X 1 f1 This is nothing but X 2 f2 and this is nothing but X 3 f3 now Add all this here. So you'll get summation X I fi you add this column and here you'll get summation fi Right. So what is X bar? X bar is this Okay, so in this case, let us solve this. So hence if you if let's go here, right? This is X I fi let us make the table once again. So the table was let us say this is X I Fi and I'll make another table X I fi So the values were five Three and two and here it was two Four and six. So what is X I fi? Then what is next case 12 and this is 12 right 5 to the 10 3 4 to 12 2 6 to 12, right? Now summation X I fi will be Summation I is equal to 1 to 3 in this case X I fi is nothing but 34 Right 10 plus 12 is 22 22 plus 12 is 34 and what about summation X I summation fi, sorry So this is I equals to 1 23 it will be 12 Isn't it? So what is X bar? X bar is nothing but summation X I fi I is equal to 1 to n divided by summation I is equal to 1 to n Fi which is let us find out these values from the table. So 34 upon 12 which is 2 point 83 Approximately, right? So Yeah, 2.83. So this is the mean. So this is the mean of or the number of average number of medals one in that Competition, okay, this is how this is what is meant by direct method of finding arithmetic mean