 Hello friends, welcome to the session of statistics and after Having studied the direct method in this method. We are going to find out the arithmetic mean of a given data Whether it is in the form of discrete data or in a class interval form We will be using something called assumed mean method Assumed mean method to find out the arithmetic mean of the data Now, why do we use this method is in earlier days when we didn't have calculators and computers It was you know and the data Let's say the value the variable values were quite large Then it would it would become very difficult and cumbersome to do calculations So hence just statisticians and mathematicians came up with a technique of reducing the you know calculation load By this assumed mean assumed mean method. So hence Let us understand what this assumed mean method is before that Let us first have a recap of what was our direct method. So if you remember our direct method of Finding the mean was x bar was given as summation fi x i divided by summation fi Right. So this was i is equal to 1 to n observations. Let us say we had i is equal to 1 to n Observations we had n observations in table like for example, it was x i and there was fi And then we used to calculate x i fi or fi x i whichever way you want to call it And let us say this was x 1 it was the frequency was f 1 then x 2 Then the frequency was f 2 x 3 and the frequency was f 3 and so on and so forth We had x n was having a frequency of f n and hence we calculated x 1 f 1 then x 2 f 2 multiplication of these two quantities x 3 f 3 and So on and so forth till x n f n Correct, and then we used to sum this column and we used to find out summation x i fi and this is from i is equal to 1 to n and This column also was summed up and it was called summation fi from i is equal to 1 to n And hence we used to get this particular relationship for mean of the data Now what we are going to do is let us say if x these x's are quite large Okay, or even for that matter f 1 is quite large Then this multiplication became becomes very cumbersome right multiplying each one of the x's With the f becomes very very difficult So hence we came up with a technique so that we can reduce the magnitude of these values So that our calculation becomes easier. So that is what we use in a zoomed mean method So what do we do? Let us see a table So you can clearly see x i's are quite large to 40 to 56 300 320 350 and if you would have done the direct method then we would have Ended up with doing all these calculations and then finally summing it up So it looks a little intimidating. So what I'm going to do is I am going to subtract a constant number Which is called the assumed mean which I have taken here as 300 now You can take any value of a here. It's it's not necessary that one of the values of x i's Only half has to be taken as a you can take anything which reduces your Calculation load so it has to be balanced so that if you take up a number Let's say thousand then again it becomes another Big cumbersome multiplication Process why because if you subtract everything from thousand or all these numbers minus thousand if you do again, there will be big Numbers so instead of that we take somewhat middle value Which is which looks like it would the mean would be somewhere near there So we take that value so hence it is called assumed mean so we are assuming that the mean is somewhere around 300 Okay, so and it could it can be you know You can find the value of assumed mean by just having a look on the data the type of data So if you see here, this is 240 to 350 it is varying So somewhere around 300 the mean must lie so hence I am taking 300 and another reason of taking 300 is that you know subtracting any number From 300 or subtracting 300 from any number will also be very easy to Calculate so if you can see now what I did I subtracted this 300 that is my assumed mean Through from all the x i's so if you see 240 minus 300 we got minus 60 Then minus 44 0 20 and 50 can you see one one of the values is zero so hence now multiplication by Zero will also be easier Now this is called the eyes. Yeah, the deviations from The individual x values how what is the difference between the individual x values and the assumed mean? so now if you see the numbers which you have Achieved are relatively smaller than where we started Right, so it is easier to multiply 60 with 7 than 240 with 7. Okay so hence now the next step is multiply these di's which you got with the Corresponding frequency values that is minus 60 has to be multiplied with 7 here and so on and so forth So if you do this you will get these values then you'll have to just sum them up Some the summation di fi so you'll get a value of minus 1 1 2 and the formula for x bar is x bar is equal to Assumed mean plus summation fi di divided by summation fi Now the question arises. Where do we get this formula from? So let us look at it now? If you see what are we doing? We are saying x i is equal to or Or rather we are saying di Di is equal to nothing but x i minus assumed mean that's what we have stated here. Can you see this? di is equal to x i minus a Isn't it so hence I can say x i will be equal to a plus di just rearranging this equation That means now what is x bar? Let us find out x bar from our direct method. We know x bar is summation fi x i i is equal to 1 to n divided by Summation fi again from i is equal to 1 to n right now Let us substitute this x i with whatever we have found out over here So I can say summation again i is equal to 1 to n and Fi stays as it is and instead of x i I will write a plus di Okay, and then divided by summation Fi from i is equal to 1 to n Okay, now there is a rule Friends that if you know you can simplify this summation. How so summation i Is equal to 1 n you open the brackets and you write you can write f1 a Plus sorry fi a and fi di Okay, this is within the summation and then summation i is equal to 1 to n Fi now There's a rule that if Summation a plus B is there. Okay, so summation a let us say i is equal to 1 to n is same as summation a from i is equal to 1 to n plus summation B from i is equal to 1 to so I can distribute the summation. Yeah, if you Are confused. Let us take an example So, let us say I had Summation let us take an example just to understand this so that it becomes clear for you just to understand this Let us take an example How so let us say we had summation i is equal to 1 to n xi plus yi What does it mean according to the definition of summation it will be x1 plus y1 Plus x2 plus y2. So I will vary next then x3 plus y3 so on and so forth Till xn plus yn. This is what is summation xi yi is Now can't I club all xs together so x1 plus x2 plus x3 plus so on and so forth till xn plus y1 plus y2 plus y3 plus so on and so forth till yn Right, so the left-hand left You know term here is nothing but summation xi from i is equal to 1 to n Plus summation yi from i is equal to 1 to n so this Proves this particular rule. Okay, so if this rule is correct then Let me just insert some space here. Yeah, so now this was just to illustrate The rule of summation. So hence now what I'll do is I will insert some more. Yeah, now Let us let us carry on from here. So hence I can write this as summation i is equal to 1 to n fi a plus summation fi di from i is equal to 1 to n divided by Summation fi from i is equal to 1 to n Now guys a was a constant. This value is a constant now. Let us give me let me give you another another rule so another rule is Rule is what is the rule with the summation summation. Let us say if you had Xi into constant, let us say c Constant term. Okay, and i is equal to 1 to n, right? So what will with this value? This is nothing but x1 into c plus x2 into c plus x3 into c plus so on and so forth till xn into c So this will be nothing but you can take c common and you can write x1 plus x2 plus dot dot dot till xn Which can be written as c times Summation xi i is equal to 1 till n So what do I mean? I mean that if there is a constant term then it can be pulled out of summation sign So if you see here a is also a constant term So it can be very much pulled out of the summation sign. So hence it can be written as a summation i is equal to 1 to n fi plus summation fi di i is equal to 1 to n and this was whole divided by summation summation fi from i is equal to 1 to n Now what we can do is What we can do is we can now split this fraction into two So we can written as it can be written as summation a i is equal to 1 to n fi divided by summation i is equal to 1 to n fi plus summation fi di divided by summation fi both from i equals to 1 to n and here also i equals to 1 to n Okay, so if you see this this term So these two are same Same so I can cancel that right and this is not zero summation of fi is not zero So I can cancel it. So it will become a plus summation fi di i is equal to 1 to n divided by summation i is equal to 1 to n fi So this is what the formula was all this was coming from where from here We started with x bar. So x bar is now when we did all this calculation We saw x bar is so hence I can summarize here and right if you're doing assume mean method So x bar will be assumed mean plus summation fi di i is equal to 1 to n and Then not whole divided by sorry this the second term divided by Summation fi only yes of a summation fi Yeah from i is equal to 1 to n n Right, so I have calculated the data in this table the mean of the data in this table and I have found out That using this formula. I am getting 295.85. You can try with the direct method You will get the same result same mean will be obtained Okay, so this is what is called assumed mean method. There is one more method called step division method Which is a further step in in in the direction of reducing the calculation complexity So we'll see that in the next session. Thanks guys