 Hello, when how are you all today? My name is Priyanka and I shall be helping you with the following question. Here the question says, find the mean and standard deviation using shortcut method. Now before proceeding on we should be well versed with the shortcut method for finding mean and standard deviation. First of all the formulas will be equal is x bar is equal to a where a is the zoned mean plus h multiplied by in the numerator summation f i by i where i is equal to 1 to n and in the denominator we have summation f i where i is equal to 1 to n. Now here what is a h and y will be more clearer to you once this method will be used by me in this question. Now let me write down the formula for variance also that is, now here we do not need to find the variance but for finding out the standard deviation also we need to firstly find out the variance and then we can only find out the standard deviation. So the formula is h square upon summation f i i is equal to 1 to n the whole square into summation f i into summation f i y i the whole square i is equal to 1 to n minus summation f i y i the whole square. So this formula looks a little bit confusing but as we will be solving it with the help of our table method and the shortcut method it will be easier for you to understand it. And lastly we will obtain standard deviation that is equal to square root of variance right. So these three formulas will be helping us in the following question and hence they are the key ideas towards this question. So let us firstly make a table in which we will write down the x i and f i's which are given to us. Now here we have x i and f i. Now in this shortcut method we will assume any observation from x i as the assumed mean. Let us take 64 as our assumed mean let us denote it by a right. And h is the difference between any two consecutive x i and that if you observe in each of the case is equal to 1 right. So what we have done over here is we have assumed mean that is a equal to 64 and h is the difference between any two consecutive x i's and that is if you subtract anyone that is 66 minus 65 let us say it is coming out to be 1. So here we have a is equal to 64 and h is equal to 1. Now we will find out y i y i is x i minus the assumed mean. So it is 60 minus 64 giving us minus 4 then 61 minus 64 minus 3 and we have minus 2 minus 1 0 1 2 3 and then we have 4 that is 68 minus 64. Now I need to multiply f i with y i and that is 2 into minus 4 minus 8 1 into minus 3 minus 3 12 into minus 2 minus 24 and so on we have the following x i f i's. Now with the help of shortcut method you are actually simplifying your calculations further we need to find out y i the whole square also. So it will be the square of all y i that is 16 then 9 4 1 0 1 4 9 and then again 16 then multiply y i with y i whole square also and on doing so we have 2 into 16 giving us 32 1 into 9 giving us 9 then we have 12 into 4 48 and so on 29 0 12 40 36 and 80. Now we need to find out the sum of f i y i the whole square that is 286 then the sum of y i square that is 60 the sum of y i f i is will be 0 and lastly the sum of f i which is 100. Now we will be finding out first of all meal that is y down the formula that is a plus h into summation f i y i i is equal to 1 till n divided by summation f i i is equal to 1 till n now here 8 is 64 h is 1 it will get multiplied by this summation of f i y i was found to be above as 0 so it will be 0 divided by 100 so this whole term will become equal to 0 so we have the mean as 64 now proceed on to find out the variance now this is the formula which is known to us now on substituting the value we have h as 1 summation of f i as 100 so it will be 100 the whole square into summation of f i is 100 will get multiplied by summation f i y i the whole square sorry square and it is found out above as 286 isn't it minus summation f i y i was 0 so its whole square will again give us 0 simplify it we have 1 upon 10000 into 286 00 now simplifying it we have the answer as 2.86 right and don't forget you need to find out the standard deviation so standard deviation is under root the variance variance is 2.86 so its square root is 1.69 so we can find the answer to the session as mean was found out to be 64 standard deviation is 1.69 right this completes my session hope you understood the whole concept well and also the formulas that were that were explained to you in the starting of the session have a very nice day ahead take care of your calculations