 Hello friends and how are you all today? The question says the diameter of circle in millimetre drawn in a design are given below. Now here the diameter and the number of circles which we are given, we need to calculate standard deviation and the mean diameter of the circle. So for this let us draw a table. Now we need to find out x i which is lower limit plus upper limit upon 2 and that is giving us 34.5. Then we have 38.5 that is 37 plus 40 upon 2 then we have 42.5 then 46.5 and 50.5. Now here let us assume the mean as the middle value that is 42.5 as e and h that is the difference between the consecutive number of x let us take it as equal to 4. So we will find y i by subtracting x i sorry by subtracting a from x i and dividing it by h that is taken to be as 4. So we have 34.5 minus 42.5 divided by 4 giving us minus 2 then in the same manner we have minus 1, 0, 1 and 2. Then we need to find out f i y i and that is 15 into minus 2 minus 30, 17 into minus 1 minus 17 and so on. Then square of y i that is 4, 1, 0, 1 and again 4 and then 15 into 4, 60, 17 into 1, 17, 21 into 0, 0, 22 into 1, 22, 25 into 4 giving us 100. Now we need to sum them so here the sum comes out to be 99 whereas here it is 25 and the sum of all the frequencies are 100. Now we need to find out mean as well as a standard deviation. So to find out standard deviation firstly we need to find out variance also. So we have their formulas as so we have mean as equal to the summation of f i x i and that is equal to a that is taken to be 42.5 plus h that is 4 into we have 25 by 100. So on simplifying we have the mean as 42.5 plus 1 giving us the answer as 43.5. Now here we have the variance equal to 4 square upon 100 square into 100 into 99 minus 25 square. Let us simplify it further. So we have 16 upon 10000 into 19900 minus 625 that is further written as 16 upon 10000 into 19275 and that is equal to 30.84. Now to find out the standard deviation we need to find out the square root of variance that is under the root 30.84 that is equal to 5.55. So the answer to the session is the value of mean is equal to 43.5 standard deviation is equal to 5.55. Right this completes the session hope you enjoyed it have a nice day.