 Hello and how are you all? The question says find the mean and variance for the data. Now here we are given the x i's and the f i's of this data. We need to find out the mean and the variance. So the formula for mean and variance when we are given the observation as well as the frequency along with it is equal to mean that is denoted by symbol x bar is equal to 1 by n summation f i x i when i is equal to 1 till n right and where n is equal to summation f i where i is equal to 1 to n. Now here the observation are the given values which we call as x 1, x 2, x 3 and so on. We need to find out the variance also and its formula will be equal to 1 upon a n summation f i into x i minus x bar the whole square where i is equal to 1 to n and n again is equal to summation f i is equal to 1 to n right. So these are the two formulas that will be helping us in this question and hence it will be our key idea. Now let us proceed on with our solution for a convenience. Let us make a table now here we have a table in which we have written down the value of x i and f i. Now to find out the mean firstly we need to multiply x i with f i to obtain f i x i. So it will be 6 into 2 that is equal to 12 10 into 4 40 14 into 7 that is giving us 98 18 into 12 giving us 21 6 24 into 8 giving us 192 28 into 4 is 112 and lastly 13 2 3 gives us 90. Now we need to find out the sum of x i f i or f i x i that is equal to 7 60 and we need to find out the sum of f i also and that is equal to 40. So the mean is equal to 1 upon n summation f i x i where i is equal to 1 till n. So on substituting the values we have 7 60 upon 40 as our mean on simplification we have the answer as 19. So 19 is the mean of the question. Then we need to find out the variance also so for that as indicated in the formula we need to firstly subtract the mean from the observation to obtain x i minus mean and then we will find out its square and multiply it by f i. So we have so now we will be subtracting 19 from each x i. So it will be 6 minus 19 giving us minus 13 10 minus 19 giving us minus 9 and so on on subtracting 19 from each and every x i we have the following results. Now I need to square them so it will be x i minus mean the whole square. So the square of 13 is equal to 169 square of minus 9 is 81 here we found out the square of minus 13. Then here we will have 25 1 25 81 and 1 2 1. Now according to the formula we need to multiply the square with f i. So 2 will get multiplied by 169. So here we are finding f multiplied by x i minus mean the whole square. So now it is equal to 338 that is 2 into 169 4 into 81 giving us 324 7 into 25 giving us 175 then 12 into 1 giving us 12 8 into 25 200 then we have 4 into 81 324 and 3 into 121 giving us 363. Now I need to find out their sum that means on adding we have the answer as 1736 right. So the next step is to find out the variance. So we have variance that is standard deviation square is equal to 1 upon n summation f i into x i minus x bar the whole square where i is equal to n 1 till n. So on substituting the values we have the summation of f i into x i minus x bar the whole square found out above as 1736. So we have 1736 divided by the total of frequency that is 40. So we have the answer as 43.4. So in the end we can conclude the session by writing down the answers that we have obtained. So it is mean is equal to 19 whereas variance is equal to 43.4. So this completes the session hope you understood the whole concept well and have a nice day ahead.