 Hello and welcome to the session. The question says, the mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On retaking it was found that observation 8 was incorrect. Calculate the correct mean and standard deviation of each of the following cases, first one is if wrong item is omitted and second is if it is replaced by 12. First let us learn the formula to calculate the variance of n observations is given by summation xi square i running from 1 to n upon n minus x bar square. So this formula is the key idea which we shall be using in this problem to solve it. Now let's start with the solution. Here n is equal to 20 since we are given 20 observations. The mean of these 20 observations which is denoted by x bar is given by summation xi i running from 1 to 20 upon 20. Now we are given that mean as 10. So we have 10 is equal to summation i running from 1 to 20 xi upon 20 which further implies that summation i running from 1 to 20 xi is equal to 20 into 10 that is 200. Also we are given the standard deviation which is denoted by sigma is equal to 2. Now the formula to calculate the standard deviation is root over summation i running from 1 to n xi square upon the number of observations which are 20 minus 10 square. Now square in both sides we have 4 is equal to summation i running from 1 to n xi square upon 20 minus 100 which further implies that summation i running from 1 to n xi square is equal to on taking 100 on the left hand side we have 104 and then multiplying 104 with 20 we get 2080. Therefore summation xi square i running from 1 to 20 is equal to 2080. So let this be equation number 2. Here also n is 20. Now the first part is if wrong item is omitted. If the wrong item is omitted then the number of observations are now 19 and the correct mean is given by summation xi i running from 1 to 20 minus 8 since 8 is omitted which is the incorrect observation upon now the number of observations are 19. Now summation of xi i running from 1 to 20 is 200. So this is further equal to 200 minus 8 upon 19 and this gives 192 upon 19 which is equal to 10.11. Also the correct variance is equal to summation i running from 1 to 19 xi square upon 19 minus 10.11 square which is by our key idea. Now here the number of observations are 19 if one observation is omitted and this is the correct mean. Now can be written as summation i running from 1 to 20 xi square minus the observation which is omitted 8 so 8 square upon 19 minus 192 upon 19 whole square and from equation 2 the value of summation i running from 1 to 20 xi square is 2080 so we have 2080 minus 64 upon 19 minus on simplifying this bracket we have 36864 upon 361 so this is further equal to 2016 upon 19 minus 36864 upon 361 which is further equal to LCM is 361 so multiplying 2016 upon 19 we get 38304 minus 36864 gives 1440 upon 361 which is equal to 3.98 and thus the correct standard deviation which is the square root of variance is equal to 1.99. So if one incorrect item is omitted that is 8 then the correct mean is 10.1 and the correct standard deviation is 1.99 so this is the answer to the first part and now let's proceed on to the second part which says if the wrong item is replaced by 12 so here again the number of observations are 20 and the correct mean in this case is given by summation xi i running from 120 minus 8 which is the incorrect observation plus it is replaced by 12 so plus 12 upon the number of observations are still 20 since 1 we have omitted and 1 we have added. Now the value of summation xi i running from 120 is 200 so we have 200 minus 8 plus 12 upon 20 and this is equal to 204 upon 20 which is equal to 10.2 and now let us calculate the correct variance let's formalize summation xi square i running from 1 to 20 upon the number of observations which are 20 minus 10.2 square which is the correct mean now this can be written as summation i running from 1 to 20 xi square minus 8 square plus 12 square upon 20 minus 10.2 whole square is 104.04. Now summation xi square i running from 1 to 20 is 2080 minus 64 plus 144 upon 20 minus 104.04 and this is further equal to 2160 upon 20 minus 104.04 which is equal to 108 minus 104.04 and this is equal to 3.96 and thus the correct standard deviation will do root over the correct variance which is 3.96 and this is equal to 1.98. Hence our answer is if it is replaced by 12 then the correct mean is 10.2 and the correct standard deviation is 1.98. So this completes the session hope you have understood it well take care and have a nice day.