 Hello and welcome to the session the question says the mean and the standard deviation of six observations are eight and four respectively. If each observation is multiplied by three find the new mean and new standard deviation of the resulting observations. Let's now start with the solution and let the six observations x1, x2, x3, x4, x5 and x6. Now we are given that mean of these six observation is eight. It is denoted by x bar and its formalized summation as I goes from 1 to 6 xi upon x. So this implies we have 8 is equal to x1 plus x2 plus x3 plus x4 plus x5 plus x6 upon 6 or x1 plus x2 plus x3 plus x4 plus x5 plus x6 is equal to 48 which we get on cross multiplying. Let this be equation number one. Also we are given that the standard deviation of these six observation is equal to 4. Thus we have 4 is equal to the formula to calculate the standard deviation of six observation is root over variance and variance is summation as I goes from 1 to 6 xi minus x bar which is the mean and it is 8 whole square upon the number of observations that is 6. Now on squaring both the sides we have 16 into cross multiplying then 6 is equal to summation xi minus 8 whole square I goes from 1 to 6. So this implies summation I running from 1 to 6 xi minus 8 whole square is equal to 96 and let this be equation number 2. Now the question says each observation is multiplied by 3 find the new mean and new standard deviation. So on multiplying 3 all the observations new observations are 3x1, 3x2, 3x3, 3x4, 3x5 and 3x6. So the new mean of these six observation is equal to summation as I goes from 1 to 6 yi upon 6 where yi is equal to 3 times of xi. So we have 3x1 plus 3x2 plus 3x3 plus 3x4 plus 3x5 plus 3x6 upon 6 taking 3 common inside we have x1 plus x2 plus x3 plus x4 plus x5 plus x6 upon 6 and from equation number 1 the value of x1 up to x6 is 48. So we have 3 into 48 upon 6 and on simplifying we get 24. Hence the new mean is equal to 24 and the new standard deviation is given by under root summation yi minus 8 whole square i goes from 1 to 6 upon 6 where yi is 3 times of xi. First we just make a table. So this is the required table. First column denotes the values of yi. Second column denotes yi minus 24 where 24 is the new mean sorry here we have to calculate the new standard deviation we have to use the new mean which is 24 and in the last column we have calculated the values of yi minus 24 whole square. Now we have to find summation i going from 1 to 6 yi minus 24 whole square. So let us add to get the values of this summation. So this is equal to summation i going from 1 to 6 3 square into xi minus 8 whole square. Now from equation 2 we have summation xi minus 8 whole square i running from 1 to 6 is equal to 96. Therefore value of this summation summation i running from 1 to 6 yi minus 24 is equal to taking 3 square outside the summation we have summation i running from 1 to 6 xi minus 8 whole square and its value is 96. So we have 9 into 96 and this is equal to 864 and thus to calculate the new standard deviation let us substitute the value of summation i running from 1 to 6 yi minus 24 whole square the new standard deviation is equal to square root summation i running from 1 to 6 yi minus 24 whole square upon 6 is equal to square root of 864 upon 6 which is equal to square root of 144 which is equal to 12. Thus the new mean and new standard deviation of the resulting observations that is when the given observations are multiplied by 3 are 24 and 12. So this completes the session hope you have understood it take care and bye for now