 Hello and how are you all doing today? My name is Priyanka and the question says find the mean and variance for the following this frequency distribution. Now here we are given the classes as well as the frequency. Now we need to apply the shortcut method for finding the mean and variance which is divided into four steps. So let us draw a table to make the four steps looking simpler for us. Now here the classes are given to us as 0 to 30, 30 to 60 and so on till 180 to 210. The frequency are given to us as 2, 3, 5, 10, 3, 5 and 2. Firstly we need to find out the Xi which is lower limit plus the upper limit divided by 2. So it is lower limit plus upper limit the whole divided by 2. So on finding we obtain 30 plus 0 divided by 2 will give us 15, 30 plus 60 divided by 2 will give us 45 and then we have 75, 105, 135, 165 and then 195 each having the difference of 30. Now we need to find Yi. Yi will be obtained if we will be subtracting the assumed mean from that. Now from Xi let us take 105 that is the middle term as our assumed mean. Now we need to subtract assumed mean from each Xi. So it will be 15 minus 105 which is giving us 90 and we will be dividing it by H with that is here we have assumed mean that is denoted by the letter E as 105 and we have taken H that is the difference between any two consecutive Xi that is let us take 195 minus 165 that is 30. So we will be dividing Xi minus A by 30. So it will be minus 90 divided by 30 giving us minus 3 and so on 45 minus 105 giving us minus 60 minus 60 divided by 30 gives us minus 2. So we have minus 3 minus 2 minus 1, 0, 1, 2, 3. Now I need to multiply Fi with Yi and it is equal to minus 6, minus 6, minus 5, 0, 3, 10 and then 6. Now find out Yi square and that is equal to minus 3 the whole square giving us 9 minus 2 the whole square 4 minus 1 the whole square 1 and so on. Now again multiply the frequency with Yi square. So it will be 2 multiplied by 9 which will give us 18 then 3 multiplied by 4 12 5 into 1, 5, 10 into 0, 0, 3 into 1, 3, 5 into 4, 20, 2 into 9, 18. Now let us sum them up. Now the sum of Fi Yi the whole square where i is equal to 1 till n is equal to 1 adding all these terms we get 76. Now here we have the summation Fi Yi i is equal to 1 till n the sum is coming out to be 2 and we need to sum up the frequency also as all these things will be used in the formula 30. So we have now the mean as the formula for mean in shortcut method when we are given continuous series is E plus H multiplied by summation Fi Yi i is equal to 1 till n divided by summation Fi and this is what we have found out above. So on substituting the value we know the value of A is taken as 105, value of H is taken as 30 then we have summation Fi Yi as 2 divided by summation Fi that is 30. So on simplifying we have the answer that is mean is equal to 107. Now this is the formula for variance now we will be substituting the values here also we have H as 30 so it will be 30 square upon summation Fi is also 30. So it will be 30 again the whole square into summation Fi is 30 summation Fi Yi square is 72 minus summation Fi Yi is 2 so it will be 2 the whole square. Simplifying we have the value as 1 here so we have further 13 to 76 giving us 2280 minus 2 the whole square is equal to 4 so we have the variance as 2276. So we can write the answer mean was found above as 107 whereas the variance is equal to 2276. So this ends the section hope you liked it have a nice day.