 Hello and welcome to the session. In this session we are going to discuss average. Let us start with the definition. An average is a single value which represents the whole set of symbols and other items concentrate around it. It is also called as measure of sensory tendency. The general formula is the average of n quantities x1, x2, x3 up to xn is given by px root of 1 upon n into x1 raised to the power p plus x2 raised to the power p plus x3 raised to the power p. raised to the power p plus and so on up to xn raised to the power p where n belongs to n the preferred natural numbers and p is an integer. It also satisfies the test of the correctness of the formula as if all individual members are equal. The px root of 1 by n into k raised to the power p plus k raised to the power p plus k raised to the power p plus and so on k raised to the power p n times which is equal to px root of 1 upon n into n into k raised to the power p which is equal to px root of k raised to the power p that is equal to k at any value from infinite integral values therefore range of choice for the definition of an average is also infinite. Now if we take equal to 1 then the formula for the average is 1 upon n into x1 raised to the power 1 plus x2 raised to the power 1 plus and so on up to xn raised to the power 1 whole raised to the power 1 upon 1 which is equal to x1 plus x2 plus and so on up to xn whole upon n this is the formula for simple arithmetic average or mean of n quantities that is mean n is equal to h1 plus h2 plus and so on up to xn whole upon n or we can also write it as m is equal to summation of x by n for simple distribution n is equal to f1 into h1 plus f2 into h2 plus and so on up to fn into hn whole upon f1 plus f2 plus and so on up to fn which can also be written as n is equal to summation of f into x upon summation of f for a frequency distribution now if the value of p is equal to 2 then the formula reduces to that is we put the value of p as 2 in this formula and we get square root of 1 by n into h1 square plus h2 square plus and so on up to xn square this formula is used to calculate standard deviation standard deviation denoted by sigma is equal to square root of x1 minus n the whole square plus h2 minus n the whole square and so on up to xn minus n the whole square whole upon n here the standard deviation is defined as the average of deviations x1 minus n x2 minus n x3 minus n and so on up to xn minus n now p is equal to 2 is chosen as square of the deviation is always positive irrespective of the positive or negative deviation if the value of x is given in centimeters then the value of x minus n the whole square will be in square centimeters and standard deviation sigma which is the square root of summation of x minus n the whole square by n will again be in centimeters now we are going to discuss weighted arithmetic average let us suppose our student scores 55 marks out of 75 in science he scores 32 out of 50 in computers and 90 out of 100 in mathematics to find the average we cannot reckon science mathematics and computers as equal therefore we have to give weight to each of the subject if we give 1 weight to mathematics and the maximum marks is 100 and we give 75 upon 100 that is 3 by 4 to science which is the maximum marks in science and we give 60 by 100 that is 1 by 2 to computers since 50 is the maximum marks in computers therefore we can say the appropriate proportional weights are 1 is to 3 by 4 is to 1 by 2 or we can also write it as 4 is to 3 is to 2 hence we can define it as if there are n quantities that is x1, x2, x3 and so on up to xn and w1, w2, w3 and so on up to wn are their respective weights then the weighted arithmetic average is given by x1 into w1 plus x2 into w2 plus x3 into w3 plus and so on up to xn into wn 0 upon w1 plus w2 plus w3 plus and so on up to wn which can be written as summation of x into w upon summation of w let us take an example suppose there are 5 men, 3 women and 2 boys and their daily wages are 100 dollars per a man 75 dollars per a woman and 50 dollars per a boy then the weighted arithmetic average of their daily wages is here we can say that the value of x1 is given as 100, the value of x2 is given as 75 and the value of x3 is given as 50 and the corresponding values for weight that is w1 is given as 5, w2 is equal to 3 and w3 is equal to 2 then weighted arithmetic average of their daily wages is given by w1 into x1 plus w2 into x2 plus w3 into x3 whole upon w1 plus w2 plus w3 which is equal to w1 into x1 that is 5 into 100 plus w2 into x2 that is 3 into 75 plus w3 into x3 that is 2 into 50 upon w1 plus w2 plus w3 that is 5 plus 3 plus 2 that is 5 into 100 that is 500 plus 3 into 75 that is 225 plus 2 into 50 that is 100 whole upon 5 plus 3 plus 2 that is 10 therefore we have 500 plus 225 plus 100 that is 825 by 10 which is equal to 82.5 therefore the weighted arithmetic average of their daily wages is given by 82.5 which is the required answer this can be for our session hope you enjoyed this session