 Hello and welcome to the session. In this session we will learn about general method for finding moving averages. Now we know that the trend of a set of data is given by moving average. And it shows out the season difference, mothering variations or daily differences. Now moving averages are studied in two different situations. Let us discuss the first one that is the odd period moving average. Now we will learn about the general method for finding out the moving averages for odd periods of the cycles. Now let us suppose the period of the cycles for the given time series is odd. Say when in the first step the values for the first three years are added their arithmetic mean. Now in the second step this mean is written against the mid-year that is the mid-year among the first three years that is the first, second and third year will be the second year. Then in the next step leaving the value of the first year of the next, now the next three years will be the second, third and fourth year. And this mean is written against the mid-year, now the mid-year between the second, third and fourth year will be the third year. So this value will be written against the third year that is the mid-year of this group. Now we proceed likewise until the mean of the last three years is written against their mid-year. Now a new series is obtained that is the series of arithmetic mean of the group of three successive years corresponding to all years except the first and the last year. This is due to the reason that the mean of the first group of three successive years is written against the second year and the mean of the last group of three successive years is written against the second last year and the values of the new series is moving averages. And in the present case the arithmetic means taken over this periods therefore they are called moving averages. Likewise we can obtain five years, seven years and nine years moving averages. So this is the journal method to obtain the odd period moving averages. Now let us discuss the journal method for obtaining the even period moving average. Now if the period of the time series is even say four years which is even then we find the moving average exactly as we have discussed the case of three yearly moving averages. Now here in the first step we will write the averages, the rows between that of the second and the third year and then the third and fourth year and so on. Then in the second step we will take the arithmetic means of the successive periods of the averages which we have obtained in step one. Now in the next step the results which we have obtained in step two are then written against the central year. That is against the third, fourth, fifth year respectively and the new averages thus obtained are called centered four year moving averages. And in either case that is when the period of time series is odd or even the final series moving averages represents the trend. And the moving averages are plotted against the corresponding years and the successive points thus obtained are joined by straight line segments. The resulting polygonal graph gives the trend of the given series. Now let us discuss the procedure for calculating the moving averages. Now let us discuss the case one when the period of moving averages is an odd number. Now let us discuss it for five yearly moving averages. For this first of all we will draw a table for the given data. Now here the year and the values in those particular years are given to us and in the next column we will find the five yearly total and in the last column we will find the five yearly moving average. Now first of all we will find the five yearly total for this add the values of the first five years that is of the first, second, third, fourth and fifth year. So we will add these five values and write the total against the middle year. Now the middle year among these five years is the third year. So we will place the total against the third year. So the total of these values is A plus B plus C plus D plus E and let it be equal to A1. So we have placed this total against the middle year that is the third year. Now in the next step we will leave the first year value that is we will leave this value and now we will add the values for the next five years that is for the second, third, fourth, fifth and sixth year. So we will add these five values and now we will place this value against the middle year for this group. So the middle year for the second, third, fourth, fifth and sixth year is the fourth year. So we will write the total against the fourth year and the total is B plus C plus D plus E plus F and let it be equal to A2. And we will continue these steps until the last year value is taken into account. Now in the next step we will leave this value and we will add the values for the next five years and we will place it against the middle year. So we will write here C plus D plus E plus F plus G that means here the last year's value is taken into account and let this total be equal to A3. And now in the last column we will find the five-yearly moving average. For this we will divide the five-yearly totals by five that is here the five-yearly moving total is A1. So the five-yearly average that is the five-yearly moving average will be A1 by five and for this case the five-yearly moving average will be A2 by five and here the five-yearly moving average will be A3 by five. This means that in the column of five-yearly total we are adding the five values and in the column of five-yearly moving averages we are taking their arithmetic mean. And this last column that is the final series of moving averages represents the trend. And then we will plot these points on the graph that is we will plot the years with the five-yearly moving averages on the graph and then join them to obtain the trend line. So this is the procedure for calculating the moving averages when the period of moving average is an odd number. Now let us discuss the case two when the period of moving average is an even number. So let us discuss it for six-yearly moving averages. For this also first of all we will draw a table for the given data. Now here in the first column the years and in the second column the values in the particular years are given to us. And in the next column we will find the six-yearly moving total. Then in the next column we will find the six-yearly moving average and in the last column we will find the six-yearly moving average centered. First of all we will calculate the six-yearly moving total. For this we will add the values of the first six years that is we will add these six values and we will place this total in between the third and fourth year. That is we will place this total which is a plus b plus c plus d plus e plus f which is equal to let it be a1 in between the third and fourth year. Now we will leave the first year's value and we will add the values for the next six years and we will place this total in between the fourth and fifth year. So we will write it here b plus c plus d plus e plus f plus g and let it be equal to a2. So we will continue these steps until this last year value is taken into account. And now we will find the six-yearly moving average for this we will divide the six-yearly moving totals by six. Now in this case the six-yearly moving total is a1. So the six-yearly moving average will be a1 by six for this case it will be and so on. Now let this be m1, this be equal to m2, this be equal to m3, then m4 and this be equal to m5. And now we have to find six-yearly moving average sentence. For this we will take the first two moving averages and then add them and divide the total by two to get the six-yearly moving average sentence b1. And we will place this value against the fourth year. That is when we take the first two moving averages we will add them and divide the total by two and place it against the fourth year and let it be equal to b1. Moving averages add them and divide the total by two the year that is the fifth year so it will be m2 plus m3 whole upon two let it be equal to b2. And we have continued these steps until the last moving average is taken into account. And this final series that is the six-yearly moving average sentence which we have obtained represents the friend. And now we will plot these points that is the years with the six-yearly moving average sentence on the graph and then the trend line. So this is the procedure for calculating the moving average when the period of moving average is an even number. So in this session you have learnt about the general method of finding moving averages and the procedure for calculating the moving averages. So this completes our session. Hope you all have enjoyed the session.