 We know how to compute mean when numbers like this are given to us and I am just writing few random numbers and if we wanted to calculate the mean of this, we would sum all these numbers and divide the sum by the number of observations that we have. So, it would simply be like 2 plus 3 plus 4 plus 5 plus 2 plus 3 plus 4 plus 5 and then I would simply divide it by 8 and this would give me the mean. But we can see that 2 is reappearing, 3 is also reappearing, 4 and 5 both are reappearing. So, now how we compute mean sometimes depend upon how is the data set given to us and what we want to study is when the data set is given in a way that we know the frequency of each observation occurring. So, for example, I just create a simple table, I will write observation and below that I will write frequency. So, if from the earlier data set we had 2, 3, 4 and 5 as the observations and each of it was occurring 2 times. So, the frequency of each of them was 2 and in such cases how do we find the mean? So, to compute mean in such cases, we multiply the observation by its frequency. So, 2 times 2 plus 3 times 2 plus 4 times 2 plus 5 times 2 divided by the total sum of frequencies and total sum of frequencies is basically 2 times 4 which is 8 and this will give us the mean in this case. Let us take a real life example. Say we have a data of unit test marks of few students. So, the test scores for some students are 8, some students got 7, some got 6, 2 and 10 and for each of those scores we have frequency of how many times is that score appearing that actually means 3 students have got 8 marks, 4 students have got 7 marks and so on. So, frequency shows number of students who got that many marks and in this case if we wanted to calculate the mean marks, what we will do is that we will multiply that test score by the frequency of it and we will sum all such cases. So, 8 times 3 plus 7 times 4 plus 6 times 1 plus 2 times 2 plus 10 times 2 divided by the total number of students who appeared for the test or the number of students for which the data is available here. So, number of students is basically the sum of frequencies and the sum of frequency is 3 plus 4 plus 1 plus 2 plus 2 that is 12 and therefore, the mean marks is equal to 24 plus 28, 6 plus 4 plus 20 divided by 12 and this is nothing but 82 divided by 12 and the mean comes out to be 6.833. Now, why is this method a little different? We could have written this data of marks like this 8 comma 8 comma 8 because 8 is appearing 3 times and then 7 appearing 4 times but this just gets tedious and people are lazy they don't want to write the data that is repeating and there is a way around it. So, when we represent the data, we just make a table and write down all kinds of observations we have and just write the frequency of it which just enables us to visualize data better and it also just changes finding the mean a little bit although the fundamental concept of finding mean is the same. We are just summing up all the scores and dividing it by the total number of observations. Note that the observations are usually represented as x i, this i refers to the position of the data. So, when I say x 1 it is 8 or the observation appearing at the first place and so basically this is x 1, this is x 2, this is x 3 and so on and the frequency represented as f i. So, what we have done in this formula is basically we have sum up, so this is a summation sign we have sum up the multiplication of x i and f i and we have divided it by the summation of f i or all the frequencies. So, this is how we calculate the mean when we have data given in table of form with frequency. We could also get the data like this and we can still use the same method. This formula works well when we have data given with frequencies and note that this data is an individual data, this is not a group data, so it is important to note that this is an ungrouped data. So, we just learned how to calculate mean for the data set given which is ungrouped and frequencies are given to us.