 We are looking at data set 2 which consists of grain size data from 2 different phases and we are doing the rank based properties and their representation. So let us continue and next what we want to do is to do a histogram plot of this data for both the phases. So this is phase 1, so it should say phase 1 and we want to do a histogram plot of phase 2 data from grains of phase 2. So let us do that and then call it as, so this is the same information we have seen it several times now. So both peak somewhere around 24 and both have a tail only to the left and this is a much longer tail and relatively probably fatter tail as compared to this. So this is what we have been noticing and so you can clearly see here in these pictures how it looks like. So you can do the box and whisker plot also. So we just need to change, let us make it horizontal true, I have to do this again and I have to do it for phase 2. You can clearly see that these data sets are really really different from what we have been seeing. So this point is generally the median value and you can see that in both cases the data point that occurs maximum number of times is actually the last value and then you have tail and here the tail is much longer and here there are lots of outliers you know this is like the box is supposed to be second and third quartile and beyond this is supposed to be fourth quartile there are no data points that is because there is no tail on that side at all and on this side you have one but lots of data points are lying outside of that and here you know everything all these points are outside. So the second and third quartile and there is no fourth quartile everything is sort of collapsed here. So this box plot also gives you more information about how the data looks. So now let us try to get the properties of these data sets. So let us get the mean of the two. So mean is 24.1 and 23.4. Now the standard deviation will tell you this values we have already seen. So you can see that for 1 it is 24.1 plus or minus 0.4 and for 2 it is 23.4 plus or minus 2. So it is not surprising because most of the values are here and the average turns out to be here. Of course the standard deviation is much larger as expected because the spread is much higher. So it is 5 times the standard deviation and variance will be correspondingly different because variance is just square of this standard deviation. So it is 2 times 4 and it is 0.4 and 0.16. So this is just square and so mean and then you can get median values that we already know from the box plots. So in both cases the median value is 24.3 and that is what you see here both cases just gives you 24.3. So quantile again the information is already there from the box plot is getting them in terms of numbers and you can see that the spread is here from 11 to 24.3 and here it is and you can see 50 percent, 75 percent, 100 percent everything falls in this 24.3 and so also here and there is some slight distribution here. I mean in fact here 25 percent onwards everything is here only the first quantile is slightly different value. Here only from the 50 percent so that is the third, fourth quantile they are all of the same value. So this is what we have seen also and as we did earlier of course let us plot everything in one go these plots are useful. So I am going to change this a little bit instead of so let us do this for the second data set also. Second data set you can see one standard deviation 2, 3, 4, 5, 6, etc on the one side because after the on the plus side there is nothing so let us do that. So there is some with the command. Okay. So you can see that we marked the mean and we marked the median and we marked 1, 2 standard deviation etc. But here we have been marking 1, 2, 3, 4, 5, 6 so up to 6 standard deviations you have to go before you actually take all the data into account. So this is really large spread and that is what is shown in these figures also. So you can see that somewhere here this red is median black is mean and about the black the first standard deviation is the green and second is blue in this case you do not even see the blue you see the blue and then this light blue and purple and yellow and I do not know what this color is but so there are several standard deviations from the mean on one side you have to go before you encompass all the data points. So we and this information we have also seen in the case of quantile. So to summarize we have looked at grain size distribution, we have looked at two different cases one was very straightforward we just had the data for one phase because it was a single phase material. In the other case it was a two phase material and we pulled out data and we separated it into phase 1 and phase 2 and we carried out the same analysis the there are three things that we did one is just plotting that is stem and leaf or dot chart or the scatter plot. Second one is to prepare rank based properties that is by doing some analysis like cumulative distribution box plot, histogram plot and things like that and the last one is to prepare summary based reports like mean median standard deviation variance quantiles etc. And of course we tried to put the scatter plot along with the summary based reports to get better handle on how the data looks like. And it is very clear from this data that if you are having something like grain size it is better not just to report the mean and standard deviation but also some information about the distribution and probably the best way to represent the distribution is by giving the histogram plot. And so that is also common histogram or cumulative distribution plot gives an idea about how the data looks like in these cases, how the grain size spread is in these cases. So we will continue with more of the analysis and this example and the previous example is to show that sometimes in materials science and engineering you come across data sets which are to be represented as distributions not just in terms of simple numbers. So thank you.