Lecture 33: Descriptive Statistics

[Music] [Music] hi everyone welcome back again to the nptl online certification course on research methodology for planning and Architectural studies in our last lecture if you recall that we discussed about interpreting data there I mentioned about something about mean median while describing the box plot so some of you probably got it very right and maybe some of you are having certain doubt about uh those terminology who are just doing it for the first time understanding this kind of Concepts so for that definitely we'll be discussing it in detail in this current lecture so the current lecture is titled as descriptive statistics lecture number 33 so what we are going to learn out of this particular lecture they are as follows we'll discuss about the role of descriptive analysis in research and how researcher can get some firsthand insight about the information or the data collected through survey through interview or through some kind of experimental setup in that case there are certain um you know typology or you can say certain typologies where uh we are going to measure the frequency of certain variables like it may be just mostly you can say that uh you know uh frequency of certain score okay exam marks or maybe the age group or like you can say the frequency of male female in a particular sample or else like it can be something where we also need to understand the measures of central tendency so out of all informations what kind of uh you know uh central tendency means what kind of average value that we are looking for or whatever the average uh value for the entire data set that we can get it from this kind of measures of central tendency so normally measure of frequency is very much useful for some sort of you can say the ordinal data that we discussed earlier in detail or maybe it is good for nominal data okay then also we'll discuss about the variability in data so either like in the box blog we have seen that how it is variable like from different kind of percentile like for 25th 50th which is considered as a median for the box plot so those kind of uh you know measurements are also there as descriptive statistics so basically it comes from description of the data and there are different means of doing it and definitely it is going to help you to you know uh give some sort of firsthand insight about the entire data set before you really go for further analysis and do some sort of you know uh Advanced uh computational thing or maybe statistical method that you will apply to bring certain useful information and then finally we'll summarize our learning at the end of this presentation now uh what is descriptive analysis so descriptive analysis basically summarizes Collective information or data to understand the patterns and Trends now if your collected information is showing that based on your data simple data collection in a particular City or in a particular Village that maximum say maybe 40 8% out of 50 they are youth population they're ranging in a youth group so maybe considering say say you can say 20 to 4 that particular age that where you are going to get the highest number among like other age group like above 60 or below 20 so this is some sort of you can see that uh kind of frequency or else it can be a pattern that okay when you study two variables together in the previous uh example like in the previous lecture we have taken one that it's all about you can see about the number of hours of study and then the score so those kind of relationship or maybe you can take other like it maybe you can say that you know population density and then the water consumption or in such cases like depending on your uh research depending on the variables that you have measured you can take up uh certain kind of representation and you can identify the those kind of descriptive statistics means or measures to get certain firsthand idea it is very much critical in making informed decision based on available information right so in in a particular sample if you simply get that okay so there are certain sort of distribution where you can say that uh maybe you can say 30% uh 20 this is 30% and then 40% these are the different distribution of the age group okay and then area of interest that this particular 30% again you can go for further dividing into among 30% there is a split of say 50% 50% of male and female so maybe if it is something the age group below 20 years and then you can do further uh analysis or you can bring such such kind of pie chart to represent their uh you know distribution as well so this is one way you can do it or else you just simply make a plot and where you can say that okay this is something where this is 30% this is 30% and then this is 40% okay the data representation may be different that we'll discuss in our next lecture that what type of data charts and other combination you can use for a better Clarity of your descriptive statistics or any statistical inferences that you can make through those graphs tables but for this time being so we can take help of different graphs to represent this kind of descriptive statistics so normally whenever you collect any uh information through experiment there are also have to give those kind of data distribution and if you have collected information from a field survey respondent survey so you have to give a respondent profile so normally male female or maybe you can say other gender age group car ownership okay income level etc etc so that you know normally we us to give a table or maybe we can represent with a p chart or a graph whatever can fit better for your understanding and for you know reading ibility of the research article that you are writing but they are going to give some basic insight about the overall data set that whether the data is too much like uh you know bias or maybe that particular information is not representing the population say in case that uh in a particular sample you will find that uh maybe it is the case that where you get 70% male and 30% female but actual the population is something different so in such cases like you can always take a look and while the survey is on you can also have the look simultaneously so that you can uh change your sampling strategy if you find that okay in my case is a stat stratified sampling and where I want to represent male female almost equal so maybe you can stop uh doing survey for the mail respondent and you increase the number for the a female that way also you can do so this kind of bar chart simple uh you know descriptive statistics can also help you in taking some decision in your survey process and the post analysis of collected information coming to the importance in architecture and planning so definitely say for example if you want to understand the historical Trends so understand architectural Trend and arban development patterns so say in this particular cas is like if this is some kind of you know uh region where you want to study and probably if you have a temporal data set so probably will find that maybe the initial settlement was something like this and then gradually it expanded over time right and then it expanded on that side even further so in such cases you can always have those kind of descriptive statistics like the person percentage of build form and the settlement where probably in that particular boundary it was as low as a 10% now maybe in this case it is going for the 20% maybe it is 40% so there is a change and increase over time you can relate them that in 10 years 20 years or then you can say that probably this is something in 10 years the growth was only 10% but next 10 years it is almost the Gap is multiplying so this is something you can make it will also help for this inform decision making it aids in material selection environmental impact assessment and research allocation for any such kind of infrastructure projects or any uh building so this is something you know where you can see the distribution of materials or the cost Etc design efficiency definitely through this you can also optimize the space space planning and usage based on past uses of the data and Trends so if you have those kind of information you can understand that okay for such kind of location where you have designer sitting like this and then another options that somewhere you have a seting like this so in both the cases like how people are like interacting with those spaces you can take a call and then through that the percentage and all those simple values can take or give you some sort of uh Direction like what to take uh ahead and you know what kind of changes that you require for your uh space planning now coming to descriptive analysis so most predominant way of uh doing this analysis one is the measure of frequency so in this particular case we can use histogram that you know in the previous lecture we have discussed it right where we simple frequency can give you those kind of information so where you will see that okay there is some kind of value where it is having a maximum number of uh you know observations they are marging at that particular time so this is one and again we have also discussed about the uh Steam and Le thing so where you know there we have taken those kind of age group example and then we can write the variables here this is one next is measures of central tendency so when you plot the histogram next stage is that you can try to identify that what type of distribution they have so where exactly it is having those kind of you know you can say the mean value or it is something like mean median and mode so it is all about like looking at the data point and where exactly like the average value or you know something in between value the middle value that you can uh do in this particular measure of central tendency now coming to the measures of dispersion and variation so here also you can uh find out certain kind of you know values where you know you you can see those kind of variation where you have the data say for example this is something where you get exactly it is a normal only distributed data but it may happen that it is not like that and there is some kind of skewness so either it is uh like skewed on the left hand side or maybe it is something on the other side side so in all those cases you find that there is some sort of you know variation That You observe in the data set so maybe if it is a it is B it is C so here you have certain kind of left side skes or you know shift and here you can get something on the uh right side qess so then in that case there is a shift of those kind of like you can say the mean value and all so whenever is perfectly fit so your mean median mode will be equal so we'll discuss that now measures of position is also important in this case what we are going to see it's about where your data lies whether it is in that particular you know at the mid value so individual data where it lies so it is like 50th percentile this is 25th and this is 75th and then probably the other values are on that in and then you can find out some sort of outlays measures of frequency so this is something you can see that measures of frequency are statistical measurement that professionals use to summarize the values in a data set and make them easier to understand so this is something a simple frequency where it is a count of number of tourist so for that definitely here uh out of all then it was found found that 182 number of tourist they are male 235 female and then also we have 27 in other category so where it is just something like you can say a nominal data right where this male female and other thing they are not having any sort of uh value difference or something so they are just some name so they are all categorical data and nominal otherwise you can also have this kind of grouped frequency so in such cases maybe you can collect information with the actual age and then you just make them in a group so age group 0 to 5 5 to 18 18 to 40 40 to 60 and 60 and above and out of total sample of the population you calculate the percentage if you add up all of them it will give you 100% but from here it is very obvious that 42% is that age group where 18 to 40 that you know that I just referred earlier and also 40 to 60 is considerably similar as like 5 to 18 but where the child population and inly population they are compared to the others they are low as low as 6 and 8% so this is hypothetical definitely in your case this may change but normally we can represent in this kind of group frequency so I can give you another uh you know example of group frequency suppose you are doing a particular uh speed survey spot speed survey through radar gun so you want to measure the instantaneous speed of any vehicle passing a particular point so and You observe 100 vehicles and their speed so you may have all and 100 values those are dis value so somewhere you may get that it is 70 if I take the round figure 70 80 okay in in a particular express way context I'm just mentioning so you can have any values like that so one way it's something like you can just take the average you can take the summation and then you can take the average or else in order to show it in a frequency you know uh table so you can always make a chart that okay less than 60 kilom per hour 60 to 80 80 to 100 and above 100 and then you make a count and then write the frequency suppose you get certain number and then from there you can also calculate the percentage so in this case it is percentage these are not actual value to get the actual value you just need to know the total population that you have surveyed you must multiply with that you get the exact number right hope this is clear to you we'll move forward to this particular slide where measures of central tendency so there are three main measures of central tendency mean median and mode okay we'll learn one by one so what is mean this is average of a data set calculated by adding all numbers together first you have to make a summation and then dividing that particular sum by the number of total count of those num so say for example if you have 10 observations 10 values so first you have to add all 10 values so that's why it is the summation you can say this is if it is stain for your case uh so there should be you know you can say X1 + X2 + X3 like this and then you finally have X1 value and divided by 10 so this is simple mean now in this particular case if it is for any number so say we can represent this is i1 to n and this is I and N is the total number so in this case definitely it is something like capital N so we have all the numbers so start with first value to the last value and we can divide it with the total number so this is also referred as a arithmetic mean now there is another alternative to this which is called geometric mean so in geometric mean it is something different so in geometric mean this should be calculated in a different manner so in this case you have to make product of all these values and then you have to divide you have to go for uh n root if there are total n number of uh observations so we have to go for n root so if there are only two data point so in that case it will be simple like square root of the product if it is like three so then cube root if it is 10 so 10th root of this product so in both the cases the values will change little bit but overall pattern will be same right which will be higher that will be higher which will be lower that will be lower in both the cases now take up one example uh for the mean say city planners they want to uh you know know average Building height in a particular District to inform zoning loss aesthetic cohesion and infrastructure need hypothetical data in this case suppose we have a small district with following building Heights normally they are varying with 15 M 12 M 18 M 20 M and 14 M so then calculation is the mean height is summation of all of them and then divided by five because this are five data points so 15.8 so the interpretation that this information can be used that okay 15.8 is the mean height of any building in that particular studied area so this will help to guide planning decision for instant to determine the new proposed building whether it is sign significantly taller or shorter than the average and how it might fit into the existing Skyline so this is just one example of calculating mean coming to median in this case the median is the middle number in a shorted list of numbers and then the shorting of the data can be done either by ascending or descending order so here we are not calculating or adding up and dividing it so we try to find it out say for example if you have simple number like 1 2 3 4 5 this is already sort it so in this case we just uh take the middle value so in this case it is odd number so the middle value is three so I can say that from 1 to 5 if these are all discrete value 1 2 3 4 5 so three is the median but case will be little different when you have you know uh some sort of uh even number so for that what we need to do as because whenever we say middle so any odd number the middle value is very easy to find because you have exact number on left hand side and exact number on right hand side so this is something you can say n by 2 N by two but for even number you have to do this kind of calculation where you have to take the value of n by 2 and then n by 2 + 1 that particular term and then this particular two things to be added and divided by two so say for example in this case if you have 1 2 3 4 5 and six right so in this case if you say the half of the number so here you have this particular thing where it is you can say say that three and four these are two numbers because this is one particular half and then it is the next you know + one so in for that particular case the median for this series will be 3 + 4 by 2 so this will be 3.5 now for uh the median of OD number it is very simple so in this case also you can say that how to calculate it is uh very easy so for that we have one to five so 5 + 1 divided by 2 so this is three right now example an architect may want to understand the typical number of floors in a residential buildings within a specific area to guide design standard or inform policy decisions for those kind of building code and all hypothetical data Maybe the number of flows in a small sample of residential buildings is 5 3 6 12 4 7 5 so first we have to arrange the data in ascending order so 3 4 5 6 7 12 so in this case also you have odd number right so odd number it is very easy to calculate so with uh the seven buildings in the list the middle value is nothing but the five right so this is something you can um easily calculate the U you know the median number of uh floats in this particular case is five so we can take a call on that so now coming to uh the next one that is the mo so this is something where the mode is the value that appears most frequently in a data set if it's nothing the average or nothing the middle number but most frequently appears a data set may have one mode only one particular value having the highest number of occurrence or frequency or that is also called as uni model or you can also have more than one mode like bu model where you have two mode in the data set or it can be multimodal or in some cases there may not be any mode at all if there is no reputation say find the mode of this particular data set where 2 3 5 5 7 9 9 9 10 12 they are there so in this case you can say that 2 3 7 10 and 12 they only occur once where five occurs twice and nine occurs three times so you can simply say that as because it is three in numbers the frequency the nine is actually the mode of this particular data set if separ for example there is another five so then it will be five and nine example an architect might be interested in a district most commonly used building materials to understand the local building Trends resource utilization and Architectural Heritage the building material used in a sample houses are brick wood brick stone brick concrete wood brick so in this case as because brick is appear most like 1 2 3 four times out of all the samples so then it is something you can see that break is that particular thing so this is just one example from just kind of you know context uh where you can make the count but definitely the mode is again a mathematical operation so where you can get any values as well right so that you can take into account coming to the measure of central tendency in this case also you can see that whenever you have such kind of mean median mode so they are very much symmetrical so both the side of mean you have similar uh distribution where you have some sort of you know skew on the left hand side so it is negatively skewed where you have skew on the other side you have the positive skew so then obviously the mean will be something greater than median and greater than Mo so this is the common uh assumptions now coming to the variability in the data in this case uh what we need to do is uh basically uh look at the range look at the variance look at the standard deviation right so in this particular case as we you know discussed earlier that the moment uh you have some sort of data dispersion uh in this particular variability so then you have certain sort of uh you know issue like definitely in this case there are certain more deviation that you can see where data is dispersed and probably if it is too much skewed so the deviation is not that much so let us understand that here the range is the difference between the highest and lowest value in a data set so here if the height is concerned so the highest value is 20 cm and then the lowest height is 150 CM so it provides a measure of spread and dispersion of the values and then you can see the dispersion along the mean so the range is very easy to calculate where it is the maximum value minus minimum value so example a quick example an architect wants to understand the diversity in the apartment size within a residential complex to CER to different Market segment to ensure variety of living option so size of apartments in square meter are 45 60 70 90 55 65 80 so in this case you can simply identify that 90 square m is the highest and then you can see the 45 m is the lowest so here the range is 45 M uh you can uh 45 sorry it's not meter it is 45 square m right uh in terms of the area the range of apartment size uh in this particular case is 45 Square met indicating a considerable Variety in sizes from smallest to largest when you have those kind of variation lce so almost things are all equal now coming to the variance variance measures how far a set of numbers is spread out from their average value it is the average of the square difference so in this particular case if you see that the equation this is the value the individual value and this is actually the mean value so whenever we uh do this calculation for the population entire population we represent this with the MU but when it is s normally we represent with the xar right so this how to calculate that already we have seen that okay for a simple one so you can just simply uh add all those value and divide it by n okay so here I to n so and this is the total number of observation so similarly for the sample there is some sort of change in the denominator where you have n minus one so here n is population in that particular you know Target population and here it is the sample so normally what happen when it is a entire population so n minus one is almost you know a huge number it is equivalent to n that's the only difference between the denominator used for this sample variance calculation and mean variance calculation and standard deviation in this particular case is nothing but the square root of this particular variance where it is uh the measure which you know make the amount of variation of a random variable expected about its mean that how it is uh varying say for example if you have certain number say abruptly I'm just uh uh giving some number 40 50 UH 60 30 and then maybe if you have the mean value is something like if you just add all of them so it will be how much so I just make a calculation here so it is uh 180 right uh so in in this particular case uh if the mean value will be 180 divided by 4 so 45 so in this case you can see that there is some sort of uh you know deviation so if you compare it with this so if you consider this formula x - x bar so here you are getting this value is having little bit like this and here you have five here you have you know you can see that this is 15 and there you have minus 15 so these are the uh you know dispersion from the mean value the individual case now coming to the variance and standard deviation example so this is is some sort of you know rent price for a commercial establishment and in, so 200 220 240 260 280 so first you have to calculate the mean if you just sum and then divide it you get 240 then you go for variance the same equation okay for your reference I'm writing it once again so the formula was x - mu squar that the square difference and divided by n here n is nothing but five so Al together it is going to give you 800 as variance and then square root of that is 28. 28 which is standard deviation and in this case this standard deviation is written as plus minus so as because some of the values may be higher than the uh mean and some values can be like uh lower than that so this is something you can say that standard deviation that may occur now coming to the skewness and uh you know you can say the kosis so in this case already we have discussed it let us draw it once again so say for example like if you have perfect uh you know distribution so they both the sight of this particular you know uh Central value so it is having almost equal distribution of data but you may have a particular skew distribution where it is something you know you can say negative and where you may have uh the other side which is positive so this skewness to the right side positive CU to the left side negative this is something can take place so a b c so in this case this is the skewness now compared to that the CIS is something where it talks about uh the measure of data pck so the distribution can be something like this or maybe the distribution is something like this or maybe distribution is something like this so in all these cases it is a b c so in this case you can say that you know the peak of that kind kind of mid value so there is nothing to skewness but you based on the data you can have certain value so whenever uh the values are uh having very low standard deviation okay so then you can see the values are around so you'll get something like this so if you get everything like all values are same like uh if in a particular exam somebody like all 30 student they scor the same marks it is nothing but just you know the line so you don't have any kind of know distribution but normally it is not ideal so we'll have but if it is around like that is uniformly distributed but the mean is quite High the standard deviation is very low so you can get something like this as well but if you see the mean is something low and then it is having some sort of spread so you have as low value as say 20 and high value as 90 but in this case it is not the uh option right now the last part of this is the measures of this particular position where you can identify where your data set it is so in this case you can say that this is the median value then you have 25th percentile which is referred as the first quartile median is at you know 50th the second quartile which normally calculated the inter quarterly range as Q3 and q1 so all the SED area and this is the 70th percentile and then beyond that definitely if the data lies beyond that particular uh range and which is not in your data consideration so then those will be like acting as a outliers and you have to remove it so uh normally you know in this particular case these are very useful in identifying the out layers and understanding the sprit and then you can compare in different data set how it is being uh populated so for maybe one particular group it is like this but for others you may have the change in the quartile it may happen that the median is not that much having high you know kosis right so this is something here we have discussed about this it can go you know higher or it can go lower as well now let us summarize this understanding I am hopeful that the concept and the different components that we discussed in this lecture they are very clear because I've given with examples you can also look at different other examples as well and if there is any confusion let me know by writing in the Forum so in this case we have understood the importance of descriptive statistics for any researcher to get the initial Insight of the collected information then there are some sort of you know aspect into it the measure of frequency measure of central tendency variability in data in terms of cuses skewes you know uh the dispersion Etc and the major of position where different kind of percentile and quartile where data lies which will give you some kind of um understanding to decide upon the outliers and then you can uh take uh those information removing out layers for your further analysis so with that I conclude here these are some uh study materials that even I shared earlier so with that uh we'll be moving forward to our next two lectures on representation of data and inferences like you might have very uh huge amount of information you might have done certain good analysis descriptive analysis but it is equally important to represent those analysis in appropriate Manner and how to infer it so in next two lectures 34 and 35 we'll be discussing on that with different pictorial representation and then you know use of tables figures Etc how to represent those kind of uh findings in a research paper or a research writing so with that thank you again for joining and we'll be meeting in our next lecture thank you [Music] bye-bye [Music] a