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Data representation and plotting — Transcript

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0:02[Music]
0:16and hi uh welcome to today's lecture so
0:20I hope you have done your you know
0:22assignments and gone through the
0:23multiple choice questions which were
0:25uploaded so today we will uh start
0:28discussing about uh data and ways of
0:31representing data so broadly speaking
0:34there are two components of Statistics
0:37one is descriptive statistics which
0:39essentially summarizes that is to
0:41basically convert raw data into some
0:44numbers so that is what descriptive
0:46statistics is about and the other type
0:48of Statistics is called inferential
0:50statistics here we want to develop
0:52procedures for finding out or making
0:56distinct conclusions from the measures
0:59that we have drawn from the sample from
1:01the population okay so there are so of
1:04course inferential statistics is the
1:06most important thing so there are few
1:10steps which we need to follow in order
1:12to understand what what are the steps in
1:14inferential statistics right so the very
1:17beginning the first thing is to identify
1:19what is your question right what is your
1:21question and who is your population
1:24let's say you want to you know you want
1:26to make you want to Market a soap then
1:30and for teenagers so what should be the
1:33look and feel of the soap so has to
1:37attract teenagers to using that so your
1:40population is a teenager the question is
1:42basically to make a soap of and identify
1:45the essential features of the soap right
1:48so now you want to have a process of
1:51selecting sample right so you know it's
1:53teenagers but teenagers from where you
1:56know what is the proportion of boys
1:59versus girls in this sample right so
2:02once you have done that and you know we
2:04had discussed in previous lecture that
2:06if your sampling is improper then you
2:09might lead to a completely wrong result
2:12okay once you select the sample you have
2:15to analyze the information right you
2:17select the sample you ask the relevant
2:19questions in the course of a
2:20questionnaire and you analyze the
2:22responses given by you know boys and
2:25girls right and based on that you want
2:27to make an inference that you can apply
2:29for the whole population of teenagers
2:32and then finally you want to determine
2:34the reliability of inference right you
2:36have come up with Okay pink soap with
2:39you know which is more elliptical in
2:40nature or oval in nature is is what
2:43people would want so but you want to
2:45test the reliability of this inference
2:47so these are the steps in inferential
2:49statistics okay but before the you know
2:52the Prelude to inferential statistics is
2:54descriptive statistics and we want to
2:56begin with them descriptive statistics
2:58okay so in Des descriptive statistics
3:01one of the M most important things is
3:04the variable right what is your variable
3:06okay so variable is a characteristic
3:09which varies with time and our different
3:11in so our body temperature can be a
3:13variable right so you want to figure out
3:15whether someone has fever or not fever
3:17right so body temperature is a variable
3:19in that case some you want to figure out
3:21what is the average height of this
3:23population right so then height
3:25similarly weight so on and so forth so
3:27this is just an
3:28example of you know data of let's say
3:32five students in a class so you have the
3:35following categories in other words you
3:36have the following variables what is the
3:39gender what is the year in which the
3:41student you have selected the students
3:43five students from you know from the
3:45hostel which year they are first year
3:47second year so on and so forth what are
3:49they measuring in be it maths be it
3:51physics be it biology so on and so forth
3:54how many courses have they already done
3:57right so a first year student would have
3:59taken can probably taken five courses
4:01already that means the second semester
4:03of the first year so on and so forth
4:04okay and what is the GPA of that
4:07particular student so what you see that
4:09the nature of the variable differs a lot
4:13okay so in case of gender it's just a
4:16category right you either have male or
4:18female in year you have a number 1 2 3 4
4:22okay major is also categories right you
4:25have distinct categories maths physics
4:27biology so on and so forth now number of
4:29courses is a variable but it is a
4:32discrete variable right you can have
4:34only you know natural numbers which is
4:37greater than zero and in terms of 1 2 3
4:40like that okay but cgpa is a fraction
4:43right it is a number which is depending
4:45on what is your you know total cgpa it
4:48can vary anywhere between 0o and 10
4:50let's say okay but you can have any
4:52variable which is between these
4:54numbers so in other words my variable
4:57can be divided into the four categories
4:59right your type of data can be
5:01qualitative so qualitative I mean that
5:04is it gender is for example male or
5:06female or you can have a quantitative
5:09variable which is essentially like cgpa
5:12or which you know which year you are in
5:15so again which year you are in is a
5:17discrete variable and your cgpa is a
5:19continuous variable okay so there are
5:21various types of variables you have to
5:23identify depending on the problem now
5:27let's say we you know go back back to
5:30another plot where you have a grade so
5:33you have you know the mids exam is over
5:36and you have graded the students and you
5:37want to find out the statistics as to
5:39who has gotten what grades okay so there
5:42are 10% in the population which has
5:43gotten grade a 30% of the population
5:46Grade B 40% grade C and 20% grade D so
5:50you can represent it in a what is very
5:52you know popularly known and used it's
5:55called a pie chart it is attractive in
5:56nature so what you clearly see 40 is C
5:59and and it has the biggest section of
6:01the pie chart so this the the area of
6:04this pie chart is proportional to kind
6:06of the relative frequency of this number
6:09okay but so pie chart is easy to
6:11represent easy to understand but it has
6:14its share of problems so we need to know
6:17what are this problems so imagine in
6:19this case there are only four grades so
6:21there are four categories it is easy to
6:24come up with the pie chart imagine a
6:26situation where there are 25 different
6:29categorize you know categories possible
6:31so in other words each of these
6:33percentage areas will keep on shrinking
6:36and shrinking so imagine you have one
6:38case which is 1% and the other one which
6:41is 41% so 41 will of course take a huge
6:44chunk of this P chart but 1 person will
6:47barely be visible so in other words you
6:50it is difficult to represent in pie
6:52charts when you volume of data increases
6:55such that there are multiple different
6:57categories possible so you can express
7:01this categorical data into Al in
7:03something another thing which is widely
7:05used is a b chart so same as before you
7:08have the percentage in your y- AIS and
7:10you have the categories a b c d okay so
7:14as before one of the weaknesses or you
7:16know deficiencies of these bar charts is
7:19that if you have too many bars it looks
7:21cluttered okay if you have few bars
7:24there you know it is easy to represent
7:28okay
7:30so this is you know coming to the few
7:32bars and again the same problem that I
7:34mentioned before for pie charts right so
7:37you have a value one which is 2% and
7:39another value which is 40% how can you
7:42represent it in the same bar and still
7:45the you know the the other person can
7:47make sense out of it the 2% for all
7:49practical purposes will look like zero
7:51so it is nearly impossible but there is
7:54a solution so what we do is when the
7:57variation in data is huge as is in this
7:59particular plot you have three
8:01categories a b c where a value is around
8:0450 and C is maybe even you know 600
8:08right so what you can do is introduce
8:10something called a break okay so you
8:12want to show that there is significant
8:15difference between a and b so you have
8:18so whatever is the you know range
8:20maximum of C till there you can have a
8:22continuous you know axis in y but after
8:26that you can introduce what is a break
8:28right so let's say this guy is 800 so
8:30you can introduce a break at 400 and
8:32then plot again so everything still fits
8:35into the same thing but the essential
8:37part of the information is there for you
8:39to gather that that this is way smaller
8:42than this is also part of the
8:44information and this is way smaller than
8:46this is also part of the information and
8:48you want to capture both these things in
8:50the same
8:51plot so let us have a simple example
8:54okay we are talking about working with
8:57quantitative data right so so this is
8:59the body mass indices of you know 25
9:02people right in a class let's say you
9:05have this entire you know of course
9:06these values are continuous variables so
9:09so that you can have all these values
9:11now we want to know how can we convert
9:13it into a way of representing it so
9:16identifying categories a b is perhaps
9:19not the good you know good way because
9:21it's not a discrete quantity but a
9:23continuous quantity but what you can do
9:26is you can
9:28identify what is the range right so in
9:31order to identify the range we want to
9:33know what is the smallest value in this
9:35population so I can go through this list
9:38and I think the smallest value is 18.3
9:41so 18.3 is the smallest value and the
9:45largest value largest value is
9:5028.8
9:5334.2 okay so 34.2 is the largest value
9:58this is
10:00smallest this is largest right so we can
10:05divide it so 18 into 34 is roughly 18 to
10:0834 is equal to you know 16 so we can
10:11have a range of four baskets so we can
10:14identify four baskets let's
10:19say okay one is 18.3 to
10:2422.3 another is
10:2622.3 so 22. 3 to
10:3126.3 okay we can have another
10:35one which is
10:3726.3 to
10:3930.3 and 30.3 to
10:4434.3 now each of these numbers would
10:47mean that you in this basket something
10:49will come in if let's say that number X
10:52is greater than
10:5426.3 greater equal to 26.3 and X is less
10:58than 30 .3 so this would make sure the
11:01same point x does not go into multiple B
11:05baskets okay so this way what we can
11:08generate is called a histogram okay so
11:11you convert the data into frequency you
11:14can then plot them as numbers or
11:16percentage and then you can have
11:18multiple distributions depending on the
11:20nature of the data okay so your
11:22histogram looks something like
11:24this so you can have these bars so in
11:27our case we have four bars so we will
11:30have these distribution so these are
11:32values and this axis is frequency or the
11:36number of them so it is possible so it
11:39is possible to convert this
11:42data now let's say you are going through
11:45this
11:47exercise you
11:50have this distribution in one case where
11:53the total number of observations were 25
11:56and another
11:57distribution
12:02okay where n is equal to 600 right is it
12:06possible to put both of these data on
12:08the same plot okay and this is where you
12:12have to do what is called as a
12:13normalization exercise so you know what
12:16is n equal to 25's total and you know
12:19each of these values frequencies so you
12:21convert it you normalize the curve in
12:24other words you divide every if let's
12:27say this is my F1 this is my FS2 this is
12:30my fs3 so on and so forth I convert them
12:34into
12:36fractions okay so the nature of the
12:38curve won't change so this value this
12:41value is now fub1 by summation fi okay
12:45so it is equal to fub1 by fub1 + FS2 +
12:50F3 +
12:52F4 so this you will get a fraction it's
12:55a
12:56fraction okay so once we have done this
13:00then it is theoretically possible to
13:02generate the following
13:05plot I have the same
13:07[Music]
13:15thing and another one let's just say
13:19hypothetically so the way I drew
13:27is okay okay so if I just if I were to
13:31draw the outlines of this curve this
13:33curve would look like this so you have
13:36one curve like this and the other
13:41curve which is like this so it is
13:43possible to plot both of them at the
13:46same time but you have to do is
13:47normalize but another caveat of this is
13:50you must ensure that the data is from a
13:53similar distribution so any of course
13:55there is greater certainty when you have
13:57sampled 600 you know individual
14:00measurements but when you are you know
14:02plotting the same thing with n equal to
14:0425 there's is the great possibility that
14:07the nature of that distribution will
14:10shift okay so another way of you know
14:14another type of plot which is widely
14:16used is called scatter plot so scatter
14:19plot is just X and Y values let's say I
14:23have X versus y I have age age as one
14:28variable
14:29and the other variable is let weight
14:31right so I can have this generation age
14:34is very you know let's say 5 years
14:36weight is 10 kgs so on and so forth and
14:3950 50 years age is you know 60 kgs so
14:42you have a range okay now depending on
14:45the nature of this data you might have a
14:47you know points which look like this so
14:50this is my X this is my y so you might
14:54have a data which looks like this or as
14:56I have plotted in this particular curve
14:58you have a kind of a reverse such
15:00Association where you have greater the
15:02increase in X the Y value decreases with
15:05a notable exception okay so this is
15:08where your you know data analysis so you
15:11know in this case do you call it a
15:13negative association or do you want to
15:15have a much more nonlinear nature of
15:17this
15:18curve so Scatter Plots are widely used
15:22so again here it is better to plot these
15:25points as scatter as opposed to connect
15:27them then it is much difficult to make
15:29sense out of this
15:31data okay but you can make so if you
15:34were to connect it then you can generate
15:36what are typically called as line plots
15:38so this is an example of a line plot
15:40where X and
15:42Y I have plotted it in a slightly which
15:45looks like a you know s in some way so
15:49these are reminiscent of bacterial
15:51growth curves but you know so you can
15:54have various functions which describe
15:56these line plots so it makes sense to
15:58connect them by line when you know that
16:00the underlying phenomena is actually a
16:02physical process which has a given time
16:05constant associated with it or a given
16:08you know mechanism by the way in which
16:10it happens so there's it's it's under
16:12control it is not a completely random
16:14Association so that is when you can have
16:16very nice linear plots so just a small
16:18detour on the type of
16:20plots you you know you are all well
16:23conversent with linear plots xal to Y is
16:26a very simple plot and you know how plot
16:28it you have X you have y you take these
16:32points and you know you take these
16:34points so let's say this is 1 comma 1
16:37you have min-1 comma minus1 you know 5
16:41comma 5 so on and so forth you have a
16:43line which goes like this and this is
16:4645° right so in general if you have a
16:49line which kind of shifts up so in in
16:52general why you know you can have a line
16:55which is like this in this case
17:00okay so there's an intercept a nonzero
17:02intercept on the y- AIS which you can
17:04call it y0 and it has a given slope so
17:07you can have M as the slope or M is
17:10nothing but tan Theta so in this case Y
17:13is equal to y + MX is your equation okay
17:19so depends so you can have multiple type
17:22of you know uh functions so these are
17:24all linear functions that I have drawn
17:26you can have something like this
17:28let's say this is an example of a
17:30parabola so this is X this is Y and Y is
17:33equal to let's say x² right so the far
17:37higher you w you have a non you
17:39nonlinear nature of the curve so these
17:41are so Y is equal to X cubed will look
17:43similar but it it it have much sharper
17:46Peak be before xal to 1 and lower Peak
17:51before this but y = x = x² is symmetric
17:55but y = x x cubed looks like so Y is
17:58equal to X Cub looks like this when X is
18:00negative your y values are negative okay
18:03so these are some of the simple Curves
18:05in polinomial you can have exponential
18:07curves which are let's say an
18:10exponential DK Curve will looks like
18:12this let's say x this is Y at so if it
18:15is y is equal to e ^ minus X in terms of
18:18DK you have at x equal to 0 you have y
18:20equal to 1 and then you have a
18:21characteristic time so in the most
18:23General case you have X by to which you
18:26know which represents the time con of
18:28the form so when you are trying to fit
18:31data let's say you have a data which
18:33looks like this then it should
18:35immediately occur to you that this has
18:38something it might look like an
18:39exponential it might be a par you know
18:41it might look like a polinomial so pols
18:44are easy to fit because they have
18:46multiple Dimensions but it is not you
18:49know Wise to always fit every function
18:51with a polinomial now what are the
18:53things that you need to keep in mind
18:55while doing these
18:56plots let us go over them one by one so
18:59of course when you make a
19:02plot first thing you have to label your
19:04variables X and Y you have to put their
19:07units ideally so let's say this is if is
19:09age then I can have years in my if this
19:12is weight I can have kg okay so I need
19:15to know what is the what are my Axis and
19:18what are my units okay
19:20and I need to choose the appropriate
19:23range let's say for example we I want to
19:26make a plot of population expansion
19:29right so if I plot like this right it
19:33gives me the impression so sorry this is
19:35kg but let's say it is the weight itself
19:38right weight which is increasing as a
19:39function of years which will also
19:40probably be a linear you know increase
19:42and then some saturation after Point
19:44okay so in this case if I want to show
19:47it is linearly increasing so let's say
19:50this maximum value is around 60 okay so
19:54I need to make sure and I am plotting
19:57till 150
19:58so this portion of my plot is completely
20:01destroyed because I am not using the
20:03space I am I am visually trying to
20:06convey that the weight is not changing
20:09much with years but in reality the
20:11weight is changing with years so I
20:13should actually rraw this plot that this
20:15is from 0 to 60 and my curve should look
20:18something like
20:20this okay so if it was like 60 then I
20:22can clearly see there is a nonlinear
20:24increase initially which means that
20:27initially when kids are growing their
20:29weight increases drastically but once
20:31they reach a certain age it starts to
20:33kind of plateau off
20:36okay again the other point of breaks as
20:39I said so in whatever you have done in
20:41bar graph you can have the break here
20:44itself again let's say you have a
20:46variable X which goes from 0 to 100 and
20:50a variable Y which goes from .1 to
20:5510,000 right so here if you if you put a
20:59linear value so all so all values which
21:02are very small will look like this just
21:04look like a mess here but what you can
21:07do is you can either plot it in log
21:10scale so if you plot it in log scale
21:13then accordingly every point so this
21:15will be one this will be 10 100 so on
21:19and so forth okay so the points will be
21:22well separated out and you can see them
21:25so it is important to choose appropri
21:28range and or again as as before let's
21:31say this is from .1 to 100 what I can do
21:35is I can introduce a break so let's say
21:37I can have 0.1 to 1 and then 80 to 100
21:40if all the data is just here and then
21:43remaining is here okay so this is how I
21:46can really make use of the whole plot
21:48and still plot my Axis so that
21:50everything is clearly
21:52visible one more thing is let's just say
21:55that you have all your data is here and
21:57there is one one outlier which is here
22:00okay all your data is re is essentially
22:02concentrated in this portion of the
22:04curve but there is one point which is
22:06way out which is an outlier so would you
22:10bother to plot the entire range or would
22:13you just bother to point this plot this
22:14inside I think it is it makes sense to
22:17plot the center then and then bloat it
22:20up okay so you make it big so then you
22:24have all this scatter and as an inset
22:26you can have this higher value where all
22:29these points are looking the same so
22:31make this whole curve as the inset so
22:33this is called a
22:36inset okay to handle
22:39Outlets okay so now these are some
22:42single y plots again let's just say that
22:46I have three variables right let's say I
22:49have three variables
22:51time
22:53age and
22:55weight okay three variables
22:59okay and I want to understand and I want
23:03to make a single plot of putting all
23:05them together so this is where you can
23:07make use what is called as a Double Y
23:12plot okay so you can have two axis so
23:16this is you can label this as y1 axis
23:19this is as Y2 axis this is X and you can
23:21plot them let's say with you know weight
23:25with time might saturate an age with
23:28time has a you know linear relationship
23:31so this if x is my
23:34time this is my
23:37uh weight and this is my age then this
23:41guy will have a linearly increasing
23:44curve okay so this is just another
23:47example of a Double Y plot so in this
23:50case I have had a reverse fight in which
23:52variable y exhibits a decrease a linear
23:56decrease with X as a function of time
23:57time and variable X actually exhibits a
24:01saturation profile so beyond a certain
24:03value of x it reaches the
24:06saturation okay so now let us solve few
24:09examples so we have the following
24:12example where you have number of visits
24:14to a dental clinic in a typical week so
24:17as you can clearly see so these numbers
24:19are all discrete numbers you don't have
24:21a fraction because the number of visits
24:23is of course a discrete
24:24number but and you want to know what is
24:27the best of plotting it okay so first
24:30you see what is the range right so we
24:32have all the way from 1 to 8 and I think
24:35when you have this kind of data it is
24:37good to sort it so if I were to write
24:40the same data together in a sorted form
24:44I have
24:45one so the frequency of one I can make
24:48the frequency of one so I have one 2 3 4
24:555 6 7 8 right right so and this is my
24:59frequency axis we have the number of
25:05visits and the frequency axis so for
25:08number one the frequency is two the
25:11number
25:12two so the frequency of two is only one
25:17right frequency of three is 1 2
25:25three frequency of four is 1 2 3 4
25:315 frequency of five is 1 2 3 4 5 6
25:387 frequency of six is only one frequency
25:44of s
25:45is
25:473 8 is 1
25:512 okay so you have the number of visits
25:54because these numbers are small there is
25:57absolutely so you so of course this axis
26:00has to be one two like that
26:03three and because these numbers are
26:06small there is absolutely no necessity
26:08to make it into a relative score you can
26:11just have these values so for example
26:13for one it is two for two it is one for
26:16three it is three for four it is for
26:19four it is
26:20five
26:22okay six it is one 7 it is three eight
26:28it is two okay so you have if I if I
26:31connect that actually I should have them
26:33as
26:37bars okay so what you see is almost that
26:41the it is not a unimodal distribution
26:43there is a you know reasonable amount of
26:46variation in the data so if this was if
26:49this uh five was slightly higher then
26:53you have a nice histogram like shape but
26:56this is different okay okay so this is
26:58of course a discrete variable and then
27:00you can I think histogram would be the
27:02easiest way to plot it okay so let us
27:06and you know histogram is the way to
27:07plot it let us take another
27:10example so I have test course of you
27:14know 20 students right I have test sces
27:17of 20
27:20students and I want to know okay so I
27:23want to know that what is the average
27:26test score right as before and what is
27:28the way of plotting it as before I think
27:30the histogram is the best way of
27:32plotting it okay so we can again go
27:35through the same process we know what is
27:37our lowest number which is around 29
27:39which is our highest number which is
27:42around 93 and we can make it into a
27:45histogram okay so again where histogram
27:48is a good way of representing this the
27:50last
27:52one uh the last one is an example so
27:55imagine so the above dat data is not
27:58test of scores of 20 students but it is
28:01test scores of 10 students in two exams
28:03so 10 students exam one exam two so now
28:06we have to plot this you know so you can
28:09make them as two separate histograms but
28:11if you want to plot it in the same plot
28:13maybe it is best to put it for the for
28:15each student the X and the Y and that
28:18might give us some correlation between
28:21how they performed in each of the exams
28:23okay so I guess that brings us to the
28:26end of this just a brief recap we
28:28discussed uh you know the nature of
28:31variables either qualitative or
28:33quantitative we discussed some of the
28:35common ways of representation which is
28:37pie chart bar chart histograms line or
28:41scatter plot and then Double Y okay so
28:44depending on the nature of the data the
28:46range the size of the data you might
28:48choose to you know use the histogram or
28:51the scatter plot as the case may be when
28:54you're trying to look for some
28:55correlation you want to preferably use
28:57plots like scatter plot okay with that I
29:00thank you for today's lecture I and I
29:02hope that you attemp the questions which
29:04we upload for the multiple choice
29:06questions thank
29:16you
29:23and

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