Spearman's Rank Correlation #2 - YouTube

hello everyone in this video I will be talking about Spearman's rank

correlation once again but that will be a separate problem from my first

video if you look into the first problem then in that particular problem there

was no standoff between the ranks was appearing the ranks were not tied up

anywhere but this is the problem where the ranks ties up and a standoff appears

amongst the values amongst the ranks and exactly how this kind of problem can be

solved let us understand the problem first of all let us suppose there is a

competition in which two judges are there and the participants it

could be a dance competition or something in which the judges are

judging the participants out of 100 and that's how to judges are giving the

marks you could clearly see the pattern that judge one is quite lenient in

giving the marks judge two is a little conservative and giving the marks and

you could see there are ten participants in total so first participant judge one

has given fifty marks out of hundred and just two has given 12 marks and so on

these paired values are appearing for each candidate, this course given by

both the judges now my job here is to find out that how alike these two judges

think or what is the you could say that what is the correlation between them

since this is again a problem of opinion a problem of converting attribute into I

would say a numerical number so the most suitable method here is Spearman rank

correlation coefficient not the Karl Pearson's coefficient of correlation so

I've talked in my first video about how to give the ranks

you can give the highest number the lowest rank or the lowest number the

highest rank and my strategy is always that the highest number should be given

the lowest rank so I'll be the first step is to award the ranks so r1 is

representing the ranks to this course of judge one so what is the highest number

here 65 so 65 I am giving it rank one now you should see what is the

second highest number it is 57 this is given rank 2 now what is

the third-highest number if I look in the series 50 is third highest number

and it is given rank three then after that I would see which is the highest

number that is 40 given rank fourth after that again I search for the

highest number here after 40 the highest number is 33 it is given rank 5 now

after 33 I should see what is the next highest number so in this particular

case I see the next highest number is where it is appearing 24th given rank 6

now after 20 say 4 if I see what is the next highest number that is 15 but 15 is

appearing three times one two and three so look at this this is from there you

have to focus a little bit more so if I look into the 24 it has been given with

rank 6 the next rank is seventh so I can give 7 to this 15 this 15 or this 15 any

of them so there are three consecutive numbers which are actually occupying the

next three ranks that means seven eight and nine

so I'm going to take the average of seven eight and nine and I am going to

alot the rank 8 to all these numbers so if I look into this thing so 15 is

appearing at 7th position 7th rank it is also bearing at 8th position

it is also appearing at 9th position all three positions 15 is appearing so what

I'm going to do is I'm going to take the average of these three three ranks 7th

8th and 9th and I can say it is 8th so I'm going to assign the rank 8 8 8 to

all 50 numbers 8 8 here and it here but the important thing is 6 7th 8th and 9th

places are occupied so the last number will be given the rank 10th if you

didn't understand I will award the rank 2 and we will see exactly how this thing

works once again so I'll see the highest numb

here once again for just two scores are two is the ranks allotted to just two so

what is the highest number if you see the 24 is the highest number given with

Rank 1 the second highest number is 20 which is given Rank 2 now if you look

into the third highest number which is 18 given rank 3 the fourth highest

number is 15 which is given rank fourth the next highest number is 12 appearing

at two places that means appearing at fifth and sixth places up to 4 the rank

has been allotted already so 5th and 6th place 12th is a be hearing I divided by

to take the average and it becomes 5 point 5 so 12 will be given a rank both

twelves will be given a rank five point five and five point five so 5th and 6th

ranks are allotted now I look into the next smallest number which is 9 here 9

here it will be given ranks 7 because 5 & 6 have been taken by 12

now after that what is the next highest number if you look into this thing that

is 6 6 this 6 is appearing at 8th and 9th rank so air 10th 9th rank again I'll

take an average of these numbers which will come out to be 8 point 5 so smooth

6 will be allotted the rank 8 point 5 and 8 point 5 the last number is so 8th

and 9th rank is occupied that NASA number is here what is left is 4 which

will be given rank 10 so I have allotted the rank 2 first judges course in second

discourse you can pause the video you can ricci it that so actually I have to

take the average of that tied up ranks not exactly of the values this is you

need to take care of now the difference between the ranks r2 minus r1 it is

hardly it hardly makes a difference whether I do R 1 minus R 2 or r2 minus

r1 because ultimately I have to calculate the square of it so but in

this case I am calculating r2 minus r1 so that means five point five minus

three it is coming up plus two point five this is zero point five five point

five minus five one minus then are two minus I 1 1 minus 4 is minus three eight

point five minus 10 that is going to give me minus 1 point 5 4 minus 8 is

minus 4 10 minus 8 is 2 2 minus 1 is 1 7 minus 6 is 1 8 point 5 minus 8 is 0.5

and 3 minus 2 is again 1 now I'm going to take their squares up so that is

going to be six point two five zero point two five three square is 9 one

point five square is 2 point two five is going to be 16 this is going to be 4 is

going to be one one that is again going to be zero point two five and that is

going to be one so I'll be requiring the summation of D square for my formula so

I calculate sum of d square that means total of all these numbers and that is

exactly how much 41 in this case so this I'll be using in the formula let's go

ahead and let's see exactly how the formula is now here is the formula to

calculate Spearman's rank of correlation one minus six d square you have just

calculated the summation d square is 41 which you have already calculated now

the catch here is about this M so M is actually that how many times a

particular value has tied up or stand off and I is representing the number of

times the particular tire Chile appears so if you look into the eye how do you

see the eye first of all so if you look into this particular problem right here

how many times I dealt with this standoff so look at this so first I

dealt off with a standoff when fifteen appeared three times second time

what happened is twelve appeared and when I have to deal with a standoff and

third time there was a six which appeared and I have to deal actually

third time with a standoff so how many times I dealt with the standoff three

times so I will become equal to three how many times I actually means took the

average of rank and allotted it so you can see it from here also so you can see

that one time two times and three times so how many times I dealt with the

problem of standoff thrice so I will become equal to three I am repeating it

how many times I dealt with the problem of standoff or tie three times in the

entire problem so I will become equal to three if I dealt this kind of situation

four times I will become equal to four if I have encountered this kind of

situation five times it will be I will become equal to five not depending on

this I how exactly the formula gets extended to and how we can solve it so

you can see the formula you can pause the video and you can resi the formula

you can note it down now I'm going to extend this formula as per my need now

look into the formula it was R is equal to one minus six

sum of d square plus sum of sorry Quentin love I'll bring it down M now I

is three so I will be ranging actually one two three so I will be ranging from

one two three now so am i I am repeating it M I square minus 1/12 and this whole

argument goes with the division of n times n square minus 1 for your

knowledge is number of paired observation which is

ten in this case I know there are ten candidates in total you could see in the

problem what is the serial number ten so n is equal to ten

now let's extend this formula depending on I so how the formula is going to go

look at this one minus six sum of D square plus M 1 M 1 square

minus 1 plus M 2 M 2 square minus 1 plus M 3 and M 3 square minus 1 whole

argument divided by 12 and further this whole argument is divided by n times n

square minus 1 now the next thing is important thing is what is M 1 M 2 and M

3 so that I am going to see from here now how many times the tie appeared how

many values were there in first I 3 so M 1 is 3 how many values are there in the

stand of two M 2 is 2 how many values are there in stand of 2 m 2 3 is also 2

that means I decides I is decided by how many times standoff appears and within a

standoff how many values are there that decides the value of M I am repeating it

how many times standoff appears that decides the value of I and within a

particular standoff how many values are there - that decides the value of M so

in first standoff the value of M 1 is 3 in second tide the value of M 2 is 2 in

third standoff the value of M 3 is 2 because there were two values two values

and three values involved M 3 1 3 M 2 and M 3 2 now the calculations are

rather simple here you could see that now it is going to be

1-6 some of d-square you have already

calculated it is exactly 41 and then plus 3 because M 1 is 3

you could see that I can show you M 1 is 3 so 3 + 3 square minus 1 3 square is 9

minus 1 plus what was M - M - was 2 so it is going to be 2 times 2 square that

is 4 minus 1 plus it is also 2 M 3 square that means 4 minus 1/12 and this

whole argument is divided by n is 10 so 10 times 10 square minus 1 now you could

see that this is a little bracket which I forgot to put in this also and I also

look into my original formula once again and I also stretch it here no it looks

better so I hope now you can make the simple arithmetic calculations you start

making the calculations and it is going to end up to one - 44 times 6 divided by

9 9 0 that gives me zero point seven three three and I'm going to write the M

values here once again M one was 3 M 2 was 2 and M 3 was also 2 so this is a

positive correlation high positive

positive sign means both the judges think in same direction that means if to

a candidate if judge one is giving high marks second is also giving high marks

and 0.73 means it's pretty highly that means both the judges are pretty

sensitive to each other thank you very much please post your comments if you

require anything thank you