So, India’s Got Talent Season 2 got it wrong. The best team did not win. What happened? The rule said judges and votes will get equal (50%) weightage. However its apparent to any Tom, Dick or Harry that the final winners were influenced much more by votes, and to a lesser extent by the judges’ scores.
I did not find any official details on what mathematics was behind the final result. However, my guess is votes were converted into scores (using percentiles for example), and averaged with judges’ scores. This would then yield a result similar to what we got, because judge scores were in the range 23 to 30, whereas prorated vote scores would be in the 0 to 30 range (might not start at zero, but the point is, it would start very low). So, while the worst performer as per judges got a score of 23, the worst voter got a very low score, say 10.
I have written about normalization before. I believe the use of this technique would have yielded more fair results. Normalization brings a set of values to a common mean, and a common standard deviation. Below is a comparison – I have taken guesses on how many votes each contestant received. First, prorating:
Voting score prorata
Manas Kumar Sahu
Next, with the use of normalization:
Nor Judges score
Nor voting score
Manas Kumar Sahu
Several improvements can be seen: Fictitious group has moved deservingly up; Teji Toko has moved down; Acroduo has moved upwards etc. Note again that the number of votes is just a guess (educated one, keeping relative popularities in mind). However, the point is, that with normalization being added to the calculation procedure the same number of votes can bring about a more fair result.
Separating your data into buckets is useful in a lot of problems especially fraud detection. How do you mathematically ‘cluster’ your data? One statistical way is the K-means clustering.
Without delving into too much statistics, here is a spreadsheet you can use to do this for your own data.
This sheet accepts pairs of two variables – for example age versus
number of sick leave applied, by a group in a year. Thereafter it
categorizes this data into buckets, the number of buckets being
specified by you. Once the sheet gives you bucket classification, you can analyse it for problems. You should see the following cases as worthy of further attention:
too many datapoints falling in a single group
only one or two datapoints in a single group
any point that does not belong to the group its in (this is only
possible if the data has a subjective background)
This method can be used to sample data for further analysis wherever
there is simply too much data to analyse. In our example it can be
used to isolate people who may be feigning sickness to take leave. In
a test for people with different levels of capability it can be used
to grade scores etc. It may also be used to solve the needle in haystack problem.
Its really easy to make a magic square: one where the horizontal totals and the vertical totals all add up to the same number. For example, here is a 5×5 magic square:
How to build one such magic square? You could build a 3×3, or a 7×7 one for example. The rules:
1. Start with writing 1 in the middle square
2. Keep moving diagonally upwards, increasing the number by 1 each time
3. If you move out of the square, roll over
4. If you hit a square already filled, move vertically downwards one square and keep moving
The readers may recall that I talked about Ramanujan sometime back. Today I introduce to my readers, a much less known Indian Mathematician.
Dattaraya Ramchandra Kaprekar, born 1905 worked on the number theory. He had no formal postgraduate training and worked as a schoolteacher in Nasik, India.
His claim to fame is the Kaprekar constant 6174. Start with any four digit number, with no repeating digits – say Z. Let A and B be two numbers formed by rearranging the digits of Z, such that A is the highest number that is possible, and B the smallest. Subtract B from A. If this is not 6174, continue the same way now taking this number to be Z. For example, starting with Ramanujan number 1729:
9721-1279 = 8442
8442-2448 = 5994
9954-4599 = 5355
5553-3555 = 1998
9981-1899 = 8082
8820-0288 = 8532
8532-2358 = 6174
7641-1467 = 6174
He also gave the world Harshad numbers: numbers that can be divided by the sum of their digits – for example 12, which is divisible by 3.
A leading Indian daily started a series on not so ordinary Indian people just before the Independence day on August 15th. On the d-day, ex Indian President APJ Abdul Kalam wrote a piece and talked about Srinivasa Ramanujan, one of the greatest mathematician of current times, from the land that created zero.
He is the person behind Ramanujan number, 1729 which is the smallest number to be a sum of two cubes in two different ways:
1729 = 9*9*9 + 10*10*10
1729 = 1*1*1 + 12*12*12
He was given books on advanced trigonometry which he mastered by the age of 13. While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan’s work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.
Ramanujan is generally hailed as an all-time great mathematician, like Leonhard Euler, Carl Friedrich Gauss, and Carl Gustav Jacob Jacobi, for his natural mathematical genius. G. H. Hardy quotes: “The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems… to orders unheard of, whose mastery of continued fractions was… beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly-periodic function or of Cauchy’s theorem, and had indeed but the vaguest idea of what a function of a complex variable was…”. Hardy went on to claim that his greatest contribution to mathematics was discovering Ramanujan.
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