Too often in life we tend to generalize and associate individual behavior with the behavior of a group. Many of our social biases stem from such a tendency and what we fail to appreciate is the wonderfully complex world we live in.
The Central Limit theorem is one of those beautifully constructed mathematical models that seek to bring about a semblance of order in a chaotic world. In its simplest formulation, it provides a way of visualizing the behavior of seemingly random events. It essentially is a mathematical model of life, a life that involves a myriad of possibilities, a plethora of possible outcomes - some good some bad.
The simplest demonstration of this theorem can be obtained by the roll of an unbiased dice. If we roll one dice then at every roll we can land up with any of the six numbers from 1 - 6 and the probability of occurrence of any of these digits is the same and equal to 16.66%
However, if we now roll two dice and evaluate the sum of the digits that show up then the total number of possible outcomes goes up to 11 with the probability of a 7 showing up now being the highest. The probability is not longer uniformly distributed. This is because 7 can show up as a result of the following combinations
1 - 6
2 - 5
3 - 4
4 - 3
5 - 2
6 - 1
If we start rolling more dice simultaneously and repeat the analysis, the distributions creeps closer and closer to a normal distribution yielding the simplest demonstration of the theorem. I have coded the algorithm for evaluating this in a spreadsheet and it is available here and you can try it out. The only input required is the number of dice that you want to roll together and hit "Calculate". The code evaluates the total number of outcomes possible and then then probability distribution is plotted.
Caution: Owing to the mathematical complexity of evaluating the total number of combinations which rises exponentially with larger number of dice, do not input more than 6 as it is likely to crash your PC ( if its a 32 bit CPU). If you have a 64 bit processor then you can try a bigger number
PS: Code could have probably been written to use recursion but got lazy and hence relied on mathematical jugglery.
Case 1: When one die is used. Number of possible combinations and outcomes is 6
Case 2 : With two dice the number of combinations is 36 and outcomes is 11
Case 3: With 6 dice the number of combinations is 46656 and outcomes is 31
The moral is that a lot of unrelated events can still lead to something that is so wonderfully symmetric and this applies to more aspects of our lives then we care to think about. From the noise that affects electronic circuits (Additive White Gaussian Noise) to how the marks/ratings will be distributed among a random selection of individuals to why individuals in a group behave differently than when alone ( Mob psychology), patters of behavior and data are often more relevant than individual behavior.
As individuals we make choices but given the random nature of individual choice, the outcomes follow a Gaussian distribution which we unwittingly contribute to. Extreme cases of success and failure are the outliers on this curve while humanity's choice as a whole is the mean. Making choices closer to the mean would imply a higher probability of favorable outcome but if everyone does it then it takes away the importance of individual choice.
Neo : "Choice, the problem is choice." - Matrix
The Central Limit theorem is one of those beautifully constructed mathematical models that seek to bring about a semblance of order in a chaotic world. In its simplest formulation, it provides a way of visualizing the behavior of seemingly random events. It essentially is a mathematical model of life, a life that involves a myriad of possibilities, a plethora of possible outcomes - some good some bad.
The simplest demonstration of this theorem can be obtained by the roll of an unbiased dice. If we roll one dice then at every roll we can land up with any of the six numbers from 1 - 6 and the probability of occurrence of any of these digits is the same and equal to 16.66%
However, if we now roll two dice and evaluate the sum of the digits that show up then the total number of possible outcomes goes up to 11 with the probability of a 7 showing up now being the highest. The probability is not longer uniformly distributed. This is because 7 can show up as a result of the following combinations
1 - 6
2 - 5
3 - 4
4 - 3
5 - 2
6 - 1
If we start rolling more dice simultaneously and repeat the analysis, the distributions creeps closer and closer to a normal distribution yielding the simplest demonstration of the theorem. I have coded the algorithm for evaluating this in a spreadsheet and it is available here and you can try it out. The only input required is the number of dice that you want to roll together and hit "Calculate". The code evaluates the total number of outcomes possible and then then probability distribution is plotted.
Caution: Owing to the mathematical complexity of evaluating the total number of combinations which rises exponentially with larger number of dice, do not input more than 6 as it is likely to crash your PC ( if its a 32 bit CPU). If you have a 64 bit processor then you can try a bigger number
PS: Code could have probably been written to use recursion but got lazy and hence relied on mathematical jugglery.
Case 1: When one die is used. Number of possible combinations and outcomes is 6
Case 2 : With two dice the number of combinations is 36 and outcomes is 11
Case 3: With 6 dice the number of combinations is 46656 and outcomes is 31
The moral is that a lot of unrelated events can still lead to something that is so wonderfully symmetric and this applies to more aspects of our lives then we care to think about. From the noise that affects electronic circuits (Additive White Gaussian Noise) to how the marks/ratings will be distributed among a random selection of individuals to why individuals in a group behave differently than when alone ( Mob psychology), patters of behavior and data are often more relevant than individual behavior.
As individuals we make choices but given the random nature of individual choice, the outcomes follow a Gaussian distribution which we unwittingly contribute to. Extreme cases of success and failure are the outliers on this curve while humanity's choice as a whole is the mean. Making choices closer to the mean would imply a higher probability of favorable outcome but if everyone does it then it takes away the importance of individual choice.
Neo : "Choice, the problem is choice." - Matrix
1 comment:
Very interesting post...enjoyed!!
Post a Comment