Saturday, December 22, 2012

Why the Gaussian Curve is bell shaped ??

The Gaussian Curve in statistics is described as Normal distribution Curve, which says if random variable X is chosen form a population ( with parameter values, mean = u and variance = σ^ 2) and plotted w.r.t to its probability then a bell shaped curve is obtained.

But why?
We certainly can look at its formula and quite easily state the nature of curve, but again most of us will fail to explain how that formula is derived..
So, I will try my best to explain the nature of the curve in simple theory..

First lets talk about the Standard Normal Distribution Curve. Its the Normal Distribution Curve but is standardized ( with parameter values mean=0 and variance = 1 ). Its a standard which helps to plot the curve in a standard way and w.r.t to which one can compare its Distribution Curve and study the probability of a variable.

So, I think it will be easier to explain things using the Standard Normal Distribution Curve.
Its says the population ( the set of all the variable {random variables} ) has  mean=0 and variance=1.
mean = 0 suggests that the curve is symmetric, this can't yet be concluded but you will be certain afterwards.
Variance = 1 mean the standard deviation = (plus,minus) 1.
This can be interpreted simply as a variable X which differs from its neighbors by 1 (not quite but is considerable) . This means X is an integer and the population is the set of all integers. Now, we know why the curve expands form -infinity to +infinity, its because X is an integer.

This quite easily explains some basic principle of the curve.

But the interesting part is the shape and is still unanswered in the post. This can quite easily be explained using theory of randomness.
Random variables have a nature. The nature of randomness can be observed in day to day life, but we never think it as the mathematicians do. I personally think, I'm not quite bright in mathematics and also lack the vision that mathematics have. Anyway :P , the random variables trace a nature quite magnificent.
If I tell you to think of any number, right this moment, I wont be able to say what you thought. But what I can  say with certainty is that, its not the number infinity ( as our brain is not even that capable of  thinking of that large..number :P ). So, there you go, the probability of you thinking that number(infinity) is none. And I also can say the the number might not have been that large, because normally, people wont go for a number that big.. and note that I said normally, not precisely. The number to be in range of hundreds is quite probable. So, there you go, even the random numbers has a nature(in this case the random thinking of our brain).

The bell shaped nature of the curve can thus be easily understood.

And it works quite well with numerous examples. The probability of a person being very good and very bad is less compared to the person being normal. That's what life is. The probability of students with very good scores and very bad scores ( in exams ) is less compared to that of the students with normal range of scores.

So, there you go..randomness has a curve, a bell shaped curve :)

This thought was in my head for long and I have to say, even if its quite basic and simple, its really a thing to think about :D 


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