Sunday, September 21, 2014

Normal Probability Distribution

Also called Gaussian distribution. OK, many things in this world tends, and should do, to be normally distributed.
Any distribution is a representation of how the information or data is distributed. We mainly look for its central tendency (mean) and variability (variance). That's why the normal distribution is usually written as:

N ~ (Mu, Sigma^2)

For example: the weight of most adult (who still youth) people will normally be centered around some values. Yes, you right there is a diversity: some are slim and some are obese.

We may expect the average weight for people (example: ages 20 to 30) to be between 70 to 74 kg. OK, let's consider it as 72 (this is the mean value).

Let x represents the weight of a random person. Thus,

Expected Value [x] = mean [x] = Mu = 72 kg

If we have a sample, we can compute the variance (sigma^2) to indicate variability. But we may here think as following:

Variance = Sigma^2 = Expected Value [(x-Mu)^2]
Standard Deviation = Sigma = square root [variance]

Got it? The variance is just the squared expected (average) difference between values of x and its mean Mu.

Assume that the weights could vary (in average) +4 or -4 kgs from the mean value. Thus, we have

Sigma approx= 4
Variance approx= 16

We may conclude, the probability distribution of youth people weight:

weight = x ~ N (72, 16)

Note: this is just an illustrative example where real information may be different depending on location or other factors.

Facts for any normally distributed data:
  • Within 1 sigma distance/difference from the mean value (to left and right), there exist about 68% of data.
  • Within 2 sigmas distance/difference from the mean value (to left and right), there exist about 95% of data.

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