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.