To understand the probability factors of a normal distribution you need to understand the following ‘rules’:
1. The total area under the curve is equal to 1 (100%)
2. About 68% of the area under the curve falls within 1 standard deviation.
3. About 95% of the area under the curve falls within 2 standard deviations.
4 About 99.7% of the area under the curve falls within 3 standard devations.
Items 2,3 and 4 are sometimes referred to as the ‘empirical rule’ or the 68-95-99.7 rule. In terms of probability, once we determine that the data is normally distributed ( bell curved) and we calculate the mean and standard deviation, we are able to determine the probability that a single data point will fall within a given range of possibilities.
A good example of a bell curve or normal distribution is the roll of two dice. The distribution is centered around the number 7 and the probability decreases as you move away from the center.
Here are the % chance of the various outcomes when you roll two dice.
2 - 2.78 % 8 - 13.89 %
3 - 5.56 % 9 – 11.11 %
4 - 8.33 % 10- 8.33 %
5 - 11.11% 11- 5.56 %
6 - 13.89% 12- 2.78 %
7 - 16.67%
Normal distributions have many convenient properties, so in many cases, especially in physics and astronomy, random variates with unknown distributions are often assumed to be normal to allow for probability calculations. Although this can be a dangerous assumption, it is often a good approximation due to a surprising result known as the central limit theorem. This theorem states that the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution. Many common attributes such as test scores, height, etc., follow roughly normal distributions, with few members at the high and low ends and many in the middle.