Normal Distribution - rFronteddu/general_wiki GitHub Wiki
X ~ N(μ, $σ^2$): "Variable X follows a normal distribution with mean mu and variance sigma squared."
Characteristics
- Frequently appears in nature
- Large enough samples can often be approximated to normal
- Graph is bell-shaped, with the majority of data centered around the mean
- Symmetrical with regards to the mean.
- Mean=median=mode
- E(X) = μ
- Var(X) = $σ^$2 is usually given when we define the distribution. We can also deduce it from E(X)
- Follows the 68,95,99.7 law:
- 68% of all outcomes fall within one σ from the mean
- 95% of all outcomes fall within one 2σ from the mean
- 99.7% of all outcomes fall within one 3σ from the mean
- This tells us that outliers are rare in a normal distribution
- To obtain f(x) used in E(X) and the variance we use the fact that a normal distribution is represented by a Gaussian. And integrate following the definition of E(X). The integration is solved using t substitution.
- To figure out a normal distribution we need lots of values or we risk outliers to drastically affect our analysis.
- The standard dev tells us how spread is the distribution while the mean tells us the center of the distribution (which is symmetrical around that point)
- A smaller std dev signifies an higher mean value and thinner tails.
Standardization
Every distribution can be standardized using the standardization operation:
~(µ, $σ^2$) ⇒ ~(0, 1)
This can be done using the z-score $Z=\frac{X-μ}{σ}$
If we standardize a normal distribution we get the Standard Normal Distribution, represented in the following graph: