Normal (Gaussian) Distribution: Everything you need to know
Normal or Gaussian distribution is one of the most widely used distribution functions in statistics. This post covers all the basics of Normal distribution you need to know.
Normal distribution plays a vital role in statistics because of the Central Limit theorem. Suppose we draw a sample of observation of a random variable from a distribution independent of each other. The average of those samples of observation converges to normal distribution.
A random variable X is said to have Normal distribution, X~N(μ, σ2) if,
Here μ is the center (or mean) and σ is the spread (or standard deviation) of the distribution. Let us plot and see various properties of normal distribution.
You can see the various properties of Normal distribution highlighted in the curve. Now let us see one of the most important property of Normal distribution. The standard deviation σ controls the spread of the distribution. The more the σ, the data are farther from the mean and the lesser the σ, the data is more tightly clustered around the mean. This is the intuitive sense of σ, let us quantify it.
Three σ rule states that these ranges cover this amount of data-
(μ-σ, μ+σ) -> 68.2% of the data
(μ-2σ, μ+2σ) -> 95.4% of the data
(μ-3σ, μ+3σ) -> 99.6% of the data
Let us say that a data distribution has mean as 6 and standard deviation as 3.6 then 68.2% of the data point will lie between 6±3.6 i.e. 2.4 to 9.6. Similarly for the other ranges.
Standard Normal Distribution
We say that X has Standard Normal Distribution when μ = 0 and σ = 1.
A random variable X is said to have Standard Normal Distribution, X~N(0, 1), if
The Standard Normal random variable is denoted by Z. The PDF and CDF are denoted by φ(z) (lowercase phi) and Φ(z) (uppercase phi).
The properties of Standard Normal Distribution are important.
Consider X ~ N(µ, σ2).
- Z = (X – µ)/ σ which also implies X = μ + σZ
P(a < X < b) = P((a – µ)/σ < (X – µ)/σ < (b – µ)/σ)
= Φ((b – µ)/σ) – Φ((a – µ)/σ)
- If Xi ∼ N(μi,σi2), i = 1,…,n are independent, then
Example: Let us take X ~ N(3, 5). Let’s find the probability of P(X > 1).
Solution: P(X > 1) = 1 – P(X < 1)
1 – P(X – µ / σ < 1 – µ / σ)
= 1 – P(Z < 1 – µ / σ)
= 1 – Φ(-0.8944) = 0.81
We find the value of Φ(-0.8944) using Z table. See this to know how to use Z table.