# Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is one of the most widely used probability distributions in statistics. The normal distribution is characterized by its mean (μ) and standard deviation (σ).

For a continuous variable X that follows a normal distribution, and has mean μ and standard deviation, σ, we can define X as:

X ~ N( μ, σ2)

μ: mean

σ: standard deviation

σ2: variance

The bell shaped curve of normal distribution is:

## Standard Normal, Z

Standard normal, Z,  is a normal distribution with a mean of 0 and σ = 1. Z ~N(0, 1).

To convert any normal distribution e.g. X to Z (standard normal), we use this formula: Z = (X -μ)/ σ

We convert a normal distribution to standard normal when μ and/or σ are unknown.

## Combining Normal Distributions

If X and Y are independent, with X ~ N( μ1, σ12) and Y ~ N( μ2, σ22)

aX ± bY ~N(aμ1 ± bμ2 , a2σ12 +b2σ22)

If x̄ represents mean of X out of n samples, and X ~ N( μ, σ2),

then x̄ ~ N( μ, σ2/ n)

## Central Limit Theorem

If x̄ represents mean of X out of n samples, where n is large, if the distribution of X is unknown, or the distribution is NOT a normal distribution, we can still say x̄ ~ N( μ, σ2/ n), by Central Limit Theorem.

## Finding Normal Distribution Using Graphic Calculator Ti84

You’ll use these functions in your graphic calculator (Ti-84):

To find probability, use normcdf.

If you are given P(X<a) = 0.3, and asked to find a, then use invnorm.

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