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, σ^{2})

μ: mean

σ: standard deviation

σ^{2}: 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( μ_{1}, σ_{1}^{2}) and Y ~ N( μ_{2}, σ_{2}^{2})

aX ± bY ~N(aμ_{1} ± bμ_{2 }, a^{2}σ_{1}^{2} +b^{2}σ_{2}^{2})

If x̄ represents mean of X out of n samples, and X ~ N( μ, σ^{2}),

then x̄ ~ N( μ, σ^{2}/ 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, σ^{2}/ 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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