The Normal Distribution¶

Properties of Normal Distributions¶

Empirical Rules¶

• 68% of the data lies within one standard deviation of the mean $(\mu \pm \sigma)$
• 95% of the data lies within two standard deviations of the mean $(\mu \pm 2 \sigma)$
• 99.7% of the data lies within three standard deviations of the mean $(\mu \pm 3 \sigma)$

What Can Be Modeled Using A Normal Distribution¶

• The variable is roughly symmetrical with only one mode
• X can take any real value
• Values far from the mean (more than 4 standard deviations away) have a probability density of practically zero

What Cannot Be Modeled Using A Normal Distribution¶

• More than one mode or no mode
• Not symmetrical
• Variables in which acceptable ranges of values ($\mu \pm 3 \sigma$) become negative when they are meant to be positive

Standardized Z-Scores¶

• Standard Normal Distribution: a normal distribution where mean is 0 and standard deviation is 1. Denoted by $Z$.
• Any normal distribution can be transformed to the standard normal distribution curve by a horizontal translation and a horizontal stretch
• $Z = \frac{X - \mu}{\sigma}$, where $X$ is a normal distribution with mean $\mu$ and standard deviation $\sigma$
• Standard Normal Table: z-score → possibility

Comparing Normal Distributions¶

Two can be compared by examinating their $\mu$ and $\sigma$

Comparing $Z$-Scores¶

• $Z$-score indicates the number of $\sigma$ that a data lies from the $\mu$ of its distribution
• The data point with greater $z$-score lies furthur from the $\mu$

Inverse Normal Calculations¶

P(X < a) $\rightarrow$ value¶

1. Find the cell in the standard normal table $\leq$ P(X<a)
2. Get z-score
3. Convert z-score back to the actual value

P(X > b) $\rightarrow$ value¶

1. Get P(X < b) = 1 - P(X > b)
2. Find the cell in the standard normal table $\geq$ P(X > b)
3. Get z-score
4. Convert z-score back to the actual value

Notice, when getting the cell, choose the one that is closest while within the limit.