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.