# 8. The Standard Normal Distribution

## l. determine the probability that a normally distributed random variable lies inside a given interval;

## m. define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution;

What is the standard normal distribution?
The *standard normal distribution* is a normal distribution with a mean of 0 and a standard deviation of 1. It is denotes as `N(0, 1)`

.

What are the standard deviations and confidence intervals for a standard normal distribution? This is related to the empirical rule. Mean = 0 First standard deviation = 1 with 68% confidence Second standard deviation = 1.96 with 95% confidence Third standard deviation = 2.58 with 99.7% confidence

What is the formula to transform a normal distribution into a standard normal distribution?
\(z = \frac{X - \mu}{\sigma}\)
- `X`

= score from the original normal distribution
- `\mu`

= mean of the original normal distribution
- `\sigma`

= standard deviation of the original normal distribution

What is a Z-score? How do you calculate it?
- The Z-score is the number of standard deviations away from the mean for a normal distribution. If the mean is 5, the standard deviation is 10, then the z-score for `x=20`

would be 1.5.
- You could use the formula `(X - mean)/std-dev`

What is a Z-table and how do you use it? - A z-table is a table of values that indicate the percentage of area under a normal distribution that lies below the given z-score (# of std dev from the mean). - The z-table has integers down the left and decimals across the top. Just match your given score down and across to find the percent of area that lies below.

##### Source:

CFA

##### Graph:

- 113.026 Statistics - Z-curve to 115.020.40.08 Reading 9 - 8. The Standard Normal Distribution
- 113.028 Statistics - Empirical Rule to 115.020.40.08 Reading 9 - 8. The Standard Normal Distribution
- 115.020.40 Reading 9 - Common Probability Distributions to 115.020.40.08 Reading 9 - 8. The Standard Normal Distribution