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115 CFA
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115.020.40.08 Reading 9 - 8. The Standard Normal Distribution

# 8. The Standard Normal Distribution

## 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.

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