Normal distribution is characterised by mean $\mu$ and standard deviation $\sigma$

No matter what $\mu$ and standard deviation $\sigma$ is,

• the area between $\mu + \sigma$ and $\mu - \sigma$ is 68%

• the area between $\mu + 2\sigma$ and $\mu - 2\sigma$ is 95%

• the area between $\mu + 3\sigma$ and $\mu - 3\sigma$ is 97%

Standard Normal Distribution

The standard normal distribution is characterised by $\mu$ of 0 and $\sigma$ is 1.

All normal distributions can be converted into standard normal distribution by subtracting the mean and dividing by the standard deviation.

Z = (X - $\mu$) / $\sigma$

Birth Weights example

If Birth weights in a population are normally distributed with a mean of 109 oz and a standard deviation of 13 oz,

Question

What is the chance of having birth weights of 141 oz and higher when sampling birth records at random ?

Solution

Z score of 141 = ( 141 - 109)/13 = 2.46

Probability of the weight 141 and lower corresponding to Z score of 2.46 is 0.99305

Probability of the weight 141 and higher 1- 0.99305=.00695 which is 0.695%

Question

What is the chance of having birth weights of 120 oz and lighter when sampling birth records at random ?

Solution

Z score of 120 = ( 120 - 109)/13 = 0.846153846

Probability of the weight 120 and lower corresponding to Z score of 0.846153846 is 0.79955 which is around 79.95%

SAT score example

For example if we wanted to know the math score that corresponded to the 90th percentile ( assuming a mean score of 500 and standard deviation of 50) ?

In the Z table we would find the Z score corressponding to the probability of 0.9 which would be 1.28.

Z = (X - $\mu$) / $\sigma$