Empirical Rules and Z-Scores Notes

• Empirical Rule: A statistical rule stating that for a normal distribution:

  • Approximately 68% of the data falls within one standard deviation of the mean.

  • About 95% falls within two standard deviations.

  • Around 99.7% falls within three standard deviations.

Z-Score: Tells you how many standard deviations the data value is from the mean.

$\text{Z} = \frac{(\text{X} - \mu)}{\sigma}$

where:

• $\text{X}$ is the raw score,

• $\mu$ is the mean of the data set,

• $\sigma$ is the standard deviation.

A z-score is particularly useful when the data distribution is mound shape and approximately symmetric.

Allows you to use the mean and standard deviation to make statements about the data values

*Used when shape of distribution is bell/mound shape and is approximately symmetric

Interpreting Z-Scores

The tavb;e classifies rnges of z-scores informallu, in term of being unusual or not.

Un-standardizing Z-Scores Original value x can be computed from z-score. Take the mean and add z standard deviations:

Population: $\text{X} = \mu + \text{Z}\sigma$

Sample: $\text{x} = \bar{\text{x}} + \text{Zs}$