Z-Scores & Z-tables

A Z-score indicates how many standard deviations an element is from the mean of a data set.

The Z-score is calculated using the formula: $$z = \frac{{X - \mu}}{{\sigma}}$$ where X is the value, $$\mu$$ is the mean, and $$\sigma$$ is the standard deviation.

A positive Z-score indicates that the value is above the mean.

A negative Z-score indicates that the value is below the mean.

The Z-score is $$z = \frac{{60 - 50}}{{10}} = 1$$ .

A Z-table shows the percentage of values below a given Z-score in a standard normal distribution.

Approximately 68% of values lie within one standard deviation of the mean.

Approximately 95% of values lie within two standard deviations of the mean.

Approximately 99.7% of values lie within three standard deviations of the mean.

The Z-score is $$z = \frac{{85 - 100}}{{15}} = -1$$ .