While the Boxplot and interquartile range measures position with the median. Another method is to measure position in relation to the mean using z-scores.
The sample z-score for a measurement x is \[z=\frac{x-\bar{x}}{s}\] The population z-score for a measurement x is \[z=\frac{x-\mu}{\sigma}\] |
Example: Suppose a sample of 2000 high school seniors' verbal SAT scores is selected. The mean and standard deviation are \(\bar{x}=550,\,s=75\). A student scored 475. What is his sample z-score?
Solution:
\[z=\frac{x-\bar{x}}{s}=\frac{475-550}{75} = -1.0\]
This means that the student scored 1.0 standard deviations below the sample mean.