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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?


\[z=\frac{x-\bar{x}}{s}=\frac{475-550}{75} = -1.0\]

This means that the student scored 1.0 standard deviations below the sample mean. 

Statistics by Matthew Cheung. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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