Z-Score

Our first tool of comparison uses the mean and standard deviation and is called a standard score or z score.

Z Score or standard score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation.

To calculate the z-score we use the formula:

\[z=\frac{\text{value}-\text{mean}}{\text{standard deviation}}\]

That is if we denote \(x\) as a data value for a sample and \(\overline{x}\) as the sample mean and \(s\) as the sample standard deviation then the formula for the sample \(z\)-score associated to \(x\) is:

\[z=\frac{x-\overline{x}}{s}\]

That is if we denote \(x\) as a data value for a sample and \(\mu\) as the population mean and \(\sigma\) as the population standard deviation then the formula for the population \(z\)-score associated to \(x\) is:

\[z=\frac{x-\mu}{\sigma}\]

A standard score or z score tells how many standard deviations a data value is above or below the mean for a specific distribution of values. If a standard score is zero, then the data value is the same as the mean.