Relative standing

Relative standing answers where one observation sits inside its own distribution. Two reports are standard and they measure different things. A percentile is a rank: the $p$th percentile is the value with $p$ percent of the data at or below it. A z-score is a distance: $z = \frac{x - \bar{x}}{s}$ counts standard deviations above or below the mean, where $\bar{x}$ (x-bar) is the mean and $s$ is the standard deviation. Neither is a property of the raw number on its own. Both need the rest of the distribution before they mean anything.

Take a raw score of 80. In a class with mean 72 and $s = 4$ it gives $z = \frac{80 - 72}{4} = 2.00$. In a class with mean 65 and $s = 12$ the same 80 gives $z = \frac{80 - 65}{12} = 1.25$. Same paper, same number written on it, two different standings, because standing is always relative to the group you are being compared against.

"A z-score of 2 puts you at the 98th percentile." That conversion is a normal-model result, not a property of z-scores. Under a normal distribution $z = 2$ does sit at the 97.7th percentile. Nothing forces a distribution to be normal. Roll a fair die: the mean is 3.5 and the standard deviation is 1.71, so the largest possible outcome, a 6, has $z = 1.46$. No outcome there reaches $z = 2$ at all, and the 6 is at the 100th percentile.

The two reports are not interchangeable in either direction. A percentile can be computed from any data set whatever its shape, by counting. Turning a z-score into a percentile needs a model for the whole distribution, and turning a percentile back into a raw value needs that model plus the mean and the standard deviation.

Topic 1.9 of the Fall 2026 course asks you to compare z-scores as measures of relative position, both within one distribution and between two, and topic 1.7 defines the percentile as the value with $p$ percent of the data at or below it.