Relative Standing vs Percentile

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.

The $k$th percentile is the value at or below which about $k$ percent of the observations lie. The grammar of that sentence is worth slowing down for: $k$ is the percent, and the percentile is the value, carrying whatever units the data carry. If the 90th percentile of a test is 628 points, then 90 percent of scores are 628 or lower and about 10 percent are higher.

Run the definition backwards and you get the percentile rank, which starts from a value and returns a percent:

Take the ten scores 52, 58, 60, 64, 65, 69, 73, 79, 86, 91. Seven of them are 73 or lower, so the percentile rank of 73 is $\frac{7}{10} \times 100 = 70$, and 73 is the 70th percentile of this set. Move the same score into a different group and its percentile rank changes, because a percentile always describes a position within one particular set of data.

Percentiles work on a smooth distribution too, where they are read off the curve instead of counted. On a normal distribution with mean 500 and standard deviation 100, the 90th percentile sits about 1.28 standard deviations above the mean, at $500 + 1.28(100) = 628$ points. That is the same 628 from the opening paragraph, and it shows the two routes to a percentile: counting observations in a data set, or finding the area to the left under a density curve.

The misreading that costs the most marks is treating a percentile as a score. Standing at the 85th percentile is a claim about the other test-takers, not about how many questions you answered correctly. On a hard test you can get 40 percent of the items right and still sit at the 85th percentile of it.

Three percentiles have names of their own: the 25th is the first quartile $Q_1$, the 50th is the median, and the 75th is the third quartile $Q_3$. Percentile and z-score are the two standard ways of reporting relative standing, and they answer different questions. A percentile says how many values sit at or below yours; a z-score says how far above or below the mean yours is, measured in standard deviations.