Median vs Percentile

Sort the values from smallest to largest and the median is whatever sits at position $\frac{n+1}{2}$ (n plus 1, over 2), where $n$ is how many values there are. When $n$ is odd that position lands on an actual observation. When $n$ is even it lands between two of them and the median is their average, so the median of an even-sized set need not be a number that appears in the data at all. It is commonly written $M$.

Take the seven quiz scores 12, 15, 15, 18, 22, 24, 91. They are already sorted and $n = 7$, so the median is the value in position 4: $M = 18$. The mean of the same seven scores is 28.14. Now change the 91 to 27. The median is still 18, while the mean drops to 19. That is resistance: the median only counts how many values sit on each side of it and ignores how far away they are, so moving one extreme value cannot shift it by more than one position.

With an even count the averaging step kicks in. For 2, 6, 8, 10 the median is $(6 + 8) / 2 = 7$, a number that appears nowhere in the list.

The most common error is reporting the position instead of the value. For those seven scores the answer is 18, not 4; position 4 only tells you where to look. Two related slips are taking the middle of the list before sorting it, and averaging the smallest and largest values, which gives the midrange rather than the median.

The median is also the 50th percentile and the center of the five-number summary. The quartiles are the medians of the halves on either side of it, with the median itself left out of each half under the TI-84 convention this site uses: for the seven scores, 12, 15, 15 gives $Q_1 = 15$ and 22, 24, 91 gives $Q_3 = 24$. Comparing the two centers is also a quick read on shape: here the mean of 28.14 sits far above the median of 18, which usually signals either a right skew or a single high value dragging the mean up. In this set it is the second of those, since the 1.5 IQR rule puts the upper fence at $24 + 1.5(9) = 37.5$ and flags 91 as an outlier.

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.