Percentile vs Relative Frequency

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.

A relative frequency turns a count into a share of the whole: divide the number of observations in a category by $n$, the total number of observations. The result always lands between 0 and 1, and multiplying by 100 reports the same quantity as a percent. When the categories are mutually exclusive and cover every observation, the relative frequencies add to exactly 1.

Forty students report how they get to school: 18 walk, 14 take the bus, 8 come by car. The relative frequencies are $18/40 = 0.45$, $14/40 = 0.35$, and $8/40 = 0.20$, and those add to 1.00. As percentages, 45 percent, 35 percent, and 20 percent.

The mistake relative frequency exists to prevent is comparing raw counts across groups of different sizes. "School B had 30 walkers to our 18, so walking is more common there" collapses if School B has 200 students: $30/200 = 0.15$ against $18/40 = 0.45$. The count is larger and the share is a third as big. A smaller slip is reporting the relative frequency as 45 when the value is 0.45. The proportion and the percent are the same quantity on two scales, so say which one you are using.

The shares only add to 1 when each observation is counted once. Let students name every method they use and someone who walks and buses appears twice, the shares total more than 1, and the denominator has stopped being the number of students. In a two-way table one word covers three different divisions, by the grand total, by a row total, or by a column total, so a relative frequency there means nothing until you state what you divided by.

Relative frequency is also how a probability gets estimated from data: run a random process many times and the share of trials producing an outcome settles near its probability as the number of trials grows. That share is an estimate built from one set of trials, not the probability itself. Summary statistics for one categorical variable are Unit 1 topic 1.3.