What is a quantile?

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Maybe visually would help. Definitions usually have a lot of mathematical jargon.

In: Mathematics

A quantile is like a bin. Think of it like sorting gravel. You have a big conveyor belt carrying rocks that you want to sort. You set up a vibrating mesh tray with specifically sized holes in the mesh to let tiny rocks through (say, less than 1 mm in diameter). It vibrates to jostle the rocks and keep things moving along. The next mesh has 2 mm holes in it, to catch small rocks. The next is maybe 5 mm, then 10 mm, then the rest is waste (too big for gravel).

What you’ve done is binned your gravel by size. The first quantile is smallest (<1 mm), the next is larger (1-2 mm), the next larger (2-5 mm), and the last is largest (5-10mm). It’s just a way of expressing sorting into groupings.

[Here is a picture of quantiles](https://upload.wikimedia.org/wikipedia/commons/5/5e/Iqr_with_quantile.png)

In this picture, the x (horizontal) axis represents the value of something. It could be, for example, the age of a person.

The y (vertical) axis would represent how many members of the sample set have that value. If we are talking about the age of people, the y-axis represents the number of people of a given age.

The curve as a whole represents the whole population (or at least some sample size of the population). But let’s say you want to divide the population up into groups by age so that each group has the same number of people in it.

Those divisions are called quantiles.

In the linked picture, we divided the group into four, which makes them *quar*tiles. From the left side of the graph to Q1 has the same size as Q1 to Q2 which has the same size as Q2 to Q3 which has the same size as Q3 to the right-side of the graph.

100 kids take a 100-question test. One kid only gets one question right, one gets two right, and so on up to 100. The kid who scored 50 scored better than 50% of the kids (because half of them scored less). The kid who scored 99 did better than 99% of the kids. You can flip that around, too. Let’s say you want to tailor classes to the kids, so you have a class with extra help for the kids that are struggling, a “normal” class, and an advanced class for the kids that are doing really well. The tutoring class and advanced class each have room for one fifth of the kids. What score do you need to be in the top 20%? You need at least an 80 to get into the advanced class. What’s the highest score you can have and still be in the bottom 20%? If you score under 20, you would be in the tutoring class.

It can be abstracted to less round numbers, less evenly distributed “scores” and a continuous variable instead of discrete test scores. Let’s say you measure the weight of 1,023 kids, to figure out who might need food assistance and who might need classes about healthy eating. Most of the kids weigh between 80-120 pounds. Maybe the area has a lot of poverty, so there are more kids that weigh less than that, say 65-80, than in the same 15-pound weight range from 120-135.

Since the weights aren’t evenly distributed or discrete like the test scores were, its harder to eyeball and say *this* is the bottom fifth and *that* is the top fifth. So you can use quantiles to calculate what the bottom and top 20% are. In this case the 20% quantile might be 70 pounds and the 80% quantile might be 120. There are the same number of kids in each group, but they have different weight ranges because the weights aren’t evenly distributed around the average of 100 pounds.

It may help to know that “quantile” is just a more general term that includes “percentile.” If you understand what percentile means (i.e. that a record in the 85th percentile has a higher value than 85% of the data), then any other quantile is the same concept, but breaking the data into a different number of pieces than 100. For example, “quartiles” break the data into 4 equally sized pieces where, e.g., a record in the 3rd quartile has a higher value than any record in the 1st or 2nd quartile. Some others:

-Tercile – 3 pieces

-Quintile – 5 pieces

-Decile – 10 pieces

-Vigintile – 20 pieces (you can thank France for this weirdo)

All of these (along with percentile) are quantiles.