Quantile — Definition, Formula & Examples
A quantile is a cut point that divides a sorted dataset into equal-sized groups. Quartiles, percentiles, and deciles are all specific types of quantiles.
Given an ordered dataset, the -quantiles are the values that partition the data into subsets of approximately equal size. The -th -quantile is the value below which a fraction of the data falls.
How It Works
To find quantiles, first sort your data from smallest to largest. Then decide how many equal groups you want — 4 groups gives you quartiles, 10 gives deciles, and 100 gives percentiles. The cut points between these groups are the quantile values. For example, the 2-quantile (also called the median) splits the data into two equal halves. When a data value falls exactly on a quantile boundary, different methods (such as interpolation) may be used depending on the context.
Worked Example
Problem: Find the 4-quantiles (quartiles) of the dataset: 3, 7, 8, 12, 15, 18, 21, 24, 30.
Step 1: The data is already sorted. There are 9 values, and 4-quantiles require 3 cut points that split the data into 4 groups of roughly equal size.
Step 2: The 1st quartile (Q1) is the median of the lower half: 3, 7, 8, 12. That gives us the average of the two middle values.
Step 3: The 2nd quartile (Q2) is the overall median, which is the 5th value.
Step 4: The 3rd quartile (Q3) is the median of the upper half: 18, 21, 24, 30.
Answer: The three 4-quantiles are Q1 = 7.5, Q2 = 15, and Q3 = 22.5, dividing the data into four groups.
Why It Matters
Quantiles are essential in AP Statistics for constructing box plots and identifying outliers. In fields like finance and public health, quantile-based measures (such as income percentiles or growth chart percentiles) communicate how an individual observation compares to a population.
Common Mistakes
Mistake: Confusing the number of quantiles with the number of groups.
Correction: The -quantiles produce cut points and groups. For instance, quartiles are 4-quantiles with 3 cut points, not 4.