The Ultimate Guide to
Quartile Algorithms
Did you know there are at least 9 different ways to calculate a quartile? We break down the Hyndman & Fan classification system used by R and Python.
Introduction: The "One Right Answer" Myth
If you ask five statisticians to calculate the quartiles of a small dataset, you might get five different answers. And technically, they could all be correct.
In 1996, statisticians Rob J. Hyndman and Yanan Fan
published a seminal paper titled "Sample Quantiles in Statistical Packages", where they classified the 9 most common methods used
by software. This classification is now the standard
used in the R programming language (type=1
through type=9).
Why does this matter?
If you are replicating a study or verifying homework, using the wrong "Type" can lead to "incorrect" answers, even if your math is sound.
The 9 Types of Quantiles
The methods are generally categorized into two groups: Discontinuous (Types 1-3) and Continuous (Types 4-9).
Discontinuous Methods (Discrete)
These methods always return a value from the original dataset (or a midpoint). They are often used for small samples or ordinal data.
- Type 1 (Inverse of Empirical Distribution Function): Used by some older software. It simply takes the value at the specific rank.
- Type 2 (Similar to Type 1 but averages at discontinuities): Used in some SAS procedures.
- Type 3 (Nearest Even Order Statistic): The default in SAS. It rounds to the nearest rank.
Continuous Methods (Linear Interpolation)
These methods interpolate between data points, meaning the result might be a decimal number that doesn't exist in your dataset.
| Type | Description | Software Usage |
|---|---|---|
| Type 4 | Linear interpolation of the empirical distribution function. | Rarely used as default. |
| Type 5 | Piecewise linear function. | Hydrology applications. |
| Type 6 | Weibull plotting position (N+1 basis). | Minitab, SPSS, Excel (QUARTILE.EXC). |
| Type 7 | Linear interpolation (N-1 basis). | R Default, Python (NumPy), Excel (QUARTILE.INC). |
| Type 8 | Approximately median-unbiased. | Recommended by Hyndman & Fan for most uses. |
| Type 9 | Approximately unbiased (Normal distribution). | Used in some financial modeling. |
Most Common Methods
While there are 9 types, you usually only encounter three in practice.
Type 7 (R Default, Excel INC)
The gold standard for modern data science. It
interpolates data linearly.
h = (n-1)*p + 1
Type 6 (Excel EXC, Minitab)
Commonly used when you want to extend the range
for small samples (Excludes endpoints).
h = (n+1)*p
Tukey's Hinges (EDA)
Not strictly one of the 9 types. Used for hand calculations and box plots. It involves splitting data by the median.
Code Implementation
In R Language
data <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
# Compare all 9 types
sapply(1:9, function(type) {
quantile(data, probs = c(0.25, 0.75), type = type)
})In Python (NumPy)
import numpy as np
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
# Method 'linear' corresponds to R type 7
q1 = np.percentile(data, 25, method='linear')
q3 = np.percentile(data, 75, method='linear')
# Other methods include: 'lower', 'higher', 'midpoint', 'nearest'Which one should you use?
For Students
Check your textbook! If you are doing hand calculations, you are almost certainly using Tukey's Hinges (Inclusive). Use our Tukey Calculator.
For Data Scientists
Stick to Type 7. It is the standard in R, Python (NumPy, Pandas), and Scikit-learn. It has the best mathematical properties for continuous data.
For Business Analysts
If you work in Excel, use QUARTILE.INC (which
is Type 7). Avoid the legacy QUARTILE function
if possible.
Confused? Let us check for you.
PlotNerd calculates your data using the 4 most common methods simultaneously, so you can see the difference instantly.
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