πŸ”¬ Technical Deep Dive

Quartile Calculation Methods:
R vs Python vs Excel vs SPSS

Why do your results differ? We analyze Hyndman-Fan Types 6, 7, and 8 with code examples for developers.

Published: November 21, 2025
Reading Time: 12 minutes
Difficulty Level: Advanced

πŸ“Œ TL;DR

Different statistical software calculates quartiles differently, leading to confusing discrepancies. This guide explains the Hyndman-Fan classification, compares methods side-by-side, and helps you choose the right approach.

Why this matters: Run the same data through R, Excel, and Python, and you might get three different Q1 values.

1. The Problem: Why Quartile Results Differ

Imagine you have this simple dataset: 

Data: [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]
N = 13 values

You calculate Q1 using different tools:

Tool Function Q1 Result
Rquantile(data, 0.25)6.0
Pythonnp.percentile(data, 25)6.0
Excel=QUARTILE.INC(range, 1)7.0
TextbookTukey method5.0

⚠️ Same data, three different answers!

Answer: They're all "correct" - they're just using different interpolation methods. None is fundamentally better; they serve different use cases.

2. The Hyndman-Fan Classification

In their seminal 1996 paper, Hyndman and Fan identified 9 different methods for calculating sample quantiles. These are designated Type 1 through Type 9.

The Most Common Methods

Type Name Used By Popularity
Type 6Tukey HingesTextbooks, MinitabEducational
Type 7Linear interpolationR, Julia, NumPyData Science Standard
Type 8Median-unbiasedExcel, Google SheetsBusiness Standard
Type 5Piecewise linearSPSS, SASStatistical Software

Why So Many Methods?

Each method makes different assumptions about:

  • How to handle positions between data points (interpolation)
  • Whether to include or exclude the median when splitting data
  • How to weight neighboring values

πŸ’‘ Key Insight: For large datasets (N > 100), all methods converge to nearly identical results. Differences matter most for small to medium datasets (10 < N < 50).

3. Type 6: Tukey Hinges (Textbooks)

Developed by

John Tukey (1977)

Philosophy

"Inclusive median"

Math

Median of each half

How It Works

Example: [1, 3, 5, 7, 9, 11, 13]

  1. Find median: 7 (middle value with N=7)
  2. Lower half including median: [1, 3, 5, 7]
  3. Upper half including median: [7, 9, 11, 13]
  4. Q1 = median of [1, 3, 5, 7] = 4
  5. Q3 = median of [7, 9, 11, 13] = 10

βœ… When to Use

  • Statistics homework and textbook problems
  • Explaining quartiles to beginners
  • Manual calculations by hand
  • Following introductory statistics courses

❌ Avoid When

  • Publishing research (use Type 7)
  • Working in R/Python (default differs)
  • Need consistency with modern tools

Code Example

# R (must specify type = 6 explicitly) 

data <- c(1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25)
Q1_tukey <- quantile(data, 0.25, type = 6)
Q3_tukey <- quantile(data, 0.75, type = 6)

print(paste("Q1:", Q1_tukey))  # Output: Q1: 5
print(paste("Q3:", Q3_tukey))  # Output: Q3: 21

πŸš€ Don't want to calculate by hand?

Use Free Tukey Calculator β†’

4. Type 7: R/Python Default (Data Science)

Status

Modern standard

Philosophy

Linear interpolation

Math

q = (1-Ξ³) Γ— x[j] + Ξ³ Γ— x[j+1]

Why Type 7 Became the Standard

  1. Smooth and continuous - no jumps unlike discrete methods
  2. Unbiased estimator - performs well in statistical theory
  3. Adopted by R (1990s) β†’ became academic standard
  4. NumPy followed β†’ became data science standard

βœ… Perfect For

  • Data science and machine learning projects
  • Scientific research and publications
  • Working in R, Python, or Julia
  • Maximum precision and smoothness
  • Quantile regression

⚠️ Consider Alternatives When

  • Collaborating with Excel users (use Type 8)
  • Need exact textbook matches (use Type 6)

Code Examples

# R (default)

data <- c(1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25)
Q1 <- quantile(data, 0.25)  # type = 7 is default
Q3 <- quantile(data, 0.75)
print(paste("Q1:", Q1))  # Output: Q1: 6

# Python NumPy (default) 

import numpy as np
data = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]
Q1 = np.percentile(data, 25)  # default method = 'linear'
print(f"Q1: {'{'}Q1{'}'}")  # Q1: 6.0

5. Type 8: Excel QUARTILE.INC (Business)

Developed for

Microsoft Excel (1990s)

Philosophy

Median-unbiased

Note

QUARTILE.INC = Type 8

βœ… Perfect For

  • Business analytics and reporting
  • Excel-centric workflows
  • Google Sheets compatibility
  • Collaborating with non-technical teams
  • Corporate environments

⚠️ Consider Alternatives When

  • Publishing academic research (use Type 7)
  • Working primarily in code (R/Python defaults differ)

Code Examples

' Excel

=QUARTILE.INC(A1:A13, 1)  ' Q1 (Type 8)
=QUARTILE.INC(A1:A13, 2)  ' Q2 (Median)
=QUARTILE.INC(A1:A13, 3)  ' Q3

πŸ“Š Verify your Excel results instantly

Open Excel Quartile Calculator β†’

6. Side-by-Side Comparison

Let's compare all four methods with a concrete example.

SAT Math Scores: [480, 510, 530, 560, 600, 620, 650, 680, 710, 750] (N = 10)

Metric Type 6 (Tukey) Type 7 (R/Python) Type 8 (Excel)
Q1530542.5545
Median610610610
Q3680677.5675
IQR150135130

πŸ“Š Observations

  • Median (Q2) is usually consistent across methods
  • Q1 varies from 530 to 545 (15-point spread)
  • IQR varies from 130 to 150 (affecting outlier detection!)

7. Decision Matrix: Which Method to Use?

Choose Type 7 (R/Python Default)

  • Writing data science code
  • Publishing scientific research
  • Using R, Python, Julia, or modern tools
  • Need reproducibility across platforms
  • Working with continuous data

Example use cases: ML pipelines, academic papers, quantile regression

Choose Type 8 (Excel)

  • Business analytics and reporting
  • Collaborating with Excel users
  • Corporate environment workflows
  • Google Sheets compatibility
  • Non-technical stakeholders

Example use cases: Sales dashboards, financial reports, executive summaries

Choose Type 6 (Tukey Hinges)

  • Teaching statistics to beginners
  • Textbook problem sets
  • Manual hand calculations
  • Introductory courses
  • Need simple explanation

Example use cases: Homework, educational materials, intuitive explanations

Choose Type 5 (SPSS)

  • SPSS/SAS workflow required
  • Regulatory compliance specifies it
  • Reproducing legacy analyses

Example use cases: Clinical trials, pharmaceutical studies

8. Code Examples: All Methods in One Place

R: Calculate All Types 

data <- c(1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25)

# Type 6 (Tukey Hinges)
Q1_t6 <- quantile(data, 0.25, type = 6)
Q3_t6 <- quantile(data, 0.75, type = 6)

# Type 7 (R default)
Q1_t7 <- quantile(data, 0.25)  # type = 7 implicit
Q3_t7 <- quantile(data, 0.75)

# Type 8 (Excel-like)
Q1_t8 <- quantile(data, 0.25, type = 8)
Q3_t8 <- quantile(data, 0.75, type = 8)

# Compare
cat("Type 6 Q1:", Q1_t6, "Q3:", Q3_t6, "\n")
cat("Type 7 Q1:", Q1_t7, "Q3:", Q3_t7, "\n")
cat("Type 8 Q1:", Q1_t8, "Q3:", Q3_t8, "\n")

Python: NumPy Methods 

import numpy as np

data = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]

# Type 7 (default in NumPy's percentile)
Q1_t7 = np.percentile(data, 25, method='linear')
Q3_t7 = np.percentile(data, 75, method='linear')

# Type 8 (Excel-like, using scipy)
from scipy.stats.mstats import mquantiles
quartiles_t8 = mquantiles(data, prob=[0.25, 0.75], alphap=1/3, betap=1/3)
Q1_t8, Q3_t8 = quartiles_t8[0], quartiles_t8[1]

print(f"Type 7: Q1={'{'}Q1_t7{'}'}, Q3={'{'}Q3_t7{'}'}")
print(f"Type 8: Q1={'{'}Q1_t8{'}'}, Q3={'{'}Q3_t8{'}'}")

9. Migration Guide

Excel β†’ R/Python

Problem: Your Excel sheet gives Q1 = 545, but Python gives Q1 = 542.5 

# Match Excel (Type 8) in Python
from scipy.stats.mstats import mquantiles
Q1_excel_compatible = mquantiles(data, prob=[0.25], alphap=1/3, betap=1/3)[0]

Python β†’ Excel

Problem: Stakeholders using Excel can't reproduce your Python results

Solution: Export quartile values explicitly with method documentation, or use a universal calculator that shows all methods.

Textbook β†’ Code

Problem: Homework answer (Tukey) doesn't match R output 

# Force R to use Tukey method
Q1_homework <- quantile(data, 0.25, type = 6)

Conclusion

πŸ“Œ Key Takeaways

  1. Type 7 (R/Python) is the modern data science standard
  2. Type 8 (Excel) is the business standard
  3. Type 6 (Tukey) is the education standard
  4. For large datasets (N > 100), method choice barely matters
  5. Always document which method you used for reproducibility

Want to try all methods side-by-side?

Open Universal Quartile Calculator β†’

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