Mathematics 1/Data Analysis/03._Quartiles

Quartiles

Mathematical Foundation

Quartiles are statistical measures that divide an ordered dataset into four equal parts, providing insights into the distribution and spread of data. They extend the concept of the median by identifying the boundaries that separate the lowest 25%, middle 50%, and highest 25% of the data. Quartiles are essential for understanding data distribution, detecting outliers, and creating box plots.

Quartile Definitions:

First Quartile (Q₁): 25th percentile - separates lowest 25% from upper 75%
Second Quartile (Q₂): 50th percentile - same as the median
Third Quartile (Q₃): 75th percentile - separates lowest 75% from upper 25%

Calculation Methods:

Position Formula: Q_k position = k(n+1)/4 where k = 1, 2, 3
Inclusive Method: Q₁ at position (n+1)/4, Q₃ at position 3(n+1)/4
Exclusive Method: Q₁ at position (n+3)/4, Q₃ at position (3n+1)/4

Interactive Quartiles Calculator

Data Values (comma-separated):

Data Visualization

Q₁

Q₂ (Median)

Q₃

IQR

Worked Example 1: Quartiles for Odd Number of Values

Find the quartiles of: 12, 7, 15, 9, 21, 18, 13

Solution:

Step 1: Sort the data: 7, 9, 12, 13, 15, 18, 21

Step 2: n = 7, positions: Q₁ at (7+1)/4 = 2nd position, Q₃ at 3(7+1)/4 = 6th position

Step 3: Q₁ = 9, Q₂ = 13, Q₃ = 18

Worked Example 2: Quartiles for Even Number of Values

Find the quartiles of: 4, 7, 2, 9, 5, 8, 10, 6

Solution:

Step 1: Sort the data: 2, 4, 5, 6, 7, 8, 9, 10

Step 2: n = 8, positions: Q₁ at (8+1)/4 = 2.25th → average of 2nd and 3rd, Q₃ at 3(8+1)/4 = 6.75th → average of 6th and 7th

Step 3: Q₁ = (4 + 5)/2 = 4.5, Q₂ = (6 + 7)/2 = 6.5, Q₃ = (8 + 9)/2 = 8.5

Worked Example 3: Interquartile Range (IQR)

For the data: 3, 7, 8, 12, 15, 18, 22, 25, find Q₁, Q₃, and IQR.

Solution:

Sorted: 3, 7, 8, 12, 15, 18, 22, 25

Q₁ = (7 + 8)/2 = 7.5, Q₃ = (18 + 22)/2 = 20

IQR = Q₃ - Q₁ = 20 - 7.5 = 12.5

Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of the data:

IQR = Q₃ - Q₁

The IQR is used to detect outliers and is more robust than the range since it excludes extreme values.

Quartile Deviation

The quartile deviation (or semi-interquartile range) is half of the IQR:

Quartile Deviation = IQR / 2 = (Q₃ - Q₁) / 2

This measures the average spread around the median.

Box Plot Construction

Quartiles are used to construct box plots:

Minimum: Smallest value (or Q₁ - 1.5×IQR for outliers)
Q₁: Bottom of box (25th percentile)
Q₂ (Median): Line inside box (50th percentile)
Q₃: Top of box (75th percentile)
Maximum: Largest value (or Q₃ + 1.5×IQR for outliers)

Practice Problems

Problem 1: Find the quartiles of: 5, 8, 12, 15, 18, 20, 22

Solution:
Sorted: 5, 8, 12, 15, 18, 20, 22
Q₁ position = (7+1)/4 = 2nd value = 8
Q₂ = 15
Q₃ position = 3(7+1)/4 = 6th value = 20

Problem 2: Find the quartiles of: 3, 7, 9, 12, 15, 18

Solution:
Sorted: 3, 7, 9, 12, 15, 18
Q₁ position = (6+1)/4 = 1.75th → average of 1st and 2nd = (3+7)/2 = 5
Q₂ = (9+12)/2 = 10.5
Q₃ position = 3(6+1)/4 = 5.25th → average of 5th and 6th = (15+18)/2 = 16.5

Problem 3: For the data: 2, 5, 7, 8, 10, 12, 15, 18, find IQR and quartile deviation.

Solution:
Sorted: 2, 5, 7, 8, 10, 12, 15, 18
Q₁ = (5+7)/2 = 6, Q₃ = (12+15)/2 = 13.5
IQR = 13.5 - 6 = 7.5
Quartile deviation = 7.5 / 2 = 3.75

Problem 4: The quartiles of a dataset are Q₁ = 25, Q₃ = 75. Find the IQR.

Solution:
IQR = Q₃ - Q₁ = 75 - 25 = 50

Problem 5: Find the quartiles of: 1, 3, 5, 7, 9, 11, 13, 15, 17

Solution:
Sorted: 1, 3, 5, 7, 9, 11, 13, 15, 17
Q₁ position = (9+1)/4 = 2.5th → average of 2nd and 3rd = (3+5)/2 = 4
Q₂ = 9
Q₃ position = 3(9+1)/4 = 7.5th → average of 7th and 8th = (13+15)/2 = 14

Key Takeaways

Definition: Quartiles divide ordered data into four equal parts: Q₁ (25th percentile), Q₂ (50th percentile, median), Q₃ (75th percentile).
Calculation: Position formulas: Q_k at k(n+1)/4 where k=1,2,3; for non-integer positions, interpolate between values.
IQR: Interquartile range (Q₃ - Q₁) measures the spread of the middle 50% of data and is robust against outliers.
Applications: Used in box plots, outlier detection, comparing data spread, and understanding data distribution.
Robustness: Quartiles are less affected by extreme values than the mean and range, making them suitable for skewed data.
Box Plot: Visual representation using quartiles: min, Q₁, median, Q₃, max (with whiskers for outliers).
Quartile Deviation: Half of IQR, measuring average spread around the median.