Inter-Quartile Range (IQR) Calculator

Use this unified calculator to find the inter-quartile range (IQR) for both ungrouped (raw) data and grouped (frequency distribution) data. IQR measures the spread of the middle 50% of data and is essential for outlier detection.

Quick Start

Inter-Quartile Range Calculator
Data Type	Ungrouped (Raw Data) Grouped (Frequency Distribution)
Enter the X Values (Separated by comma,)
Type of Frequency Distribution	DiscreteContinuous
Enter the Classes for X (Separated by comma,)
Enter the frequencies (f) (Separated by comma,)

Results
Number of Observations (N):
Ascending order of X values:
First Quartile (Q₁):
Second Quartile (Q₂) / Median:
Third Quartile (Q₃):
Inter-Quartile Range (IQR):

Understanding Inter-Quartile Range (IQR)

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

IQR = Q₃ - Q₁
Robust to outliers (not affected by extreme values)
Contains 50% of all data points
Used for box plot construction
Essential for outlier detection

Why Use IQR?

Advantage	Explanation
Robust	Unaffected by outliers or extreme values
Interpretable	Shows where central 50% of data lives
Box Plot	Foundation for box plot visualization
Outlier Detection	Standard method using 1.5×IQR rule
Skewness Resistant	Symmetric measure independent of distribution shape

Formula

Inter-Quartile Range

$$IQR = Q_3 - Q_1$$

Where:

Q₃ = Third quartile (75th percentile)
Q₁ = First quartile (25th percentile)

Outlier Detection Rule

Lower outlier threshold: Q₁ - 1.5 × IQR
Upper outlier threshold: Q₃ + 1.5 × IQR
Values beyond these thresholds are considered outliers

Worked Examples

Example 1: Ungrouped Data

Data: Test scores: 45, 52, 58, 63, 72, 81, 85, 90, 95

Solution:

Q₁ = 55, Q₂ = 72, Q₃ = 87.5

$$IQR = 87.5 - 55 = 32.5$$

Answer: IQR = 32.5

Example 2: Outlier Detection

Using the example above:

Lower threshold = 55 - 1.5(32.5) = 6.25
Upper threshold = 87.5 + 1.5(32.5) = 136.25

Result: All scores are within range (no outliers)

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