Interquartile Range Calculator
Interquartile Range Calculator
Enter your data to calculate quartiles and detect outliers:
- Q1, Q2 (Median), Q3 quartile values
- Interquartile Range (IQR = Q3 - Q1)
- Outlier detection using 1.5×IQR rule
- Box plot visualization
- Complete statistical summary
Formula: IQR = Q3 - Q1, Outliers: < Q1-1.5×IQR or > Q3+1.5×IQR
Calculate the interquartile range (IQR) and analyze data distribution with quartiles. Perfect for outlier detection and statistical data analysis.
What is Interquartile Range?
The Interquartile Range (IQR) measures the spread of the middle 50% of data values, calculated as the difference between the third quartile (Q3) and first quartile (Q1).
Understanding Quartiles
Quartile Definitions
Q1 (First Quartile): 25th percentile - 25% of data is below this value
Q2 (Second Quartile): 50th percentile - the median of the dataset
Q3 (Third Quartile): 75th percentile - 75% of data is below this value
IQR: Q3 - Q1, represents the middle 50% spread
Data Distribution
25% of data is below Q1
25% of data is between Q1 and Q2
25% of data is between Q2 and Q3
25% of data is above Q3
Outlier Detection Method
1.5 × IQR Rule:
Lower Fence:
Q1 - 1.5 × IQR
Values below this are potential outliers
Upper Fence:
Q3 + 1.5 × IQR
Values above this are potential outliers
Note: The 1.5×IQR rule is a common method, but not the only way to detect outliers. Context and domain knowledge are important for outlier interpretation.
IQR Applications
Data Analysis
Quality Control: Manufacturing defect detection
Finance: Identifying unusual transactions
Healthcare: Abnormal test results
Sports: Performance outlier analysis
Research
Psychology: Behavioral data analysis
Education: Test score distributions
Marketing: Customer behavior patterns
Science: Experimental data validation
Business Intelligence
Sales Analysis: Performance benchmarking
Risk Management: Identifying anomalies
Operations: Process variation control
HR Analytics: Salary range analysis
Box Plot Interpretation
Box Plot Components:
- Box: Represents the IQR (Q1 to Q3)
- Median Line: Divides the box at Q2
- Whiskers: Extend to minimum and maximum non-outlier values
- Outliers: Points beyond the whiskers
What Box Plots Show:
- Skewness: Asymmetric box or median position
- Variability: Width of box and whiskers
- Outliers: Unusual data points
- Quartiles: Data distribution percentiles
Distribution Types:
Symmetric: Median centered, equal whiskers
Right-skewed: Long right whisker
Left-skewed: Long left whisker
Uniform: Very small IQR
IQR vs Other Measures
| Measure | Formula | Robust to Outliers | Use Case |
|---|---|---|---|
| IQR | Q3 - Q1 | Yes | Outlier detection, robust spread |
| Range | Max - Min | No | Simple spread, sensitive to extremes |
| Standard Deviation | √(Σ(x-μ)²/n) | No | Normal distributions, precise calculations |
| MAD | Median |x - median| | Yes | Very robust alternative to SD |
Step-by-Step Calculation
- Sort Data: Arrange values in ascending order
- Find Q1: 25th percentile position = 0.25(n-1)
- Find Q2: 50th percentile (median) = 0.5(n-1)
- Find Q3: 75th percentile position = 0.75(n-1)
- Calculate IQR: IQR = Q3 - Q1
- Find Fences: Lower = Q1-1.5×IQR, Upper = Q3+1.5×IQR
- Identify Outliers: Values outside fence boundaries
IQR Example
Sample Calculation
Data: 2, 4, 4, 4, 5, 5, 7, 9
Q1: 4 (25th percentile)
Q2: 4.5 (median)
Q3: 6 (75th percentile)
IQR: 6 - 4 = 2
Lower Fence: 4 - 1.5×2 = 1
Upper Fence: 6 + 1.5×2 = 9
Outliers: None (all values within fences)
Using the IQR Calculator
- Input Data: Enter at least 4 numbers separated by spaces or commas
- Calculate: Click the "Calculate IQR" button
- Review Quartiles: Check Q1, Q2, and Q3 values
- Analyze IQR: Understand the middle 50% spread
- Check Outliers: Review detected outliers and their impact
- Interpret Results: Use box plot visualization for insights
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