🎓 Statistics & Data • Descriptive Statistics
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IQR Calculator (Interquartile Range & Outliers)
Calculates the middle 50% spread (IQR = Q3 - Q1) and identifies outliers beyond 1.5 × IQR fences.
Descriptive Statistics Engine
Dataset Analysis & Five-Number Summary
Input comma or space separated numbers. Computes central tendency, spread, shape moments, Tukey fences, and interactive plots.
Raw Data Values10 observations parsed
Presets:
Mean (x̄)24.7Sum / N
Median (Q₂)23.550th percentile
Sample Std Dev (s)8.7819s² = 77.1222
IQR (Q₃ - Q₁)12Middle 50%
Interactive Data Visualization
IQR Box (Q₁ to Q₃) Median (23.5)✕ Mean (24.7) Outlier (Outside 1.5×IQR)
Comprehensive Metrics Table
Count (N)10
Sum (Σx)247
Mode22
Range28 (12 - 40)
Sample Variance (s²)77.1222
Pop Variance (σ²)69.41
Standard Error (SE)2.7771
Coef of Variation (CV)35.55%
10% Trimmed Mean24.375
First Quartile (Q₁)18
Third Quartile (Q₃)30
Tukey Fences [LF, UF][0, 48]
Skewness0.319 (Symmetric)
Excess Kurtosis-0.525 (Platykurtic)
Detected OutliersNone detected
Single Value Evaluator (Z-Score & Percentile)Standardized deviation from sample mean: z = (x - x̄) / s
Standardized Z-Score+0.034 σ0.034 standard deviations above mean
Percentile Rank55th PercentileBetter than or equal to 55% of sample
Step-by-Step Statistical Derivations
1. Sorted Ascending Dataset:
12, 15, 18, 22, 22, 25, 28, 30, 35, 40
2. Central Tendency:
- Mean: x̄ = Σx / n = 247 / 10 = 24.7
- Median: Ordered middle position average of items 5 & 6 = 23.5
- Mode: 22 (Frequency = 2)
3. Sum of Squared Deviations & Variance:
- SS = Σ(x_i - x̄)² = 694.1
- Sample Variance: s² = SS / (n - 1) = 694.1 / 9 = 77.1222
- Sample Standard Deviation: s = √s² = √77.1222 = 8.7819
4. Five-Number Summary & Outlier Fences:
- Min = 12, Q₁ = 18, Median = 23.5, Q₃ = 30, Max = 40
- IQR = Q₃ - Q₁ = 30 - 18 = 12
- Lower Fence = Q₁ - 1.5×IQR = 18 - 18 = 0
- Upper Fence = Q₃ + 1.5×IQR = 30 + 18 = 48
What is the Interquartile Range (IQR)?
The interquartile range is the spread of the middle 50% of values, used as a robust measure of statistical variability immune to extreme outliers.
Formula & Step-by-Step Calculation
IQR = Q₃ - Q₁, Lower Fence = Q₁ - 1.5·IQR, Upper Fence = Q₃ + 1.5·IQR
Tukey’s outlier detection rule.
Worked Step-by-Step Examples
Example 1
Dataset with Q1 = 20 and Q3 = 50. Find IQR and outlier fences.
Solution: IQR = 30; Outlier Fences: [-25, 95]
• IQR = 50 - 20 = 30; Lower = 20 - 1.5(30) = -25; Upper = 50 + 1.5(30) = 95
Common Real-World & Academic Use Cases
- ✓ Data cleaning in machine learning pipelines
- ✓ Outlier filtering in clinical datasets
- ✓ Robust finance risk management
How to Use the IQR Calculator (Interquartile Range & Outliers)
1
Enter Numbers
Input observations.
2
Check Fences
Inspect IQR and any flagged outliers.
Frequently Asked Questions
Q: Why is 1.5 used in Tukey fences?
In a normal distribution, ±1.5 IQR encompasses approximately 99.3% of data, making values beyond it rare outliers.