Dispersion Calculator
Print PageA Dispersion Calculator (also known as a Statistical Dispersion & Variability Calculator, Measures of Data Spread Utility, Mean Absolute Deviation [MAD] Generator, or Coefficient of Relative Dispersion Analyzer) computes both Absolute Measures of Dispersion (Range R = Max - Min, Interquartile Range IQR = Q3 - Q1, Quartile Deviation QD = (Q3 - Q1) ÷ 2, Mean Absolute Deviation MAD = ∑ |xi - X̄| ÷ n, Sample Variance s2, and Sample Standard Deviation s) and Relative Measures of Dispersion (Coefficient of Variation CV = [s ÷ X̄] · 100%, Coefficient of Range (Max - Min) ÷ (Max + Min), Coefficient of Quartile Deviation (Q3 - Q1) ÷ (Q3 + Q1), and Coefficient of Mean Deviation MAD ÷ X̄).
In manufacturing assembly line quality control, financial risk assessment, medical laboratory assay testing, and educational grading, **dispersion** quantifies the extent to which individual observations scatter, spread out, or stretch away from the central average mean.
Our free online Dispersion Calculator provides instant calculations across all absolute and relative variability parameters:
- Range (R):
Range = Max - Min(Distance between extreme boundary values). - Interquartile Range (IQR) & Quartile Deviation (QD):
IQR = Q3 - Q1,QD = ( Q3 - Q1 ) ÷ 2. - Mean Absolute Deviation (MAD):
MAD = ∑ | xi - X̄ | ÷ n(Average absolute distance from the mean). - Sample Variance (s2) & Standard Deviation (s):
s2 = ∑ ( xi - X̄ )2 ÷ ( n - 1 ),s = √[ s2 ]. - Coefficient of Variation (CV Percentage):
CV = ( s ÷ X̄ ) · 100%(Standardized relative dispersion metric). - Coefficient of Range:
Coefficient of Range = ( Max - Min ) ÷ ( Max + Min ). - Coefficient of Quartile Deviation:
Coefficient of QD = ( Q3 - Q1 ) ÷ ( Q3 + Q1 ). - Coefficient of Mean Deviation:
Coefficient of MAD = MAD ÷ X̄(orMAD ÷ Median).
Master Statistical Dispersion Reference Table (Assembly Line Production Output: n = 5 Days, Mean X̄ = 100.0 Units)
The table below displays daily production output for two factory assembly lines with identical average means (X̄ = 100.0 units/day), highlighting how absolute and relative measures of dispersion identify machine instability:
| Dispersion Metric Category | Mathematical Formula | Machine Line A Output (Stable) | Machine Line B Output (Volatile) | Comparative Machine Stability Evaluation |
|---|---|---|---|---|
| Daily Output Dataset (units) | Raw n = 5 Observations | 100, 102, 99, 101, 98 | 70, 130, 80, 120, 100 | Line B Has Massive Daily Fluctuations |
| Arithmetic Mean (X̄) | ∑ x ÷ 5 | X̄A = 100.00 units | X̄B = 100.00 units | Identical Averages (Hides True Variance!) |
| Absolute Range (R) | Max – Min | RA = 4.00 units (102 – 98) | RB = 60.00 units (130 – 70) | Line B Range is 15 Times Wider |
| Coefficient of Range | (Max-Min) ÷ (Max+Min) | 0.0200 (2.00%) | 0.3000 (30.00%) | Relative Range 15x Larger on Line B |
| Mean Absolute Deviation (MAD) | ∑ |x – X̄| ÷ 5 | MADA = 1.20 units | MADB = 20.00 units | Line B Off-Target by 20 Units Average |
| Sample Variance (s2) | ∑ (x – X̄)2 ÷ 4 | sA2 = 2.50 units2 | sB2 = 650.00 units2 | Line B Variance 260 Times Greater |
| Sample Standard Deviation (s) | √[ s2 ] | sA = 1.5811 units | sB = 25.4951 units | Line A Standard Deviation 16x Tighter |
| Coefficient of Variation (CV) | ( s ÷ X̄ ) · 100% | CVA = 1.5811% (1.58%) | CVB = 25.4951% (25.50%) | Line A Achieves Superior 1.58% Six Sigma Control |
Step-by-Step Manufacturing Line Dispersion Calculation
To calculate measures of dispersion for Line A (100, 102, 99, 101, 98) and Line B (70, 130, 80, 120, 100):
Part A (Line A Dispersion Metrics):
Step 1 (Calculate Mean X̄_A): X̄_A = (100 + 102 + 99 + 101 + 98) ÷ 5 = 500 ÷ 5 = 100.0 units
Step 2 (Calculate Range R_A): R_A = 102 - 98 = 4.0 units; Coeff of Range = 4 ÷ 200 = 0.0200 (2.0%)
Step 3 (Calculate MAD_A): MAD_A = (|0| + |2| + |-1| + |1| + |-2|) ÷ 5 = 6 ÷ 5 = 1.20 units
Step 4 (Calculate Variance & SD): s_A^2 = (0 + 4 + 1 + 1 + 4) ÷ 4 = 10 ÷ 4 = 2.50 units^2 &implies; s_A = √2.50 = 1.5811 units
Step 5 (Calculate CV_A): CV_A = (1.5811 ÷ 100.0) · 100% = 1.5811% ≈ 1.58%
Part B (Line B Dispersion Metrics):
Step 1 (Calculate Mean X̄_B): X̄_B = (70 + 130 + 80 + 120 + 100) ÷ 5 = 500 ÷ 5 = 100.0 units
Step 2 (Calculate Range R_B): R_B = 130 - 70 = 60.0 units; Coeff of Range = 60 ÷ 200 = 0.3000 (30.0%)
Step 3 (Calculate MAD_B): MAD_B = (|-30| + |30| + |-20| + |20| + |0|) ÷ 5 = 100 ÷ 5 = 20.00 units
Step 4 (Calculate Variance & SD): s_B^2 = (900 + 900 + 400 + 400 + 0) ÷ 4 = 2,600 ÷ 4 = 650.0 units^2 &implies; s_B = √650.0 = 25.4951 units
Step 5 (Calculate CV_B): CV_B = (25.4951 ÷ 100.0) · 100% = 25.4951% ≈ 25.50%
Thus, while both lines produce 100 units/day average, Line A is 16 times more stable and consistent than Line B (1.58% vs 25.50% CV).
Dispersion Measures Comparison: Absolute vs. Relative Measures of Dispersion
Below is a comparative reference chart detailing when to use Absolute versus Relative measures of dispersion:
| Dispersion Classification | Specific Statistical Metrics | Measurement Units | Primary Practical Application |
|---|---|---|---|
| Absolute Measures of Dispersion | Range, IQR, Mean Absolute Deviation (MAD), Standard Deviation (s), Variance (s2) | Same physical units as original raw data ($, kg, mm) | Measuring physical spread within a single dataset. |
| Relative Measures of Dispersion | Coefficient of Variation (CV), Coeff of Range, Coeff of QD, Coeff of MAD | DIMENSIONLESS RATIOS or PERCENTAGES (%) | Comparing dispersion between datasets with different scales or units. |
History & Mathematics: 1893 Karl Pearson to 1914 Arthur Bowley
1893 Karl Pearson & Standard Dispersion
In 1893, English mathematician Karl Pearson formally defined standard deviation and established mathematical measures of dispersion in Philosophical Transactions of the Royal Society of London.
1914 Arthur Lyon Bowley & Relative Dispersion Coefficients
In 1914, British statistician Arthur Lyon Bowley introduced quartile deviation coefficients and relative measures of dispersion in Elements of Statistics, enabling cross-industry economic comparison.
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Frequently Asked Questions (FAQ)
What is the difference between absolute and relative measures of dispersion?
Absolute measures (like standard deviation or range) carry the physical units of the original data (e.g. dollars or grams). Relative measures (like the coefficient of variation) are unitless percentages that allow direct comparison between datasets with different units or scales.
What is Mean Absolute Deviation (MAD)?
Mean Absolute Deviation (MAD) is the average of absolute differences between each data point and the mean: MAD = ∑ |xi - X̄| ÷ n.
Which measure of dispersion is best?
Standard deviation is the gold standard for normal distributions without outliers. Interquartile Range (IQR) is best for skewed data with extreme outliers.