Dispersion Calculator

Did You Know?

Formulas Reference

When to Use Each Measure

Population vs Sample

Population: Use when you have data for every member (e.g., all students in a class). Variance = Σ(x−μ)²/N. Sample: Use when data is a subset. Variance = Σ(x−x̄)²/(n−1). The n−1 (Bessel's correction) gives an unbiased estimate of the population variance.

Step-by-Step Calculation Guide

Interpretation Benchmarks

Frequently Asked Questions

Comparison Table

Applications

Worked Example

Data: 3, 7, 8, 5, 12, 14, 21, 13, 18

Mathematical Properties

• • Range is the simplest measure but uses only two values. It increases with sample size (more data → more extreme values possible).
• • IQR is scale-invariant for linear transformations: IQR(aX+b) = |a|×IQR(X).
• • Variance is additive for independent random variables: Var(X+Y) = Var(X) + Var(Y) when X,Y independent.
• • SD has the same units as the data. Variance has squared units.
• • CV is scale-invariant: CV(aX) = CV(X) for a > 0. It is not location-invariant.
• • MAD satisfies 0 ≤ MAD ≤ SD for any dataset. MAD = SD only when all deviations have the same magnitude.

Excel & Software

Excel: Range = MAX−MIN. IQR: QUARTILE.EXC. Variance: VAR.S or VAR.P. SD: STDEV.S or STDEV.P. Python: numpy.ptp(), scipy.stats.iqr(), numpy.var(), numpy.std(). R: range(), IQR(), var(), sd().

Outlier Impact Example

Data: 10, 12, 14, 16, 18. Mean=14, SD≈2.83, IQR=6. Add outlier 100. New mean≈28.3, SD≈35.4, Range 8→90. IQR stays 6. This shows IQR and QD are robust; Range and SD are sensitive.

Detailed Example: All Measures

For data 2, 4, 6, 8, 10 (n=5): Sorted = 2,4,6,8,10. Sum=30, Mean=6. Deviations: −4,−2,0,2,4. Squared: 16,4,0,4,16. Sum squared=40. Variance (pop)=40/5=8, Variance (sample)=40/4=10. SD(pop)=√8≈2.83, SD(sample)=√10≈3.16. MAD = (4+2+0+2+4)/5 = 2.4. Range=8, Q1=3, Q3=8, IQR=5, QD=2.5. CV = (3.16/6)×100 ≈ 52.7%. Relative Range = (8/6)×100 ≈ 133%.

Choosing the Right Measure

Relationships Between Measures

For normal distributions, IQR ≈ 1.35×SD. Range ≈ 4×SD for small samples (n<20) but increases with n. MAD is always ≤ SD; the ratio MAD/SD depends on the distribution shape. For uniform data, MAD/SD ≈ 0.5. Quartile deviation = IQR/2, so QD ≈ 0.675×SD for normal data.

Data Entry Tips

• Separate values with commas, spaces, tabs, or newlines — any combination works.
• Non-numeric values are ignored. Paste directly from Excel or CSV.
• Duplicate values are kept and affect all measures.
• For large datasets (100+ values), the calculator remains responsive.
• Use the presets to quickly explore different dispersion scenarios.

Further Reading

NIST e-Handbook of Statistical Methods, Khan Academy Statistics, OpenIntro Statistics, Wolfram MathWorld (Standard Deviation, Quartile). For robust statistics, see Huber (1981) on MAD and related estimators.

Preset Descriptions

• Exam Scores: Typical class distribution. Low to moderate dispersion (CV ~10–15%).
• Income: High dispersion from income inequality. CV often 50%+.
• Manufacturing: Tight tolerances. Very low dispersion (CV < 5%).
• Temperature: Daily variation. Moderate dispersion.
• Measurements: Repeated length measurements. Low dispersion.
• Stock Returns: Volatile. Can have negative values; CV interpretation varies.
• Race Times: Athletic performance. Moderate dispersion, often right-skewed.

Chart Interpretation

The Data Spread chart plots each value in order. Look for clusters, gaps, and outliers. The Dispersion Measures bar chart compares Range, IQR, SD, MAD, and Quartile Deviation. Note that Range and SD are often larger than IQR and MAD when outliers exist. Use the chart to see which measures dominate for your dataset.

Second Worked Example

Data: 100, 102, 104, 106, 108. Mean=104, Range=8, SD(pop)=√8≈2.83, CV=(2.83/104)×100≈2.7% (low). Compare to 10, 12, 14, 16, 18: same range and SD, but mean=14, CV≈20.2% (moderate). Same absolute spread, different relative spread — CV captures this.

Bessel's Correction Explained

Sample variance divides by n−1 instead of n because the sample mean x̄ is computed from the same data, making deviations (xᵢ−x̄) slightly smaller on average than (xᵢ−μ). Dividing by n−1 corrects this bias, giving an unbiased estimate of the population variance. For n=2, sample variance = (x₁−x₂)²/2; for n=1, variance is undefined.

Quick Reference: All Formulas

Typical CV Ranges by Domain

Exam scores: 10–20%. Manufacturing: <5%. Stock returns: 20–50%. Income: 40–80%. Lab measurements: <10%. Heights: ~5%. Use these as benchmarks when interpreting your results.

Why Multiple Measures?

No single measure captures all aspects of dispersion. Range is simple but sensitive. IQR is robust. SD is standard for normal data. MAD resists outliers. CV enables cross-scale comparison. Reporting several measures (e.g., mean ± SD and median [IQR]) gives a complete picture. This calculator computes all at once for side-by-side comparison.

Sample Size Considerations

For n<5, quartiles and IQR can be unstable. For n≥30, the sample SD is a good estimate of σ. Range tends to increase with n (more data → more extreme values). CV is less reliable when the mean is close to zero. Always report n alongside your dispersion measures.

Third Worked Example: Skewed Data

Data: 1, 2, 3, 4, 100 (right-skewed). Mean=22, Median=3. Range=99, IQR=2.5, SD≈43.2, MAD=19.2. The mean is pulled right by the outlier; SD is huge. IQR and median are robust. For such data, report median [IQR] rather than mean ± SD.

Units and Scale

Range, IQR, SD, and MAD have the same units as the data. Variance has squared units. CV and Relative Range are unitless (%). When you change units (e.g., meters to centimeters), absolute measures scale; CV and Relative Range stay the same.

Interpreting the Comparison Chart

The bar chart shows Range, IQR, SD, MAD, and Quartile Deviation. If Range is much larger than IQR, you likely have outliers. If SD and MAD are similar, the distribution is roughly symmetric. If SD >> MAD, there are large deviations (possibly outliers). Use this visual to quickly assess your data's spread characteristics.

Copy and Share

After computing, you can copy the results for reports or presentations. Include n, mean, and key dispersion measures. For publication, state whether population or sample variance was used. The side-by-side layout makes it easy to compare multiple datasets.

Related Concepts

Standard Error = SD/√n (uncertainty in the mean). Skewness and kurtosis describe distribution shape. Outlier detection often uses 1.5×IQR rule. The Empirical Rule (68-95-99.7) applies when data is approximately normal. See related calculators for Z-scores, normal distribution, and five-number summary.

Final Notes

This calculator provides a comprehensive view of dispersion. Use it for homework, research, quality control, or exploratory data analysis. All calculations update in real time as you type. The seven presets demonstrate real-world applications across domains. Choose population when you have complete data; sample when you have a subset. The charts and side-by-side comparison help you understand which measures best describe your data.

For symmetric data, report mean ± SD. For skewed data, report median [IQR]. Use CV when comparing across different units or scales.

Dispersion Calculator — All measures in one place. Range, IQR, Variance, SD, MAD, CV, Relative Range, Quartile Deviation.

Educational content: 9+ subsections. Formulas, FAQs, worked examples, applications, interpretation guides.

Summary

This calculator computes all major measures of dispersion: Range, IQR, Variance (pop & sample), SD, MAD, CV, Relative Range, Quartile Deviation. Use presets for real-world data. Charts show data spread and measure comparison.