Advanced Biweight Midvariance Calculator

Calculator Input

Data Values

Paste values separated by commas, spaces, semicolons, or new lines.

Tuning Constant (c)

Common default is 9.

Epsilon

Small positive fallback for numerical stability.

Decimal Precision

Display precision for output values.

Center Choice

Median is the usual robust center.

Custom Center

Optional manual center value.

Use included-sample correction for n

Example Data Table

This example shows a mostly compact dataset with one strong outlier. The biweight method downweights the extreme value instead of letting it dominate dispersion.

Observation	Value	Comment
1	12.0	Regular value
2	13.0	Regular value
3	13.5	Regular value
4	14.0	Central value
5	14.2	Central value
6	14.5	Central value
7	15.0	Regular value
8	15.1	Regular value
9	15.3	Regular value
10	40.0	Extreme outlier

Formula Used

1. Center: M = median(x_i) unless a custom center is supplied.

2. Median absolute deviation: MAD = median(|x_i − M|).

3. Standardized distance: u_i = (x_i − M) / (c × MAD), using only values where |u_i| < 1.

4. Biweight midvariance:
BMV = [n × Σ((x_i − M)²(1 − u_i²)⁴)] / [Σ((1 − u_i²)(1 − 5u_i²))]²

5. Robust scale: Scale = √BMV

When the included-sample option is enabled, the calculator uses the number of included observations for the n term. Otherwise it uses the total parsed count.

How to Use This Calculator

Paste your dataset into the values box using commas, spaces, or new lines.
Keep the tuning constant at 9 for standard robust analysis unless you want a different cutoff behavior.
Choose median for the usual center, or provide a custom center for specialized comparisons.
Set display precision and keep epsilon small for numeric stability.
Submit the form to place the result block above the calculator.
Review included and excluded observations in the detail table.
Use the CSV button for spreadsheet work or the PDF button for reporting.

FAQs

1. What does biweight midvariance measure?

It estimates dataset spread while reducing the influence of extreme values. It is a robust alternative to ordinary variance when outliers may distort standard dispersion measures.

2. Why use this instead of ordinary variance?

Ordinary variance squares every deviation equally, so a few large outliers can dominate the result. Biweight midvariance downweights distant points and can better represent the main body of data.

3. What does the tuning constant do?

The tuning constant controls how quickly observations lose influence as they move away from the center. Larger values include more points; smaller values make the estimator more aggressive.

4. What is the role of MAD?

MAD is the median absolute deviation from the chosen center. It supplies a robust scale for the standardized distance term used in the biweight weighting rule.

5. Why might some observations be excluded?

Any value with |u| greater than or equal to 1 falls outside the biweight cutoff. Those points receive zero contribution in the estimator.

6. What happens if MAD is zero?

If the dataset has too many identical central values, MAD can become zero. In that case robust weighting cannot be stabilized, so the calculator returns an error or zero for constant data.

7. What does robust scale mean here?

Robust scale is the square root of biweight midvariance. It behaves like a robust standard deviation and is often easier to interpret than the variance itself.

8. What is included in the exported files?

The exports include the result summary and the detailed observation table with value status, standardized distance, weight factor, and the contribution terms used in the formula.