SMp(x) Distribution Calculator

Apply the SMp(x) transformation to non-negative values, generating normalized weights proportional to x^p.

Non-negative values (separated by spaces or commas)

Exponent (p)

p > 1 emphasizes larger values; 0 < p < 1 flattens the distribution.

Total weight: 100.0000% (≈ 100%)

x	x^p	Weight	Cumulative
1.0000	1.000000	3.545%	3.545%
3.0000	5.196152	18.423%	21.968%
5.0000	11.180340	39.640%	61.608%
2.0000	2.828427	10.028%	71.636%
4.0000	8.000000	28.364%	100.000%

How to Use This Calculator

Provide non-negative data values that you want to convert into a probability-like distribution.
Select an exponent p to control emphasis on larger or smaller observations.
Review normalized weights to understand each value's contribution under SMp(x).
Use cumulative weights to evaluate thresholds or percentile cutoffs.

Formula

SMp(x) weight_i = x_i^p / Σ x_j^p

0 ≤ weight_i ≤ 1 and Σ weight_i = 1

The SMp(x) family generalizes proportional weighting by raising values to a power p before normalizing, yielding tunable emphasis.

Full Description

SMp(x) (sometimes referred to as the “softmax with power p”) transforms non-negative data into a distribution-like weighting. By adjusting p, analysts can highlight dominant contributors (p > 1) or smooth differences across values (0 < p < 1). The transformation is useful in portfolio weighting, resource allocation, feature scaling, and exploratory data analysis.

This calculator computes x^p, normalizes weights to sum to 1, and tracks cumulative contributions to support threshold-based decisions.

Frequently Asked Questions

Can values be negative?

No. SMp(x) assumes non-negative inputs so that x^p remains real for fractional p and represents magnitudes.

What happens when p = 1?

Weights become proportional to the original values (standard normalization).

How do extreme p values behave?

Large p values concentrate weight on the largest inputs; small p (approaching 0) equalizes weights across inputs.