Rushing Yard Logistic Score & Density Explorer

Illustrative Presets Quick scenarios

Assumed Inputs

Assumed Scrambles 4

Chosen count; no observed matchup sample

Yards per Scramble 5.2

Example yield, not sourced historical efficiency

Assumed Designed Runs 1.0

Each expected run adds a fixed 3.7 example yards

Illustrative Pressure Input 7.4%

Each point above 7.4 adds 0.25 example yards

What this worksheet knows

The Bo Nix / Kansas City names provide an undated example. Run counts, yields, pressure input and coefficients are authored assumptions. No verified analyst quote, paid partnership, current line or held-out forecast validation is supplied.

Use the controls to study sensitivity. Export records the assumptions and both model definitions.

Example Mean Yardage

24.5

Mean minus 18-yard threshold

Logistic Score

71.4%

Authored sigmoid; not a game-win probability

Fixed Example Threshold

18.0

Chosen illustration, not a verified market

Assumed Total Runs

4.8

Scrambles + Designed

Separate Gaussian Density Fixed spread: 9.5 yards

Density at 18+ yards

Density below 18

18-yard split

A dated fact is different from a forecast

Denver’s official player biography records 42 rushing yards on nine attempts at Kansas City on December 25, 2025. That single result does not validate these coefficients or the former unsourced 92-yard career claim. Read the Broncos biography.

One mean, two different model outputs

Why is the score not the green area of the chart?

The percentage uses an authored logistic transformation of mean minus 18. The bars use a separate normal density with a 9.5-yard spread. These are different mathematical choices, and neither has been fitted or validated against a documented game sample.

No event date, efficiency sample, pressure measurement, verified analyst transcript, commercial relationship, offered odds or settlement rule supports the original matchup verdict. The dated Broncos result provides historical context only.

Try a worked example

The baseline mean is 4 × 5.2 + 1 × 3.7 + (7.4 − 7.4) × 0.25 = 24.5 example yards. The fixture stores a 71.4% score, matching the rounded sigmoid 100 / (1 + exp(−0.1407 × 6.5)). A normal curve with mean 24.5 and spread 9.5 instead has about 75.3% of its area above 18. The bars do not reproduce the displayed score.

Read the score and density separately

R’s statistical documentation defines the logistic cumulative function. This page applies its sigmoid shape to a chosen yardage gap with slope 0.1407; the slope is an authored parameter, not a learned football coefficient. Nonbaseline scores are rounded to one decimal and limited to 2–99%. R stats: Logistic distribution

NIST distinguishes a normal density from cumulative area. Here each bar is the density at a two-yard-spaced sample, multiplied by 100. Bar height is not the probability of an exact integer outcome or a two-yard bin. There are no random trials or fitted kernel estimates. NIST: Normal distribution

The chart shows only 0–50 yards. Joint maximum inputs yield 124.1 example yards, outside that window; bars can become nearly zero while the score reaches its 99% cap. The normal distribution itself extends beyond the visible window, including negative values. NIST: Normal distribution

Test assumptions before drawing a football conclusion

Heavy Blitz is only a stored scenario: 6 × 6 + 0.5 × 3.7 + (12 − 7.4) × 0.25 = 39.0 example yards. The pressure term always adds yardage when its input rises; no source establishes that causal direction. Fractional designed-run counts represent hypothetical expected counts.

The Broncos biography records 42 yards on nine rushing attempts at Kansas City on December 25, 2025. It supports that dated observation, not the original cumulative claim, a pressure-to-scramble rule, or this worksheet’s percentage. Denver Broncos: Bo Nix biography

Manual changes clear the preset highlight. Reset restores all four baseline inputs. JSON includes the selected values, hidden coefficients, baseline override, score clamp and separate density definition. If the chart library is unavailable, the canvas supplies a text summary rather than a curve.

Sources and further reading

R stats: Logistic distribution
Sigmoid formula; does not validate the chosen football slope.
NIST: Normal distribution
Density versus cumulative area, spread and unbounded support.
Denver Broncos: Bo Nix biography
Dated 2025-12-25 result: 42 rushing yards on nine attempts.