Hysteresis Calculator
The formula
What the hysteresis tells you
The scaling law behind hysteresis executed with its exponent printed: a·x^n, plus the sensitivity that makes power laws unforgiving.
Calculation: Hysteresis = a × x^3.5 (empirical power law). Enter your inputs and every substitution is shown below the result.
Reading the result
The elasticity line is the point: with exponent 3.5 a 10 % measurement error costs 40 % of the answer.
Limitations and best practice
Power laws hold within a fitted range — extrapolating far outside it silently misleads.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
Where does the exponent come from?
Why log scales?
Source & methodology
Published empirical scaling relation.
Hysteresis: the number, the method and the caveats
You are here because you need a specific number, and you want to see how it was produced before you rely on it. There are plenty of pages about Hysteresis; almost none of them show the working. This one does: the formula, every input, a fully computed example, and a sensitivity analysis calculated live from this engine’s own defaults.
In Physics Laboratory, units and significant figures carry meaning; keeping them attached to every output is the quiet professional standard this page maintains.
In Science & Research the same quantity is often quoted under two or three different conventions. The formula printed above fixes one convention and applies it consistently, which is what makes results from this page comparable with each other.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to hysteresis = 11.314 on the default inputs, and the sensitivity section ranks the inputs that move the result — Input quantity first, at ≈39.6 % of the headline for a ten-percent error either way in that input. Work down the page and by the end you could re-derive the result in a spreadsheet from memory.
The Hysteresis formula, explained
One formula, published in full: Hysteresis = a × x^3.5 (empirical power law). Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Input quantity (example value 2) is a measured number in the unit named by the label; the formula does not convert for you. Count from the record, not the estimate; whole-number inputs are cheap to verify and expensive to get wrong.
Scaling constant a (example value 1) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Reference/output unit scale (example value 1) is a measured number in the unit named by the label; the formula does not convert for you. If the true figure falls between counts, run both and read the spread as your uncertainty.
Hysteresis with real numbers, step by step
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Input quantity, Scaling constant a, Reference/output unit scale. The computed hysteresis comes out at 11.314, which the interpretation band labels “Steep exponent — small input errors blow up the output; measure x carefully”.
- x^3.5 = 11.3137
- Result = 1.00 × 11.3137 = 11.3137
- Elasticity: +1 % in x moves the result 3.5 %
Log₁₀ result — 1.05. Read next to the headline, this is the figure that shows the shape of the trade-off.
Sensitivity (Δ% per +1 % x) — 3.5 %. If your inputs change, this is usually the first companion figure to move.
Double x gives — 128.000. This is the number experienced users quote when the headline alone would mislead.
Re-running with your own figures is instant; re-running by hand once is the best way to know the tool is telling you the truth.
Scenario table: Hysteresis at ±10 % and ±20 % of the key input
The honest version of any estimate is a corridor, and this one is cheap to produce: Hysteresis was recomputed at 5 points along the dominant input, with every other field left untouched.
| Input quantity setting | Hysteresis |
|---|---|
| 80 % of the example value | 5.181 |
| 90 % of the example value | 7.824 |
| example inputs | 11.314 |
| 110 % of the example value | 15.794 |
| 120 % of the example value | 21.416 |
The corridor runs from 5.181 to 21.416 across that band — which is the range worth planning inside rather than the single 11.314 point estimate. Reproduce it in the form above by editing Input quantity alone; the substitution steps will follow your figures exactly.
Sensitivity: which inputs change Hysteresis the most
Generic advice says “it depends on your inputs”; this section says exactly how much. Each field was perturbed by ±10 % in isolation and the hysteresis recomputed, so the ranking below is measured rather than asserted. For Hysteresis the ranking is measured rather than guessed, and it belongs to these defaults: shift the starting point and the order can shift with it.
- Input quantity +10 % → Hysteresis = 15.794 · −10 % → 7.824 (the hysteresis shifts ≈39.6 %)
- Scaling constant a +10 % → Hysteresis = 12.445 · −10 % → 10.182 (the hysteresis shifts ≈10.0 %)
- Reference/output unit scale: ten percent up gives 12.445, ten percent down gives 10.182 — ≈10.0 % of the hysteresis either way
Input quantity is the dominant lever here: ten percent either way moves the result by about 39.6 %, from 7.824 to 15.794. Spend your verification effort there first, and treat Scaling constant a as the second priority (≈10.0 % per ten percent). The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
Interpreting the Hysteresis verdict
Nobody acts on a bare figure — they act on a figure with a comparison attached. The verdict attached to the default calculation — “Steep exponent — small input errors blow up the output; measure x carefully” — is one band on a ladder your own figure may sit somewhere else on. Before treating the band as fact, note its width: ten percent either way on Input quantity already spans 7.824 to 15.794. The practical reading rules: read the band as a starting hypothesis, then test it: change the two dominant inputs by ten percent and see whether the verdict survives. Verdicts that survive stress are the ones worth acting on.
The comparison worth making is not with somebody else’s Hysteresis but with the version of your own situation measured last quarter — same formula, same conventions, different inputs.
How people get Hysteresis wrong
Power laws hold within a fitted range — extrapolating far outside it silently misleads. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Confusing a large number with an important one. Hysteresis can be dominated by one input whose absolute value looks unimpressive, which is why this page ranks influence separately from magnitude.
Attention is also misallocated between fields. Having measured this engine, we can say where it belongs: Input quantity first, everything else second.
Another is comparing across contexts: a Hysteresis computed with one set of conventions should never be lined up against a figure computed with another. Check the inputs behind both before ranking them.
A quick check for any Hysteresis figure
- Verify each input against its original source document, not memory.
- Note the date and the inputs beside the number so the figure can be audited later.
- Put Input quantity under a magnifier; the sensitivity table above is the reason why.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Confirm the example values are plausible for your own situation before replacing them.
- Repeat the ±10 % test on the inputs that matter and see whether the decision — not just the number — still holds.
How to use the Hysteresis tool properly
Using a calculator well sounds trivial, but the difference between a decision-grade figure and a throwaway one lives entirely in the routine. The routine for Hysteresis takes a minute to learn and applies to every input above.
Step 1 — collect, then type. Everything this tool asks for is listed above (input quantity, scaling constant a, reference/output unit scale). Pull each figure from something you can point at — paperwork, a dashboard, a label — before touching a field. Guesswork at the input stage cannot be repaired at the interpretation stage. A calculation you cannot explain to somebody else is a calculation you cannot defend.
Step 2 — start from the case you can defend. Enter pessimistic-but-honest figures first; the optimistic run is a ceiling, not a plan. A conclusion that holds on conservative inputs holds everywhere. When two sources disagree, it is almost never the arithmetic that differs.
Step 3 — isolate your variables. Change one field, observe, revert. The live sensitivity list does the ranking for the default inputs; verify it against your own, because dominance shifts when the starting point shifts. Note the assumption you are least sure about; that is where the real risk lives.
Step 4 — compute a corridor, not a point. Three runs — worst, likely, best — turn Hysteresis from a snapshot into a range. Plan against the pessimistic end and treat the optimistic end as the upside story. Precision is a promise you make about your data; make sure your inputs can keep it.
Step 5 — save the story with the figure. A number without its inputs and its date is unusable within a quarter. Record 11.314, the assumptions behind it, and today’s date; your future self will be auditing a stranger’s arithmetic. A figure nobody can reproduce is a figure nobody will act on twice.
The real-world conversations Hysteresis settles
Numbers earn their keep in conversations, and Hysteresis is no exception. In Science & Research discussions around Physics Laboratory, this figure decides whether an option stays on the table: it converts an opinionated debate into an arithmetic one, and arithmetic is the only referee both sides accept. Re-running a number costs seconds; retracting a decision costs considerably more.
The companion outputs — Log₁₀ result and Sensitivity (Δ% per +1 % x) — exist for the second stage of that screen: the headline sizes the prize, while the companions describe the trade-off you are accepting to get it. Repeating a calculation with the same numbers proves only that the arithmetic is reliable — which is worth knowing.
Re-running this page costs nothing and catches the slow drifts that never announce themselves. In Science & Research the expensive mistakes are rarely sudden; they are small changes in Input quantity that nobody recalculated for a year. Repeating a calculation with the same numbers proves only that the arithmetic is reliable — which is worth knowing.
The questions that come up most about Hysteresis
What affects Hysteresis the most?
Measured on this engine, Input quantity is the dominant lever: moving it ±10 % shifts the hysteresis across 7.824 to 15.794 — about 39.6 % of the headline, with Scaling constant a next at ≈10.0 %. The ranked list in the sensitivity section is computed live, not asserted from a template.
Which matters more: scaling constant a or input quantity?
Input quantity, by a factor of about 4.0. Concretely: ten percent on Input quantity is worth ≈39.6 % of Hysteresis, the same nudge on Scaling constant an only ≈10.0 %. That ranking is re-computed for each page from that page’s own defaults.
What is a typical Hysteresis?
With the example inputs on this page the result is 11.314, labelled “Steep exponent — small input errors blow up the output; measure x carefully”. “Typical” varies with situation, which is why every result here ships with its verdict band and its sensitivity list: they turn one figure into a range you can locate yourself inside.
How do I calculate Hysteresis by hand?
Use the stated model — Hysteresis = a × x^3.5 (empirical power law) — and substitute in the order the tool prints (it starts from x^3.5 = 11.3137). On the example inputs the hand route lands on 11.314, exactly what the calculator shows. A spreadsheet reproduces it in two or three cells; doing it once by hand is what makes the automated answer trustworthy.
What happens to Hysteresis if input quantity changes by 10 %?
The engine was re-run with that field moved by ten percent while everything else stayed fixed: the hysteresis goes across 7.824 to 15.794, a swing of roughly 39.6 % around the 11.314 baseline. That ratio is the practical definition of how much this input matters.
Is the Hysteresis result of 11.314 good or bad?
The tool answers “what”, not “whether”: 11.314 carries the verdict “Steep exponent — small input errors blow up the output; measure x carefully” for the default case, which is a starting interpretation rather than a verdict on your situation. Re-run with your own figures and compare the bands.
How many significant figures are reliable in Hysteresis?
About as many as your least precise input. The tool computes at full precision and displays rounded output; report 11.314 to a decimal place your inputs can actually support.
How do I calculate Hysteresis manually?
Write down the inputs, then follow the printed steps in order — Hysteresis = a × x^3.5 (empirical power law). A spreadsheet replicates it with two or three cells. The manual route is worth doing once to build intuition; after that the calculator saves time and eliminates transcription slips.
Where does the exponent come from?
Empirical fits to observations or physics (homology relations); cite the range of validity when reporting.
Why log scales?
Quantities spanning many orders of magnitude compare as straight lines on log axes.
Does this Hysteresis tool store my numbers?
No. The entire calculation runs in your browser session; nothing entered above is transmitted or saved, which is the correct behaviour for sensitive Science & Research figures and the reason you can re-run this page as often as you like.
What the labels on this page mean
- Log₁₀ result
- 1.05 for the Hysteresis worked case; it sits next to the main result because it carries context the headline leaves out.
- Sensitivity (Δ% per +1 % x)
- 3.5 % on the example inputs — a companion figure printed beside the headline of Hysteresis so that no result is read in isolation.
- Double x gives
- 128.000 in the default Hysteresis calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Hysteresis
- The primary output of this tool: 11.314 for the default inputs, computed exactly from Hysteresis = a × x^3.5 (empirical power law).
- Worked example
- The step-by-step substitution printed above, so the arithmetic can be re-derived by hand or in a spreadsheet.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “Steep exponent — small input errors blow up the output; measure x carefully”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Hysteresis moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
The provenance of this Hysteresis calculation
Everything above runs on one stated formula, applied without hidden coefficients: Hysteresis = a × x^3.5 (empirical power law). Source of record: Published empirical scaling relation. Specialist software layers extra coefficients on top of the same algebra; none are hidden here, and that transparency is what makes a disagreement traceable to a specific input instead of a black box.
Nothing here replaces a qualified professional, and nothing here is approximate either — the arithmetic is exact, and the honesty is in the inputs you supply. The design constraint was set by usefulness rather than by comfort: an estimate you can interrogate beats an answer you have to trust.
After the number: what to do with it
Re-run the calculation with your own figures, stress the two most sensitive inputs by ±10 %, and only then act on the result.
Whoever you are (people who need one trustworthy number before their next decision), the next step is the same: replace the example inputs with your own, note the date, and re-run whenever a figure moves. In Science & Research, a dated figure is a trend, and an undated one is a rumour.