How do I convert probability to log-odds?

Divide p by one-minus-p for odds, then take the logarithm; 0.5 maps to 0, 0.9 to about 2.2.

What is an odds ratio?

The multiplier on the odds from a one-unit change: a logistic coefficient of 0.7 means the odds roughly double per unit.

Why does logistic regression output an S-curve?

Because its linear score is a log-odds, and converting an open scale back to zero-to-one produces the S shape.

Where will I meet log-odds in practice?

Inside logistic regression outputs, scorecard points, and any diagnostic that talks about odds ratios for risk factors.

AI Glossary / AI and machine learning / Intermediate

Log-odds

Definition

Log-odds is the logarithm of the odds of an event: a way of stretching probability from its cramped zero-to-one scale onto an open number line where adding and subtracting evidence becomes simple arithmetic.

What it means

Probabilities are awkward numbers: they jam against 0 and 1, and combining evidence multiplies and divides them in ways heads cannot do quickly. Odds improve things: three-to-one is more workable than 75 per cent.

Log-odds goes one better. Taking the logarithm of the odds spreads the scale out: 50 per cent becomes 0, 90 per cent becomes about 2.2, 99 per cent about 4.6, and certainty stretches away to infinity on both sides.

The payoff is that evidence adds: in this space, each new piece of information shifts the log-odds by an amount you can simply add, which is why logistic regression's weighted sum is literally a log-odds score.

The transformation also explains logistic regression's shape: its famous S-curve is just log-odds converted back to probability, so the model's internal arithmetic lives on the open scale while its outputs stay familiar.

For a business, log-odds is the secret behind readable scorecards: points added for risk factors are log-odds arithmetic in disguise, and understanding the scale explains why each factor shifts the odds by a fixed multiple.

The scale is symmetric around the neutral point: a shift of plus one and minus one are equally strong evidence in opposite directions, a tidiness the probability scale cannot offer.

How it works

Convert probability p to odds, p over one-minus-p, then take the logarithm to reach log-odds.

Add evidence on the log-odds scale: each factor contributes a fixed shift, positive or negative.

Convert back to probability with the logistic function when a human-readable number is needed.

Read a logistic regression's coefficients as log-odds shifts: each field moves the score by its weight per unit.

Translate to odds ratios for communication: a coefficient of 0.7 means each unit multiplies the odds by about two.

Three practical examples

A credit scorecard where each factor adds or subtracts points, log-odds in work clothes, and the total maps to a risk grade.
An analyst combining three independent signals on the log-odds scale, where their contributions simply add.
A medical screen whose coefficient of 1.1 per symptom means each present symptom triples the odds.

Think of it this way

Think of the Richter scale for earthquakes: each step up is ten times the energy, so enormous ranges become manageable numbers you can add and compare. Log-odds does the same for probability: the cramped zero-to-one world becomes an open scale where evidence is arithmetic.

The limit: manageable numbers hide real squeezes. Adding five points at the extremes moves probability by almost nothing, while the same five points near the middle swings it hugely, so shifts on this scale mean different things in different regions.

Worked example

reading the scorecard

Suppose a bank's scorecard gives an applicant a total of 610 points, mapped to a 2 per cent default probability. The figures are illustrative.

In log-odds terms, each factor's points are additive shifts: income stability added 40 points, a missed payment history removed 65.

A reviewer can decompose any score into its pushes and pulls in seconds, because the arithmetic is genuinely additive.

When a factor's data is corrected, rescoring is a single subtraction and addition, and the audit trail writes itself.

Illustrative case study

A fictional insurer communicated model factors as probabilities to its underwriters, and the team spent a year misjudging how factors combined, assuming 10 per cent plus 10 per cent makes 20 per cent.

Retrained on odds ratios, the same factors became sayable: this factor doubles the odds, that one halves them, and the underwriters' combined judgments aligned with the model at last.

The invented lesson: probabilities multiply, log-odds add. Communicate factors in the currency people can actually combine.

Common misconceptions and mistakes

Log-odds is a different model. It is a different scale for the same probability; nothing about the underlying claim changes.
Zero log-odds means no risk. It means odds of one-to-one, a 50 per cent chance, the scale's neutral point.
The scale is only for statisticians. It is the working currency of every scorecard, and the additive intuition is immediately usable.

Business use

Communicate risk factors as odds ratios, which multiply cleanly, rather than as probability shifts, which do not.

Decompose scores into additive contributions in reviews: the log-odds scale makes the arithmetic honest.

Remember the squeeze at the extremes: equal log-odds shifts mean very different probability moves at 50 per cent and at 99 per cent.

Check your understanding

Why is evidence easier to combine on the log-odds scale than on the probability scale?

Answer: Because independent pieces of evidence shift log-odds by fixed amounts that simply add, while on the probability scale they interact through awkward multiplications.

Related terms

Learn next

AI for Business Owners covers how to read and challenge the numbers behind a risk score: https://aiforbusinessandlife.com/academy/ai-for-business-owners. [C2]

Sources

[S2] Google, Machine Learning Glossary. Full URLs are in the source ledger. Verified 11 October 2026.

Previous: Log-normal distribution Next: Log odds ratio