Drift in accepted values of constants

Why do successive measurements of a constant drift toward the previous answer instead of scattering around the truth?

Plot the published values of a physical constant against the year they were measured, and you expect a cloud: points scattered around the true value, tightening as instruments improve. What you often see instead is a staircase. Values sit close to one another for a decade, then march steadily in one direction, each new result landing just a little past the last.

Feynman pointed at exactly this in his 1974 Caltech commencement address, using Millikan's electron charge as the case. Millikan's value was slightly too low, because he had used a poor figure for the viscosity of air. The measurements that followed did not jump to the correct value. They crept toward it. Why would a sequence of independent experiments do that?

Start by asking what "independent" is doing in that sentence. The apparatus is independent. The observers are independent. But is the decision to stop independent?

Consider what actually happens in a laboratory. You run the experiment, reduce the data, and get a number. It disagrees with the accepted value by three times your quoted uncertainty. What do you do next?

You do not publish. You go looking for what went wrong — and you are right to. A three-sigma discrepancy is far more often a systematic error in your own setup than a revolution in physics. You check the calibration, you re-derive a correction, you find that a temperature coefficient was applied with the wrong sign. You fix it. The number moves closer. Now you publish.

Now run the counterfactual, because this is the step that matters. Suppose your first number had landed right on the accepted value. Would you have gone hunting for that same sign error?

Almost certainly not. And there is the whole mechanism. The error hunt is real, competent, and honest — but it is triggered by disagreement and called off by agreement. The stopping rule is conditioned on the outcome. Every individual step is good practice; the sequence is biased.

Notice that no one has to be dishonest for this to work, which is why the effect is so durable. What is doing the work is not deception but incentive: the cost of publishing a discrepant number is high — you will be asked to explain it, your systematics will be scrutinised, you may be wrong in public — while the cost of publishing an agreeing number is nearly zero. Asymmetric costs applied to a symmetric search produce a systematically asymmetric result.

Two further pressures push the same way. Referees and collaborators apply the same asymmetry from outside; a paper reporting an anomaly gets harder questions than one reporting a confirmation. And there is the file-drawer effect: discrepant results that survive scrutiny may still be quietly shelved as "probably something we do not understand," so the published record is not the record of what was measured.

There is a diagnostic, and it is the reason we can be confident this is happening rather than merely suspect it. If a sequence of measurements were genuinely independent draws around a true value, successive results should differ from one another by roughly the size of their combined uncertainties. In these staircase sequences they do not. They agree with each other far too well for their quoted error bars, and then the whole cluster later turns out to be displaced from the modern value by many of those error bars. Too little scatter, followed by a large jump, is the signature. The Particle Data Group's own reviews discuss this history openly and show such trend plots for particle properties.

It is tempting to read all this as a charge of misconduct. It is not. The clearest version of the problem involves no one behaving badly at any point; it is a property of the procedure, and it survives replacing every person in it.

Nor does it mean published constants are untrustworthy. It means their quoted uncertainties are, in this specific way, optimistic — the scatter of the community's results underrepresents the real uncertainty, because the results are correlated through the literature rather than independent. This is precisely why some ongoing discrepancies are taken seriously as physics rather than dismissed as sloppiness: the discrepant measurements are ones where the groups deliberately insulated themselves from the expected answer.

Which points at the fix. Blind analysis: the analysis pipeline is written and frozen while the final number is hidden from the analysts — offset by an unknown constant, or with a random fraction of the data withheld — and unblinded only once the systematics are settled and everyone has agreed to publish whatever appears. Now the error hunt cannot be conditioned on the answer, because nobody knows the answer while hunting. Blinding is standard in particle physics and increasingly common elsewhere. Pre-registration of analysis plans in other fields is the same idea wearing different clothes.

The honest caveat: how large the residual drift is in any particular constant is a matter of judgement, not measurement. You cannot compute the size of a bias whose mechanism is "searches that were not performed."

• Blind analysis in practice: how a collaboration adds a secret offset and what it gives up in exchange.
• Whether the same asymmetric stopping rule explains parts of the replication crisis outside physics.
• The proton radius and neutron lifetime puzzles, where two methods disagree persistently and neither camp is drifting toward the other.

Every term the collection defines is gathered in the glossary.