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### History of measurement

History of tau-mass measurement

I picked up two figures from a PDF file delivered by PDG.

This plot shows a historical perspective of
measured values of mass of tau-lepton as a function of published data. The first measurement reported 1806± 20 MeV/c^2. The second one did 1784 ± 4 MeV/c^2.

The center values of the measurements seem drifted. One should, however, realize importance of error bars. The first measurement has 20 MeV as error for the center value, which means expected true value can be located between 1806-2*20=1766 MeV/c^2 and 1806+2*20=1846 MeV/c^2 in 95% of time. The most up-dated one, 1776.84 MeV/c^2 is actually located in this range. So whole measurements seem consistent.

On the other hand, the second plot, which shows another history of measured values of life time of neutron, is bit funny.

In 60’s, it was measured as 1110 +/- 30 sec. In the beginning of 70’s, one experiment reported the center of value was different from previous ones systematically as 920±15 sec. In 90’s, better
observations appeared and finally it is 885.7 ± 0.8 sec.

The difference between 1110 and 885.6 is around 225, which is 7 times larger than 30, the error of 1110. From the statistics consideration, it is very low probability to have such a shift. It seems the first cluster of experiments must have not only statistical error but also systematic error. No one knows, however,at this point what happend there.

These examples are quite instrutive in the following poit;

The center of values themselves from measurements has meaningless. Just the value with a bandwidth constructed by error bar is important.

この絵をみるとずいぶんと質量の値が変わってきているように見えます。でも、実は中央値の次の誤差の値に注目しなければいけません。最初の値には20MeVという誤差がついていますが、その意味は、τ粒子の質量の真の値は95%の確率で、1766 MeV/c^2（=1806-2*20）と1846 MeV/c^2(=1806+2*20)の間にあるだろうということです。最新の値、1776.84±0.17 MeV/c^2はその中に入っていますから、τの質量の測定は一貫して矛盾なく、そして精度がどんどんとよくなっていることを示しています。

τ粒子の質量のときのように測定誤差を考えても、最初の1110秒と885.6秒とでは、約225秒の違いがあり、これは1110秒の時の誤差30秒の7倍もあります。統計的にはこんなに中央値が変わってしまう確率はものすごく小さいことになります。今となってははっきりとはわからないのですが、60年代の測定にはなにか系統的な測定誤差があったのかもしれません。

この２つの例は以下のような点でとても教訓的で、たぶん、そのためにPDGはこういった測定の歴史的推移の図を載せているのだろうと思います。