Estimation and intervals

Idea: Give a range

A point guess is almost always wrong. A range shows how much you know.

• Like a quote: the width is the honest uncertainty
• Too wide and nobody trades, too narrow and you lose when wrong

Idea: Bounds you can prove

Round every factor down for the low end and up for the high end.

• Low: $20\times 50\times 400=400000$
• High: $30\times 60\times 500=900000$
• Exact $23\times 58\times 410=546940$ lies between

Common mistake: Rounding a quotient the same way

For $9300÷48=193.75$, $9000÷40=225$ and $10000÷50=200$ miss it.

Instead Low end: small top over big bottom. High end: big top over small bottom.

Formula: The log-width score

$\text{score}=1-{\log }_{10}\left(\frac{H}{L}\right)$

A range from L to H that holds the value scores this. A miss scores $0$, and so does a range of $10$ times or wider.

Example: Score the proven range

The range $400000$ to $900000$ holds the value. What does it score?

1. $\frac{900000}{400000}=2.25$
2. ${\log }_{10}(2.25)\approx 0.35$
3. Score: $1-0.35=0.65$

Idea: When a range counts

In this trainer a range counts as right from a score of $0.5$.

• That is a ratio $\frac{H}{L}$ of at most $\sqrt{10}\approx 3.16$
• A sharper $500000$ to $600000$ would score about $0.92$

Quick check: Widen a Fermi chain

Central estimate $900000$, with $2$ links each off by a factor of $2$ either way. What is the high end over the low end?

Idea: Calibration

A $90\%$ range should miss about once in $10$ tries.

• Missing $3$ in $10$: too narrow, widen
• No miss in $20$: probably too wide

Idea: In the room

State the method, give both ends, name the link you trust least.

• Offer to tighten it with one more fact