Bounding uncertainty 

Part of approximate calculation is figuring out how uncertain you are.
An estimating... 
How many lightbulbs are there in the US? Well, say there are 10^{9} people in the US, and for each person there are 10 bulbs (more than 1, less than 100), thats about 10^{10} bulbs. 
A bounding... 
How many lightbulbs are there in the US? Well, there are between 10^{8} and 10^{9} people in the US. For each person there is atleast 1 bulb, and definitely less than 10^{3}. So between 10^{8} and 10^{12} bulbs. 
Bestguess estimating gives us an estimate, but doesn't tell
us how good the estimate is.
So we do bounds estimates too.
To to find this box, one does worstcase estimation,
rather than bestguess estimation.
At each step, you do whatever makes for the worst estimate.
The overestimates are multipled together to give the worst possible overestimate.
The underestimates are multipled to give worst underestimate.
And when dividing, the big overestimate is divided by the small
underestimate to give the worst possible overestimate. And so on.
Worstcase boxes tend to get big quickly. Uncertainty at each
step magnifies the surrounding uncertainty. There isn't the
compensation one gets with a bestguess estimate, where a highish
estimate here can be balanced with a lowish estimate there.
But you end up sure of where you stand.
What are the largest&smallest likely values.
What box would you be surprised if the true value was outside of.
The estimation is a compromise between bestguess and worstcase.
One does the same kind of highbound/lowbound estimating as with
worstcase, but you use the surprisebounds rather than the worstcase
bounds. And if you would find it surprising for all your estimates to
bee off in the same direction, you can fudge the box a bit smaller
every few steps. Since there is less uncertainty at each step, and
perhaps less uncertainty in the combination, the total uncertainty
grows slower than the worstcase.
possibility  likely  best guess  all together 
not possible, too big  
Largest possible value  
possible, but unlikely  
Largest likely value  
likely  
Best guess  
likely  
Smallest likely value  
possible, but unlikely  
Smallest possible value  
not possible, too small 
Some other ways to draw it... and [Tufte, Quant, 1235]
Comments encouraged.  Mitchell N Charity <mcharity@lcs.mit.edu> 