The estimation tool

In our discussion of the use of math in science, we have considered the value of estimation to help you get a sense of scale. 

Estimation as an end in itself

You'll have to do estimations in homeworks, quizzes, and exams. You'll often see the advice: "This is an estimation problem. Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer." The icon we've chosen here is your own hand saying, "Oh, it's about yay big." This emphasizes that in doing estimations, you are using your own often everyday knowledge in what might be new (and sometimes surprising) ways.

How should you present an answer to an estimation problem?

The idea behind estimation is not just to pick some arbitrary number. Rather, you have to convince your reader that

• you have enough personal information to get reliable starting numbers,
• your assumptions are reasonable, and
• your calculations have been done correctly.

"Showing your work" is critical on estimation problems!

Watch out for "I remember this from high school"! In my experience, remembering a number from a long time ago almost always comes up with an incorrect value. Try to use whatever personal knowledge you have (and think you can trust) to build numbers in complex situations.

Estimation (often tacit) provides the underlying basis for more explicit mathematical modeling. The real art of building a model is deciding what you need to include and what you can reasonably ignore. Building good intuitions for the scale of effects relies heavily on your ability to estimate.

Here are some tips for playing the estimation game:

• Develop intuitions for large numbers (Use scientific notation!)
• Decide what it is you need to calculate (equations and principles).
• Be careful!
• Avoid memorized (one-step) numbers (except in a few specific circumstances).
• Find things you know you can trust to help you figure things out and build crosslinks when possible. 
• Write enough words so the grader can follow your reasoning (but don't pad your text with excess verbiage).
• Cross-check your equations to be sure they’re correct (for example: dimensional analysis).
• Do your estimations from things you know (Know your basics!) and tell how you got them.
• Keep units throughout and USE THEM to make sure you’ve done the correct calculation (and haven't, for example, mixed meters and centimeters).
• Don’t keep more than 2-3 sig figs anywhere in the calculation.
• NEVER write an untrue equation. (“= as → ”)
• Be sure you answer all the questions asked!
• Throughout, treat your numbers as MEASUREMENTS and check them for reasonableness!

Estimation as a tool in more complex problems

While you might want to estimate to get a result (like to estimate how many people are in a long line for tickets to see if it's worth the wait), estimation plays a crucial role in almost all of the problems we solve in science. (Not just in physics!)

Since we can never describe every relevant factor in a real world situation, we always have to decide, "which of the many factors I can see as potentially relevant are going to matter for what I want to know?" This is the essential element in mathematical modeling, though often, it is not made explicit. "Do I need to include air resistance? Is friction important? The mass of the spring is pretty small compared to the object I'm looking at. Do I need to consider it?" 

Often we make these sort of decisions using our intuition, but estimation skills are valuable tools to confirm those intuitions or to help us decide what we can ignore when we are not certain.

For a good example of this, see the discussion in the example problem, Example: Oscillator calculations.

Joe Redish 2/8/19