Estimation and Uncertainty / Original lecture by H. Scott Matthews, CMU Sept 24, 2003

Fermi Problems  Estimating an unknown quantity is sometimes called a “Fermi problem,” after physicist Enrico Fermi  Wanted to show students they had the power to do estimation  His first problem: “How many piano tuners are there in Chicago?”

Sample Fermi Problems  How much tea is there in China?  How may pounds of human hair are cut every day?  How many leaves are there on all the trees in the world?  If you got a penny for each time someone said “Damn!" in the United States, how long would it take you to become a billionaire?  What area of the Earth would it take to supply the U.S. with all its energy needs if solar energy could be converted with 1% efficiency? Solar energy at Earth is about 1 kW/m 2.

Cobblers in the US – Method 1 zCobblers repair shoes zOn average, assume 20 min/task zThus 20 jobs / day ~ 5000/yr yHow many jobs are needed overall for US? zI get shoes fixed once every 4 years yAbout 280M people in US zThus 280M/4 = 56 M shoes fixed/year y56M/5000 ~ 11,000 => 10^4 cobblers in US zSensitivity: yAm I representative? yAre all shoe repairs done by cobblers? yDo cobblers work 8 hours per day?

Cobblers in the US – Method 2 zGreater Pittsburgh Yellow Pages has 36 entries under “Shoe Repairing” zAssume each repair shop has two employees. 72 in greater Pittsburgh zPopulation of greater Pittsburgh = 2.3 million (2000 Census) = 0.82% of U.S. zNumber of cobblers in U.S. = 72/ = 8780 zSensitivity: yIs Pittsburgh representative? yIs “greater Pittsburgh” the right area for the Yellow Pages? yAverage number of employees of a shoe repair shop

Cobblers in the US zMethods 1 and 2 give “close” answers: y11,000 v zActual: Census Dept says 5,120 in US yDepends on accuracy of job counting in Census yListing of occupations yFull-time vs. part-time yNumber of responses received

Problem of Unknown Numbers  If we need a piece of data, we can:  Look it up in a reference source  Collect number through survey/investigation  Guess it ourselves  Get experts to help guess it  Often only ‘ballpark’, ‘back of the envelope’ or ‘order of magnitude needed  Situations when actual number is unavailable or where rough estimates are good enough  E.g. 100s, 1000s, … (10 2, 10 3, etc.)

Methodology  First develop an upper bound and a lower bound. This will allow to do a “sanity check” on the answer  Use at least two independent methods of estimation and compare the answers  Identify sensitivity to errors in the data. For sensitive data, but sure you have good values

In the absence of “Real Data”  Are there similar or related values that we know or can guess? (proxies)  Example: registered voters v. population  Are there ‘rules of thumb’ in the area?  E.g. ‘Rule of 72’ for compound interest  r*t = 72: investment at 6% doubles in 12 yrs  Set up a ‘model’ to estimate the unknown  Linear, product, etc functional forms  Divide and conquer

Methods zSimilarity – do we have data that might apply to our problem? zStratification – segment the population into subgroups, estimate each group zTriangulation – create models with different approaches and compare results

‘How much disk space to store every word you hear in a lifetime?’ zHow many words per day can you hear? y12 hours per day, 120 words per minute = 86,400 words/day y= 33 million per year zHow much disk space to store them? yAverage word < 10 characters, 330MB/year zAverage lifetime? 75 years? zAnswer: < 25GB, less than the size of a laptop

‘How much energy used by lighting in US residences?’ zAssume 25 light fixtures per house zAssume each in use avg 2 hours per day zAssume average fixture is 50W zThus each fixture uses 100Wh/day zEach house uses 2500Wh/day z100 million households would use 250 million kWh/day y91,300 million kWh/yr

‘How much energy used by lighting in US residences?’ zOur guess: 91,300 million kWh/yr yDOE: “lighting is 5-10% of household elec” yhttp:// z2000 US residential Demand ~ 1.2 million million kWh (source below) y10% is 120,000 million kWh y5% is 60,000 million kWh y2000 demand source: epmt44p1.html

How many TV sets in the US?  Can this be calculated?  Estimation approach #1: Survey/similarity  How many TV sets owned by class?  Scale up by number of people in the US  Should we consider the class a representative sample? Why not?

TV Sets in US – Method 2  Segmenting  work from # households and # tvs per household - may survey for one input  Assume x households in US  Assume z segments of ownership (i.e. what % owns 0, owns 1, etc)  Then estimated number of television sets in US = x*(4z 5 +3z 4 +2z 3 +1z 2 +0z 1 )

TV Sets in US – By Segmentation  Assume 50 million households in US  Assume 19% have 4, 30% 3, 35% 2, 15% 1, 1% 0 television sets  Then 50,000,000*(4*.19+3*.3+2* ) = M television sets

TV Sets in US – Method 3  Estimation approach #3 – published data  Source: Statistical Abstract of US  Gives many basic statistics such as population, areas, etc.

How well did we do?  Most recent data = 1997  But ‘recently’ increasing < 3% per year  TV/HH tvs, StatAb – 229M tvs,  % error: (229M – 125.5M)/125.5M ~ 82%  What assumptions are crucial in determining our answer? Were we right?  What other data on this table validate our models?

Some handy/often used data zPopulation of US million zNumber of households ~ 100 million zAverage personal income ~$30,000

Good Assumptions  Justify and document your assumptions  Have some basis in known facts or experience  Do not allow bias toward the answer affect your assumptions  Example: what will the inflation rate be next year?  Is past inflation a good predictor?  Can I find current inflation?  Should I assume change from current conditions?  We typically use history to guide us

Notes on Estimation  Move from abstract to concrete, identifying assumptions  Draw from experience and basic data sources  Use statistical techniques/surveys if needed  Be creative, BUT  Be logical and able to justify  Find answer, then learn from it.  Apply a reasonableness test