How do you know what you know?

How do you know what you know?

1)Maybe you can measure something directly.

2)You can interpret what you have measured within the context of some physical or statistical model.

3)You can accept the authority of some published work or the authority of some person whom you trust.

Just because you read it in a Wikipedia article, is ittrue?

Natural science is based on measurable data interpretedwithin the context of certain assumptions (e.g. the Earthrevolves about the Sun).

There can be problems with data, however. As RichardFeynman said, “Any theory that fits all the data must bewrong, because some of the data is probably wrong.”

How can data be wrong? 1) bonehead errors; 2) calibrationerrors; 3) small number statistics; 4) fraud.

Examples: faster-than-light neutrinos, incorrect periodsfor light curves of variable stars, cold fusion

Modern statistics was foundedby the German mathematicianand astronomer Carl FriedrichGauss (1777-1855). Prior tohis work scientists often handpicked the “best” data pointsto derive the “most accurate”results.

Gauss showed that the most robust and fair minded conclusions can be obtained from the use of the entiredataset.

Example of a Gaussian distribution (bell shaped curve)

In the previous example, the mean value is 15.4 andthe “standard deviation” of the distribution is +/- 4.4.

68.3 percent of the data points are within one standarddeviation of the mean value for a Gaussian distribution.

Data points that are more than 3 standard deviations fromthe mean are usually considered “outliers” because theywould occur less than 0.5 percent of the time.

It is possible that a dataset could be characterized by morethan one overlapping distribution, each of which has adifferent mean and/or standard deviation. Example:test scores with one peak at 60% (people who didn’t study)and another peak at 85% (people who did study).

Not all frequencies of events can be described by abell-shaped curve. For example:

1)The net worth of a people vs. how many peoplehave that net worth (Pareto’s law of income distribution)

2) The frequencies of word use in a given book, or in agiven language (Zipf’s law)

3) The number of authors (N) publishing n paper overthe course of a lifetime (Lotka’s law)

Scientific notation and significant figures

A number such as 6378 can be represented as6.378 X 103.

Similarly, 0.0005193 is the same as 5.193 X 10-4.

In the first case, because the number is greater than 1,the exponent is positive. In the second case, becausethe number is between 0 and 1, the exponent is negative.

A number represented in scientific notation is ofthe form

n.nnnn X 10A

7.553 X 102 is the same as 755.3. The exponentof “2” means that you move the decimal pointtwo places to the right to write the number inregular notation.

Similarly, 1.234 X 10-2 = 0.01234. The decimalpoint of “1.234” is moved two places to the leftbecause the exponent is negative.

For a number in scientific notation represented as

n.nnnn X 10A

the digits “n” aren't necessarily the same number,but the fact that there are 5 of them means thata number such as 2.9979 X 108 has 5 significantfigures. A number such as 3.00 X 108 has only3 significant figures, and therefore has less precision.

When multiplying or dividing two numbers, theaccuracy of the result depends on the accuracy ofthe two numbers. The result cannot have moresignificant digits than the less accurate number.

For example, 6.378 X 103 times 1.123 X 10-3 isbest represented as 7.162 rather than as 7.162494.

Similarly, 6.378169 times 1.1 is best representedas 7.0 instead of what your calculator might tell you.

Numbers such as = 3.14159265358979.... have essentially an infinite number of significant figuresbecause they are mathematical constants.

6.7 / will have a different number of significantfigures than 6.70000 / because the numeratorin the first case has less precision than in the secondcase.

Imagine a political poll conducted in 2016 involved1000 registered voters. Say 42 percent preferred Donald Trump for President, while 46 percent preferred Hillary Clinton, and there were 12 percent undecided. You might be told that the uncertainty of these numbers is +/- 3 percentage points.

Where does the +/- 3 % come from? It turns out thatit's simply 1/sqrt(1000), or the square root of thereciprocal of the number of voters asked for a preference.

Now say an astronomer is using a telescope to measurethe brightness of a distant quasar. If 1000 photons attributable to the quasar are counted, how accuratelycan you say you have measured the brightness ofthe quasar?

It turns out that it's the same relative error as in thepolitical poll: 1/sqrt(1000) ~ 3 percent. In orderto measure the brightness of the quasar to +/- 1percent you need to measure 10,000 photons, so yourintegration time has to be 10 times as long, or youneed to use a telescope with a collecting area 10 timesas big.

If we took an opinion poll consisting of fair mindedquestions and we wanted the results to be good to+/- 1 percent, how many people would we have topoll in our survey?

A.100B.1000C.10,000D.100,000

What is the light brown hat from Switzerland primarily made of?

a.suede (some treated animal skin)b.felt (some vegetable product)c.man-made synthetic materiald.pressed mushrooms