Scientific notation and significant figures
A number such as 6378 can be represented as
6.378 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, because
the number is between 0 and 1, the exponent is negative.
A number represented in scientific notation is of
the form
n.nnnn X 10A
7.553 X 102 is the same as 755.3. The exponent
of “2” means that you move the decimal point
two places to the right to write the number in
regular notation.
Similarly, 1.234 X 10-2 = 0.01234. The decimal
point of “1.234” is moved two places to the left
because 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 that
a number such as 2.9979 X 108 has 5 significant
figures. A number such as 3.00 X 108 has only
3 significant figures, and therefore has less precision.
When multiplying or dividing two numbers, the
accuracy of the result depends on the accuracy of
the two numbers. The result cannot have more
significant digits than the less accurate number.
For example, 6.378 X 103 times 1.123 X 10-3 is
best represented as 7.162 rather than as 7.162494.
Similarly, 6.378169 times 1.1 is best represented
as 7.0 instead of what your calculator might tell you.
Numbers such as π = 3.14159265358979.... have essentially an infinite number of significant figures
because they are mathematical constants.
6.7 / π will have a different number of significant
figures than 6.70000 / π because the numerator
in the first case has less precision than in the second
case.
Errors of Measurement
The modern theory of statistics owes a lot to the
German mathematician, physicist, and astronomer
Carl Friedrich Gauss (1777-1855).
You all have heard of the “bell-shaped curve”
or Normal distribution. If you make many
measurements of some quantity there will be
some average value and the values will exhibit
a distribution about the mean as follows:
The width of the
curve at half its
height is +/one “standard
deviation”.
Consider the number 625.0 +/- 12.5. The standard
deviation of 12.5 means that 68.3% of the values
fall between
625.0 – 12.5 = 612.5
and
625.0 + 12.5 = 637.5
95% of the measurements are between +/- 2 standard
deviations of the mean value. 99% of the measurements
are between +/- 3 standard deviations of the mean.
Imagine a political poll conducted in 2012 involved
1000 registered voters. Say 42 percent preferred Newt
Gingrich, while 46 percent preferred Barak Obama, and there
were 12 percent undecided. You would likely have been
told that the uncertainty of these numbers is +/- 3 percentage
points.
Where does the +/- 3 % come from? It turns out that
it's simply 1/sqrt(1000), or the square root of the
reciprocal of the number of voters asked for a preference.
Now say an astronomer is using a telescope to measure
the brightness of a distant quasar. If 1000 photons
attributable to the quasar are counted, how accurately
can you say you have measured the brightness of
the quasar?
It turns out that it's the same relative error as in the
political poll: 1/sqrt(1000) ~ 3 percent. In order
to measure the brightness of the quasar to +/- 1
percent you need to measure 10,000 photons, so your
integration time has to be 10 times as long, or you
need to use a telescope with a collecting area 10 times
as big.
If we took an opinion poll consisting of fair minded
questions and we wanted the results to be good to
+/- 1 percent, how many people would we have to
poll in our survey?
A.
B.
C.
D.
100
1000
10,000
100,000