162
11 Errors
to undertake differential photometry, in which you compare two stars, one with a
known magnitude, and measure the difference in brightness. If the magnitude of your
known stars were one magnitude out, your observations could be very precise but not
accurate. When you use the words accuracy and precision, use them correctly and
with care; it is a very common mistake to use them as if they are interchangeable—
they are not.
Uncertainties can take two forms: random and systematic. Random uncertainties
are caused by random and unquantifiable effects, for example cosmic rays and sky
brightness. Systematic errors, on the other hand, cause a constant but unknown shift
in the data; for example, it might be the bias on a CCD or to some extent the dark
current. We don’t know what the actual value is, but we can allow for it.
11.2 Decimal Places and Significant Figures
Another common mistake is to confuse significant figures with decimal places. The
number of significant figures is a measure of precision, while the number of decimal
places reflects accuracy. Hence, 11.230 is a result to five significant figures but three
decimal places, while 0.001 is to three decimal places but only one significant figure.
When analysing your data, you should use only a level of accuracy that reflects
your equipment and your ability to read it. For example, the count of a pixel is always
an integer; hence to talk about a fraction of a count is meaningless.
The are several rules for dealing with significant figures and decimal places. For
significant figures, all nonzero numbers are significant, so 12.23 is significant to four
figures. A zero between nonzero digits is also significant, as are zeros after a decimal
point; hence 101.25 and 101.00 each have five significant figures. Zeros between a
decimal point and a nonzero number when the whole number is less than one are not
significant. Hence, 1.002 has four significant figures, but 0.002 has just one (although
it has three decimal places).
Mathematical processes in regard to decimal places and significant figures are
straightforward. When you multiply two numbers, your result should always have
the same number of significant figures as the smallest number of significant figures
contained within your multipliers. So, for example, 2.11 × 3.13 × 10. is equal to
66.043. However, our least significant figure is 10.,
1 with two significant figures, so
our result should be written as 66. If the first digit dropped is five or greater, round
up; otherwise, round down.
When you add or subtract figures, your result should have the same number of
decimal places as the least accurate figure has. Hence while 2.11 + 3.11 + 10. equals
15.22, since 10. is accurate to only two decimal places, our result should be written
15.00. As with multiplication, round the highest-value dropped figure up if it is five
or more, down if it is less.
1 Note that 10. has two significant figures because of the decimal point, whilst 10 has one significant
figure, and 10.0 has three.
11 Errors
to undertake differential photometry, in which you compare two stars, one with a
known magnitude, and measure the difference in brightness. If the magnitude of your
known stars were one magnitude out, your observations could be very precise but not
accurate. When you use the words accuracy and precision, use them correctly and
with care; it is a very common mistake to use them as if they are interchangeable—
they are not.
Uncertainties can take two forms: random and systematic. Random uncertainties
are caused by random and unquantifiable effects, for example cosmic rays and sky
brightness. Systematic errors, on the other hand, cause a constant but unknown shift
in the data; for example, it might be the bias on a CCD or to some extent the dark
current. We don’t know what the actual value is, but we can allow for it.
11.2 Decimal Places and Significant Figures
Another common mistake is to confuse significant figures with decimal places. The
number of significant figures is a measure of precision, while the number of decimal
places reflects accuracy. Hence, 11.230 is a result to five significant figures but three
decimal places, while 0.001 is to three decimal places but only one significant figure.
When analysing your data, you should use only a level of accuracy that reflects
your equipment and your ability to read it. For example, the count of a pixel is always
an integer; hence to talk about a fraction of a count is meaningless.
The are several rules for dealing with significant figures and decimal places. For
significant figures, all nonzero numbers are significant, so 12.23 is significant to four
figures. A zero between nonzero digits is also significant, as are zeros after a decimal
point; hence 101.25 and 101.00 each have five significant figures. Zeros between a
decimal point and a nonzero number when the whole number is less than one are not
significant. Hence, 1.002 has four significant figures, but 0.002 has just one (although
it has three decimal places).
Mathematical processes in regard to decimal places and significant figures are
straightforward. When you multiply two numbers, your result should always have
the same number of significant figures as the smallest number of significant figures
contained within your multipliers. So, for example, 2.11 × 3.13 × 10. is equal to
66.043. However, our least significant figure is 10.,
1 with two significant figures, so
our result should be written as 66. If the first digit dropped is five or greater, round
up; otherwise, round down.
When you add or subtract figures, your result should have the same number of
decimal places as the least accurate figure has. Hence while 2.11 + 3.11 + 10. equals
15.22, since 10. is accurate to only two decimal places, our result should be written
15.00. As with multiplication, round the highest-value dropped figure up if it is five
or more, down if it is less.
1 Note that 10. has two significant figures because of the decimal point, whilst 10 has one significant
figure, and 10.0 has three.