You aren't given enough information to gauge the errors in many of the
quantities in this lab. One place in which you can estimate error is
in your determination of m_{V}. How accurately do you think
you can measure m_{V} from the light curves? To a tenth of a
magnitude? More? Less? How much does your result for the distance
*d* change if you change m_{V} by 0.1 mag? Assuming that
all the other quantities are accurate, try to estimate the uncertainty
in your calculated result for the distance *d*.

How does this uncertainty in the distance affect your results for the Hubble constant and the age of the universe?

The number of digits you use to express a result implies a certain level of accuracy. You can be misled when your calculator displays a 10-digit result. The relationship between number of digits and implied accuracy is a topic called ``significant figures,'' and will be covered in only a cursory manner here.

If someone handed you a typical metric ruler and asked you to measure the dimensions of a certain book as accurately as possible, you could probably measure it to the nearest tenth (or few tenths) of a millimeter. You wouldn't claim that one dimension of the book is 21.517564 cm; you couldn't measure it that accurately. You would be more likely to report the dimension as 21.52 cm. There would be some uncertainty in the last digit, but 21.52 cm would best represent your measurement.

What accuracy can you claim when you record your results in this lab?

In the sample calculation for the distance *d* at the end of the
page The distance to M100, the result was
expressed in scientific notation, and given as

d = 11.6 x 10^{6}pc, or d = 11.6 Mpc.

(The prefix ``mega'' in the quantity ``Mpc'' is just another form of
scientific notation, since ``mega'' tells you to multiply by
10^{6}).

Barring the factor of 10^{6}, only 3 digits appear in the
result. By convention, when a result is given in scientific notation,
and the errors are not explicitly stated, only the last digit is
presumed to be uncertain. The above result thus implies that you know
the distance *d* to an accuracy of +/- 0.5 Mpc. In truth, you
cannot claim to know the distance that well. The professional
astronomers who worked with this data stated their derived distance to
M100 as 16.1 +/- 1.3 Mpc (Ferrarese *et al.,* 1996). They
examined 70 Cepheids, not 8, and analyzed the data in greater depth.
You might be able to determine the distance to an accuracy of 2 Mpc.
Even if you obtained a distance of 16.1 Mpc, the uncertainty in your
result is greater than that of Ferrarese *et al.*

If the distance to M100 is only known to an accuracy of two digits (using scientific notation), and you derive the Hubble constant and the age of the universe from this value of the distance, can you claim to know the Hubble constant and age of the universe with any greater accuracy? Nope. The results you get out are only as good as the numbers you put in.

So what does all this mean? Strictly speaking, your final results for
*d*, *H _{o}* and the age of the universe should be
expressed in scientific notation, and contain only two digits. If you
followed the format on the lab sheet, you recorded more digits. Is
that bad? Not really. You don't want to round off too much until you
have completed all your calculations. You use your values for

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