A note on scientific error

No scientific result is exact. A scientist is obligated to present an estimate of the uncertainties or errors in his or her results. This lab will not stress error analysis, but you should think a bit about how much you can trust the numbers you derived.

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 mV. How accurately do you think you can measure mV 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 mV 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 106 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 106).

Barring the factor of 106, 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, Ho 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 d, Ho and your first estimation of the age of the universe in subsequent calculations. It's perfectly fine to record these intermediate results on your lab sheet using as many digits as you'd like. However, somewhere on your lab sheet you should note your ``final'' values for d, Ho and the age of the universe using only 2 digits. Consider the accuracy of these values when you interpret your results!

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