http://www.ck12.org Chapter 6. Planning and Conducting an Experiment or Study
Another reason to avoid a census is when it is destructive to the population. For example, many manufacturing
companies test their products for quality control. A padlock manufacturer might use a machine to see how much
force it can apply to the lock before it breaks. If they did this with every lock, they would have none to sell! It would
not be a good idea for a biologist to find the number of fish in a lake by draining the lake and counting them all!
The US Census is probably the largest and longest running census. The Constitution mandates a complete counting
of the population. The first U.S. Census was taken in 1790 and was done by U.S. Marshalls on horseback. Taken
every 10 years, a new Census is scheduled for 2010 and in a report by the Government Accountability Office in
1994, was estimated to cost $11 billion^2. This cost has recently increased as computer problems have forced the
forms to be completed by hand^3. You can find a great deal of information about the US Census as well as data from
past censuses on the Census Bureau’s website: http://www.census.gov/.
Due to all of the difficulties associated with a census, sampling is much more practical. However, it is important
to understand that even the most carefully planned sample will be subject to random variation between the sample
and population. As we learned in Chapter 1, these differences due to chance are calledsampling error. We can
use the laws of probability to predict the level of accuracy in our sample. Opinion polls, like the New York Times
poll mentioned in the introduction tend to refer to this asmargin of error. In later chapters, you will learn the
statistical theory behind these calculations. The second statement quoted from the New York Times article mentions
the other problem with sampling. It is often difficult to obtain a sample that accurately reflects the total population.
It is also possible to make mistakes in selecting the sample and collecting the information. These problems result
in anon-representative sample, or one in which our conclusions differ from what they would have been if we had
been able to conduct a census.
Flipping Coins
To help understand these ideas, let’s look at a more theoretical example. A coin is considered “fair” if the probability,
p, of the coin landing on heads is the same as the probability of landing on tails(p= 0. 5 ). The probability is defined
as the proportion of each result obtained from flipping the coin infinitely. A census in this example would be an
infinite number of coin flips, which again is quite impractical. So instead, we might try a sample of 10 coin flips.
Theoretically, you would expect the coin to land on heads 5 times. But it is very possible that, due to chance alone,
we would experience results that differ from the actual probability. These differences are due to sampling error. As
we will investigate in detail in later chapters, we can decrease the sampling error by increasing the sample size (or
the number of coin flips in this case). It is also possible that the results we obtain could differ from those expected if
we were not careful about the way we flipped the coin or allowed it to land on different surfaces. This would be an
example of a non-representative sample.
At the following website you can see the results of a large number of coin flips - http://shazam.econ.ubc.ca/flip/.
You can see the random variation among samples by asking for the site to flip 100 coins five times. Our results for
that experiment produced the following number of heads: 45, 41 , 47 , 45 ,and 45,which seems quite strange, since
the expected number is 50. How do your results compare?