Page 59 - Petrology of Sedimentary Rocks

The value t is found in a table with the levels to be selected (5%, IO%, 20%, etc.)
plotted against the “degrees of freedom” (which in this case simply means the number
of values minus I); the value of t to be used lies at the intersection of our chosen
confidence level and the degrees of freedom. For example, with 21 values, at the 5%
(.05) level, t = 2.09, as shown in t table on page 59.

Populations and Probability

In the tests that follow it is essential that we understand the two concepts of
“populations” and “probability.”

Population is the term used for the data we are sampling; if we are measuring
porosities, the “population” that we are sampling is the vast number of porosities
present in every cubic inch of that formation, of which we test what we assume are a
representative few; if we are measuring the mica content of a bed, the “population” is
the percent of mica in each minute part of that bed; if we fill a jar with 500 black and
1,000 red beans and proceed blindfolded to pull out 50 beans, then we hope that our
sample of 50 is representative of the true “population”-- i.e., the 1,500 total beans. The
population is the vast amount of numerical data available, of which we take only a
smal I sample.

In many statistical tests, we try to answer the question, “what are the odds that we
could have obtained as great a difference or greater by chance sampling of the same
population?” This gives us an insight whether or not there is a real difference in the
properties of the two formations we have sampled, or whether we could have gotten
just as large a difference in, say chert content, by chance sampling of a single
formation (a “homogenous” population). Let’s think about it this way. We have a square
jar and a round jar, each filled with a certain mixture of 1,000 black and white beans.
The round jar has a population with a certain number of blacks, the square jar has a
different proportion of blacks. You are now blindfolded and asked to pull 50 beans from
one of the jars; let’s say you got 30 blacks and 20 whites. Now, still blindfolded, you are
asked to reach again into a jar (you still don’t know which one you are reaching into) and
pick 50 more beans. This time you get 22 blacks and 28 whites. Statistics enables us to
anwer the question, “on the second drawing, did I take from the same jar as I did the
first time; what is the probability that I drew both sets of beans from the same jar?”
Or phrased differently, “what are the odds that I drew both samples of 50 from the
same population (i.e., the same jar?)” This is essentially what we do when we obtain
numerical data from rocks. We take only a small sample of the numerical data
available in the formation, then sample another formation and see if there is any
difference between the two formations--and difference in the population of beans in
the jar, so to speak.

Probability. Many statistical tables have a critical value called “P” as an
important part. P stands for probability--in most cases, P, represented as a percent,
stands for the probability of a certain event happening. In the jar experiment above,
after running through the computations of the statistical test, we enter a table and
come out with a certain value for P. In this case let us say P came out to be .I0 (10%).
This means that if we had repeatedly sampled the same jar, only once in every ten
times would we have gotten as large, or larger, a difference as we did on the two draws.
Or, there is only one chance in ten that we have sampled the same population.
Although it is not technically correct to say it this way, we may state the corollary that
there is a 90% chance that we have sampled two different jars. Most statisticians do
not accept an experiment as significant unless P reaches the 5% level (.05 on the table,
pages 59 and 60).

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