**What is the purpose of this exercise?** First this exercise is meant to make you think about grain size, a fundamental geologic variable, more complexly. A second purpose of this exercise is to teach you how to
construct and analyze histograms, a fundamental first
step in the analysis of a single variable (univariate analysis), which we will focus on next week.
You will also learn about some basic univariate
statistical descriptors (that is a mouthful). Finally,
this exercise will give you more experience with the idea of approximation
and the importance of assumptions in analysis.

**Why is the grain size distribution of geologic
materials important?** Grain size distribution
data is widely used in geology and engineering. Think of sand, silt and mud. The grain size
of a particle is crucial in determining whether it is transported
in a current or when it comes to rest as a current diminishes.
This in turn is important for engineering purposes, such as considering
what size rip-rap to put in place to armor a channel bank. Too small and the current takes it away.. Grain
size distribution is also a fundamental determinant of intergrain permeability. Grain size is important in an array of chemical
reactions because smaller grain sizes increase surface area and
aggregate reactivity. Grain size distribution also carries information
as to origin - consider the difference between plutonic and volcanic
rocks. Example abound where grain size is crucial variable in
geoscience and other endeavors. Can you think of some additional
examples?

*Wentworth grain size scale from United States Geological Survey Open-File Report 2006-1195 . Web source: https://en.wikipedia.org/wiki/File:Wentworth-Grain-Size-Chart.pdf .*

**Background for exercise**: A sphere is an elegant shape, one that has a radius, diameter
and a distinct middle that is easily found (do you remember how
from your high school geometry class?). A perfect sphere has no
variation in its radius or diameter. Operationally, one typically measures
the radius as the distance from the center to the periphery, and
the diameter as the length of a line that passes through the center
and goes from one side to the other side of the circle. However,
an individual sand or cobble grain shows significant departures
from a spherical geometry (see the image of a sand grain below),
and this creates an object with variance in the diameter one would
measure. The next better geometric approximation for a real grain
shape might be an ellipsoid. An ellipsoid has 3 primary axes,
each of which would be a diameter by our definition. They intersect
at the center. There are also 'other' diameters that exist for
other lines that pass through the middle. Real grains of course
show departures from a perfect ellipsoid form. Lets define our
variable of interest as the diameter of a single grain; i.e. the
lengths of lines that pass from the center to the periphery.

Crucial to the concept of diameter is the identification of the
grain center. It turns out that finding the center is not a trivial
thing. One can take two perpendicular lines and move the intersection
about the grain interior until the center point is equidistant
from the margins as measured along a straight line. Then this
operation can be repeated with two other perpendicular lines multiple
times to hone in on the best center approximation. Another approach
would simply be to compute the point where the average distance
from that point to points on the grain boundary is a minimum. This point
would define the center and hence the radii in a different way.
This can be solved by brute computer force. By the end of this
course you would be able to set this exercise up in Excel if you
wanted to, but one might argue about if it is worth the time. However, many grains are
close enough to spherical that the concept of grain size as measured
by the grain diameter is still meaningful and useful. Experience indicates
the approximation is useful.

**Assumptions**:
In order to simplify we will consider a cross section of a quartz
grain in thin section. We will assume that this cross section
is representative of the grain and that it passes through the
middle of the grain and that we can identify the center. Can these
assumptions be challenged? It turns out that they can. Imagine
an egg shaped grain. When it comes to rest statistically it will
likely have its long axis horizontal, just like an egg on a counter
top. So the direction of the cross section cut is important. Additionally,
whether this cut went through the side or center of the grain
is indeterminate since it was basically a random cut. By picking
one of the larger grains in the thin section it is more likely
that this section passes near the middle. It is not uncommon that
assumptions are not perfectly met. It is also important to recognize
and analyze one's assumptions. The level of precision needed is
an important consideration here. Since we simply want to get some
idea of how the diameter of a single grain tends to vary these
assumptions are probably closely enough met. If it becomes important
to be very precise about the grain diameter measure then you might
want to proceed differently. Can you think of how? There is an
interesting trade off. To get an increase in precision of 10%
you may have to expend 100% more effort.

- Select which of the three grains above you would like to do.
- Select your grain center by eye.
- Measure carefully the two diameters associated with two perpendicular lines you draw through the center.
- Bisect each quadrant (by eye) with a diameter, and measure the length of these 2 additional diameters. Bisect each angle left with a diameter and measure these (should be 4 more). Bisect once again to get 8 more measurements of diameters. Keep going until you have >50 diameter readings (enough to get a decent histogram shape).
- For each diameter measurement mark a box with an X above the bin interval the value falls within, on the provided graph paper or within the provided Excel sheet template. Note the bins are each .01 mm wide. The scale of the copies of the grain cross sections handed out in class will be so that 10 cm = 1 mm, or 100:1. Most bins will have more than one reading, and so you stack your x boxes as you have more than one radius that falls within a bin.
- Which bin interval has the most boxes? This
is an estimate of the
**mode**. - What is the difference between your highest
and lowest measurement? This is known as your
**range**. - What will be the difference in the shape or form of the histogram for a very spherical grain versus a very elliptical grain?
- Can you think the histogram expression other gain shapes might have?
- Remember, as always, to ask questions.

@Harmon D. Maher Jr., 2010.