ES 25: Quantitative Thinking
Final Lab Assignment = 25 points ( “2.5x Regular Lab”)

10/25 points: All graphs (labeled, with trendlines and gridlines) are due Tuesday,
May 29th (HARDCOPY, IN CLASS). This deadline is intended to dissuade you
from waiting until the last minute to do this final lab (and to get you some semi-easy
points). These tasks are in blue.

15/25 points: The final lab, with all graphs pasted into the word document, is due
by HARDCOPY (printed out), in class Tuesday, June 5th. Calculus work (finding
derivatives and integrals) can be done neatly by hand in the hardcopy document).
PLEASE SHOW ALL WORK CLEARLY AND INCLUDE UNITS IN ALL ANSWERS.
Exploring the Sierra Snowpack
Climatologists predict that global climate change will cause warmer, wetter winters for
the Sierra Mountains. This may be problematic for California residents, who rely on
the winter snowpack to supply water for human consumption and for agriculture
during the summer months (and for snowboarding). Lets explore trends in the winter
snowpack for the Mammoth Mountain region.
The following information is taken from:http://www.or.nrcs.usda.gov/snow/about/swe.html
Snow Water Equivalent (SWE) is a common snowpack measurement. It is the amount of
water contained within the snowpack. It can be thought of as the depth of water that would
theoretically result if you melted the entire snowpack instantaneously.
For example, say there is a swimming pool that is filled with 36 inches of new powdery snow
at 10% snow water density. If you could turn all the snow into water magically, you would
be left with a pool of water 3.6 inches deep. In this case, the SWE of your snowpack would
equal 36" x 0.10 = 3.6 inches.
The National Resource Conservation Service measures SWE at many remote SNOTEL sites
and uses the data for stream flow forecasting. However, many and recreationists
(snowboarders and skiers) are interested in snow depth instead of SWE. The relationship
between the two values is explained as follow
To determine snow depth from SWE you need to know the density of the snow. The density
of new snow ranges from about 5% when the air temperature is 14° F, to about 20% when
the temperature is 32° F. After the snow falls its density increases due to gravitational
settling, wind packing, melting and recrystallization. THINK about what “Density” is (It is
NOT the density of water!! Note that density has no units… think about why this is…)
Given:
SWE = Density* Snow Depth
Snow Depth and SWE are typically measured in inches.
Density must be in decimal form. For example: 25% = 0.25

Import the following snowpack data file for the Mammoth Lakes region:
o http://cdec.water.ca.gov (if it takes you to “River Data,” delete extra typing
after .gov in the web address…)
o Next, click on “snowpack status”, then “Individual snow course historical data”
o We will explore the Mammoth Snow Course (208), which drains into the
Owens River. Select the month “February” and the years 1980 to 2006, and
choose “comma-delimited-output”
o Once you have the data in Excel, label the two columns of numbers clearly
(note: depth= “Snow Depth” and water = “Snow Water Equivalent or SWE”)
 Include units!
#1. Use Excel to graph “Snow water equivalent” (y) as a function of “snow depth” (x).
a. What does each data point on your graph represent?
b. Add a linear trendline, and report the regression equation and R2 value. Think
about what the y-intercept means, conceptually. NOTE: you can choose “set
intercept to zero” in the “trendline” menu, IF you think it should be zero.
c. According to your estimated equation (not the data itself), what is the average
snow density for the time period 1980 to 2006? HINT: look at the “given”
equation on page 1 and think about what the slope means.
d. Examine the data point "30-Jan-06. SWE = 19.1, Depth = 49.4” For the
Snow Depth that occurred, does this data point have a lower, or higher SWE
than would be expected from your linear regression equation? Explain your
logic.
e. Calculate the residual, or error, associated with this data point.
#2. Make a new column called “Snow Density”. Use the “Given” equation (above) to
calculate the snow density for each measurement date.
a. Snow Density is not a constant, it varies by location and many different
weather variables (temperature, humidity, etc). Use the “summary statistics”
tool in Excel (tools-data analysis-descriptive statistics) to calculate the
following values for the distribution of snow densities:
i.
ii.
iii.
iv.
Mean:____________________________
Standard deviation:__________________________________
Median:________________________________
(Note: you can also calculate these values manually without
“summary statistics,” see Lab 4.doc for commands)
b. Make a histogram of “snow density.” (see Lab 4 if you forgot how) Do you
think that the variable “snow density” is normally distributed? Why/why not?
c. IF we assume that the snow density for Mammoth is normally distributed,
with mean and standard deviation equal to those calculated above, create a
plot of the probability density function for Snow Density, using the
equation:
(HINT: this is very similar to the in-class worksheet that we did for the temperature
simulation data. Note: solution to that side of the worksheet is available on website)
d.
Add gridlines to your plot: choose “Chart Options/Grindlines/click both X
and Y major gridlines. If you want a smaller or larger distance between
gridlines, click on the axis, choose “Format Axis” then “Scale” then you can
change the “major unit.”
e. Estimate (using the plot above) the probability that Mammoth’s average snow
density next year will be 0.5 or greater, p (>0.5). Shade in the area that you are
finding on the graph.
f. Does your estimate agree with the 68/95/99.7 rule? Explain.
g. If the snow density next year is 0.51, would you believe this high density
snow occurred merely by chance? [In other words, is this observation
“statistically significant?” You will need to choose the “significance level”
()]. Why/why not? If you don’t believe it, what might have caused the high
value (you only need to offer one plausible explanation… if you are interested
in reading more about Mammoth Mountain, check it out on Wikipedia:
http://en.wikipedia.org/wiki/Mammoth_Mountain.)
#3. Plot snow density as a function of time. Set time= 0 to be 1980.
a. Add two trendlines to your graph: “Linear” and “2nd order polynomial”
i. Include the estimated equation and R2 value for each trendline.
ii. Add gridlines to your graph and print it out.
b. Define y and x for your equations (above) with units.
c. Explain the shape (straight vs. curved, increasing, decreasing, etc) of each
trendline, in terms of snow.
d. Which trendline do you think is more accurate? Can you think of a physical
explanation for the trend that seems most accurate?
e. Use each trendline equation to predict what the snow density will be in the
year 2020. (you will have two predictions)
f. Use the graphical approach to find the derivative of y, with respect to x, at:
Time = 5 years after the start (1985) Explain what your answer, y’(5),
means, in terms of snow. Do this for both equations.
g. Use calculus to find the equation for dy/dt (also written y’(t)) for each
trendline. What does this equation tell you, in words? What are its units?
i. Calculate y’(5) using your equations, above. Show your work.
h. According to your “polynomial trendline:”
i. When was the snow density a minimum?
ii. What is the minimum snow density value?
1. Mark these two values clearly on your graph.
2. Verify the values using calculus. (see workbook page 90).
Show work clearly.
i. Which trendline “agrees” with the global climate change prediction of
warmer, wetter Sierra winters? What do YOU believe?
i.
CONGRATULATIONS! You have used real data, graphs, statistics, and calculus to
examine trends in snow!