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What Does the Mean Mean? Describing Eruptions at Riverside Geyser, Yellowstone National Park

We can calculate the mean (or average) eruption interval of Riverside Geyser in Yellowstone National Park, but what does it tell us about

how frequently this geyser erupts? Is there a better way?

Thomas C. JusterDepartment of Geology, University of South Florida, Tampa, FL 33620© 2011 University of South Florida Libraries. All rights reserved.

SSACgnp.GB1198.TCJ1.3

Core Quantitative Literacy TopicsMean, Median, Mode

Supporting Quantitative Literacy TopicsWorking with real data, histograms

Core Geoscience SubjectHydrothermal features

This material is based upon work supported by the National Science Foundation under Grant Number NSF DUE-0836566.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Getting started

After completing this module you should be able to:

•Explain how geysers work, and why they are fragile

•Explain the difference between the mean, median, and mode

•Calculate the mean, median, and mode of a set of numbers

You should also know where Yellowstone National Park is

Wyoming

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Prerequisites

This module is the last in a series of three modules on Yellowstone National Park’s thermal areas. Although you can do them in any order, there are certain skills and concepts you must have in order to complete this module:

•You must know how to use Excel’s histogram feature, which means you must know how to determine if the Analysis ToolPak add-in is loaded, and how to load it if it isn’t;•You must know how to maneuver around long spreadsheet files.

These skills are taught in the second module of this series, Don’t Mess with Old Faithful!. If you have already completed that module, and feel you have mastered the background material about Yellowstone and geysers, you can proceed directly to Slide 12, The problem: conclusion.

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The setting – Yellowstone National Park

Yellowstone National Park is the world’s first national park, established in 1872 to protect the natural wonders found there, especially its hot springs, boiling mud pots, pools, and geysers. Sprawling over three states and 3,472 square miles, Yellowstone contains more geysers than the rest of the planet combined! The geysers are fueled by heat from one of the world’s largest volcanoes, which simmers quietly beneath the surface. In addition to its spectacular geologic wonders, Yellowstone is famous for its pristine ecosystem, which supports herds of wild bison, endangered wolves, and the southernmost concentration of grizzly bears in North America.

Photo from USGS

Photo from USGS Photo from USGS

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Yellowstone’s geysers

Approximately 11,000 geothermal features, including 300 active geysers, are present in Yellowstone National Park. They are concentrated in about nine “geyser basins”, most of which occur inside the Yellowstone caldera, a giant semi-circular depression that formed when the ground above the volcano collapsed following the last mind-bogglingly large eruption 640,000 years ago. Subsequent “minor” eruptions of lava and ash have largely filled the depression, obscuring its boundaries to the untrained eye. Magma still is accumulating beneath the volcano, causing the ground above to swell and occasionally break (producing earthquakes), and producing heat that escapes through fumaroles (gas vents), boiling mud pots, hot springs, and geysers.

Map of Yellowstone’s thermal features (shown in red) and caldera boundary (heavy black line), from the USGS

Q1. What is the average distance across the caldera in kilometers? Now using the formula for the area of a circle (A = πr2) estimate the area of the Yellowstone caldera in km2. Look up the area of your home town. How many of them would fit inside the Yellowstone caldera? (Click here If you don’t know how to convert area in square miles to area in square km.)

(Go direct to End-of-Module Questions)

A geyser needs a way for the water to rise from inside the Earth—a plumbing system. The plumbing system must be special, otherwise the water would just come out as a hot spring. A main component of this special system is rhyolite, a volcanic rock with a high silica (SiO2) content.

Earthquakes affect geysers in many ways. First, they fracture the rock, creating fissures that become plumbing systems. Second, they can increase or decrease pressure, which can change the periodicity of a geyser. Third, seismic waves can both seal or clear a geyser’s channel. Even distant quakes can have local consequences.

There are two types of geysers, fountain and cone. Hot springs do not have constrictions—the water flows to the surface unimpeded.

As the hot water rises through the rhyolite, it dissolves some of the silica and carries it along. The silica lines the underground passageways, making a tight seal strong enough to withstand high heat, great pressure, and violent eruptions. This form of silica is called sinter, and over centuries builds up to form a geyser’s cone.

Porous rock

Plumbing system

Water from rain and snow

Rhyolite

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Geysers: how do they work?

Heat source

What do you need to make a geyser? 1. Heat Source. Yellowstone is a hotspot—an area of the Earth’s surface that shows signs of long-lasting volcanic activity. In the past two million years, three massive and at least 30 smaller eruptions have occurred in this area. Partially molten rock is relatively close to the ground surface and supplies the heat for Yellowstone’s geysers and hot spring.

Heat source

Water

Earthquakes

Plumbing

3. Earthquakes. From 1,000 to 3,000 earthquakes typically occur each year within Yellowstone National Park and its immediate surroundings. Although most are too small to be felt, these quakes reflect the active nature of the Yellowstone region, one of the most seismically active areas in the USA. Each year, several quakes of magnitude 3 to 4 are felt by people in the park.

This animation, adapted from the Yellowstone National Park website, shows how a geyser forms and erupts.

B.P.A geyser’s plumbing also needs tight places, or constrictions within its underground passageway.

The water becomes superheated from areas of partially molten rock that lie only a few km below the surface of Earth.

A lot of the water remains in its liquid form because of great pressure from the overlying rock and water. This pressure allows the water’s temperature to rise far beyond its boiling point on the Earth’s surface.

FountainCone Hot Spring

2. Water. Most of the water in a geyser first arrives in Yellowstone in the form of rain or snow. It seeps deep into the earth and then makes its way back up again. But did you know this round trip may take hundreds or even thousands of years?

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Here’s a typical geyser at Yellowstone National Park. The geyser’s channel is coated with sinter. The sinter may accumulate as a mound above the ground (a cone), and may cause constrictions in the channel.

A geyser's eruption is a chain reaction. As superheated water rises, steam bubbles begin to form and expand. When the bubbles encounter the constriction, they become trapped because they're too large and numerous to squeeze through the tight spot.

As more water and steam in a geyser build up behind the constriction, the temperature and pressure of the whole system increases. The trapped bubbles move about violently--so violently, in fact, that some of the underlying water is lifted above the bubbles, and shoved through the constriction.

When that happens there's a sudden drop in pressure. A lot of the water in the system flashes into steam. This takes up so much space that the remaining water is forcefully thrown from the geyser's vent. That's what we call an eruption!

Watch it happen next.

Eruption!

Sinter

Constriction

Water lifted above constriction

Geysers, con’t.

Q2. One of the biggest threats to geysers in Yellowstone National Park is from people throwing things into them—coins, rocks, bottles, etc. In many cases, geysers have been completely extinguished by this careless behavior. How do you think debris thrown into the vent of a geyser can shut it down?

(Go direct to End-of-Module Questions)

Image from USGS

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Collecting data on geysers

The water that spurts from a geyser is so hot it will badly scald anyone who touches it—so the vent itself is definitely not the place to stand and make observations! In the past, scientists monitored geyser eruptions from afar, with their eyes or, later, cameras. This worked, but required frequent vigilance, attention, and didn’t generally work as well at night or in the dead of (a very cold) Yellowstone winter. More recently, the Geyser Observation and Study Association (GOSA) has been monitoring several geysers with temperature sensors placed in their output channels, where the erupted water flows away. By monitoring the temperature of the water, scientists can determine precisely when a geyser eruption starts and finishes.

Output channel for Till Geyser , Midway Geyser Basin, Yellowstone National Park

Eruption starts now!

The red dot represents the temperature in the output channel

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The problem

The National Park Service was created to conserve spectacular and rare natural resources—to protect them from being changed by unnatural (i.e., human) activities (Endnote 1). Among the rarest and most spectacular of these resources are the geysers of Yellowstone National Park. Scientists are interested in these geysers because they act as a sensitive links between three large forces: the escape of heat from the Yellowstone volcano, the movement of water beneath the ground, and the tectonic stresses that produce faults and earthquakes. For example, in 2002 geysers in Yellowstone temporarily changed the pattern of their activity following a large earthquake in Denali, Alaska!

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The problem, con’t.

Unfortunately, geysers are also very fragile and can be easily destroyed by human activities. For example, Beowawe, Nevada, used to contain over 30 geysers, some of which erupted over 30 feet. All of these geysers have been destroyed by the construction of geothermal power plants in the 1980’s, which robbed the geysers of the hot water they needed and turned them off for good. The National Park Service has the responsibility to make sure that never happens to Yellowstone’s geysers.

Beowave geysers in 1971 (photo from Desert Magazine)

Beowave geysers today(photo from the Elko Rose Garden Association)

Although protected within a national park, Yellowstone’s geysers are not without threats. There are two areas immediately adjacent to the park that overlie rock that is anomalously warm, and capable of providing geothermal energy (called KGRAs, for “Known Geothermal Resource Areas”). This energy would have to be extracted by circulating water through deep wells, and would have a significant and permanent impact on the hydrologic system of the entire area. With the US trying to wean itself from the use of fossil fuels, there will be increasing pressure to develop alternative energies, including geothermal energy.

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The Problem, con’t.

In fact, development almost occurred at Corwin Springs in 1986. The Church Universal and Triumphant was given permission to drill a 450-foot-deep hydrothermal well, which significantly decreased the flow to an adjacent spring (not in the park). In response, Congress hastily passed amendments to the Geothermal Steam Act in 1988 stopping the project, and a more permanent ban was established in 1994 with the signing of the Montana-NPS [National Park Service] Water Compact. Although this compact bans hydrothermal development in areas adjacent to the park in Montana, it says nothing about Idaho.

Yellowstone National Park

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The problem, conclusion

One of the reasons for studying Yellowstone’s geysers is to establish a baseline of activity from which deviations can be measured. This is important not only so that human threats can be detected and avoided, but also to understand how the activity of geysers varies naturally. The question is: what is ‘normal’ activity?

Geysers are intermittent eruptions of hot water from the ground. The interval between their eruptions ranges from minutes to days, is never constant, and often changes naturally over a period of years. The variation in the eruption interval is random—i.e., it cannot be predicted from other information—even though some eruption intervals are more common than others. Scientists use various mathematical tools to describe the “most likely” interval between eruptions. This information is important because it is one measure of how “unimpaired” geysers behave, and allows scientists to detect changes that might be due to human activities.

The BIG QUESTIONS:

1. How best can we characterize the natural eruption behavior of Riverside Geyser in Yellowstone National Park using the mean, median, and mode of its eruption interval?

2. How large would a change in the eruption behavior have to be to be considered “abnormal”?

The key idea in this module is to understand the three main measures of the “most likely” value in a list of random numbers: the mean, the median, and the mode. You are undoubtedly familiar with the mean, or average (isn’t that the first question students ask after an exam is returned? What is the mean?). However, as you will learn in this module the other means of “most likely” have their strengths too, and in some cases better tell the story you are trying to tell.

Measures of central tendency: mean, median, mode

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The mean is just another word for the average, which you probably know is computed by adding together a group of values and dividing by their number.

The mean (average)

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Mean

The distance between the seven numbers and the mean are plotted on a number line below Notice that the total length of the lines less than the mean (in red) is equal to the total length of the lines greater than the mean (in green). Click for a demonstration.

Also notice how the mean is strongly influenced by “outliers”—numbers that are far from the mean.

What is the outlier number in this set of seven numbers?

1 2 3 4 5 6 7 8 9 10 11

Excel also makes it easy to compute the mean: with the function =AVERAGE(range)

The median

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The median is the value that divides the data into two groups of equal number: those larger than the median, and those smaller. It can be found by first sorting the data, and then looking for the value half way down the list. This is shown graphically on the number line below, with the numbers less than the median shown in red and the numbers larger in green.

1 2 3 4 5 6 7 8 9 10 11

Median

3 numbers less than median

3 numbers greater than median

Once again, Excel makes it easy to compute the median, with the function =MEDIAN(range).

The mode is the value that appears most often. If no value appears more than once, there is no mode.

The mode can be found by sorting the data and looking for the value that appears most often, as shown in the example.

The mode

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The number 2 appears twice, which is more than any other number—so it’s the mode

As you might expect, Excel also provides a function for calculating the mode: =MODE(range).

However . . . It is often very useful to determine the mode visually from a frequency histogram of the data. For example, here is the histogram for the previous example:

The mode and the frequency histogram

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You can quickly scan the frequency column for the largest number . . . the mode

You can also quickly see the mode from the frequency histogram(it’s the highest bar)

The Geyser Observation and Study Association (GOSA) collects and catalogs eruption data on many of the geysers in Yellowstone National Park. You can see images, background, and the eruption history of all of the geysers they monitor here. For this module we will be using data from Riverside geyser, which is located about two miles away from Old Faithful in the Upper Geyser Basin of Yellowstone National Park.

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Here is what the first three days for 2007 look like:

Click on the icon to open the Excel file, then immediately save it with a unique name. The first worksheet provides space for the answers to your End-of-Module questions. The second worksheet lists the eruption intervals in minutes for Riverside Geyser from January 1, 2007 through December 31, 2008 in two separate worksheets. The data have been cleaned up to remove erroneous or missing entries (e.g., due to equipment failures). Note the spreadsheet is very long, over 1,000 lines for each year!

Riverside geyser eruption interval data

“EI” stands for Eruption Interval

Note the two tabs, one for each year

Riverside Geyser Student

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Characterizing the “most likely” eruption interval from Riverside Geyser, 2007

Now recreate the file shown below, which calculates the mean, median, and mode for the 2007 eruption intervals from Riverside Geyser:

Using the MIN(range) and MAX(range) functions, what is the range of eruption intervals for 2007?

(data continue)The formula in this cell is =COUNT(B2:B10000). Why does this work?

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Histogram for eruption intervals, Riverside geyser, 2007

Now using a bin interval of one minute, plot the frequency histogram for all the 2007 Riverside eruptions, and save in a new window. Your histogram should look like this:

Print out the histogram, and place little arrows marking the value of the mean, median, and mode.

This kind of a plot is called a “bimodal distribution”. What do you think this means (remembering, as “bipeds” we are capable of walking on two feet . . .)

(Go direct to End-of-Module Questions)

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End-of-Module Assignment

1. Answer Q1 on Slide 5 and Q2 on Slide 7.

2. Hand in the histogram you printed for Slide 20, along with the location of the mean, median, and mode for Riverside Geyser’s 2007 eruptions.

3. Based on your annotated histogram from Slide 20, which of the three measures of “most likely” best represents the data from Riverside Geyser for 2007? Or is a combination better? Explain.

4. Now using the data for 2008, compare the eruption behavior of Riverside Geyser in 2008 to that in 2007:

a) How many times did Riverside Geyser erupt in 2008?

b) Determine the range, mean, median, and mode of the 2008 eruptions.

c) How does the mean eruption interval for 2008 compare to 2007? How can you explain this change? (HINT: look at the frequency histograms!)

5. Tom and Judy are walking Yellowstone at 3:35 p.m. and come across a bunch of people waiting for Riverside Geyser to erupt. The sign says it’s predicted to erupt at 4:05, plus or minus 30 minutes. Judy wants to take some pictures of Morning Glory Pool, a short distance away, but is worried she’ll miss seeing Riverside erupt if she does so. Tom tells Judy, “No worries . . . sign says 4:05, we still have 30 minutes before this thing is going to below . . . let’s go!” What do you think about Tom’s logic? Make reference to the frequency plot in your answer!

To convert square miles to square kilometers we need a conversion factor C that will multiply square miles:

The conversion factor C must have a value of one, because we don’t want to change the length when we convert it, and multiplying a number by one leaves it unchanged. The conversion factor should also be such that the result of the multiplication with mi2 is km2. If you remember about multiplying fractions, you can infer that the conversion factor therefore should have km2 in the numerator (top) and mi2 in the denominator (bottom) so that when we cross-multiply:

The conversion factor we seek should know something about the relative sizes of miles and kilometers, which is that there are 0.61 miles per kilometer. We express this as a fraction,

Now the important thing to know about this number is that it’s also one (since 0.62 miles are equal to one km, the top and bottom of the fraction represent the same length, and thus are equal). It’s not exactly what we need (because it has mi and km, not mi2 and km2, and the km part is on the bottom, not the top) but because it’s one we can:

• Multiply, divide, or raise to any power the numerator, as long as we do the same to the denominator;

• Flip the fraction, so the numerator becomes the denominator, and vice versa.

Using these tricks we can convert the fraction into the conversion factor we need:

Return to Slide 5.

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How to convert mi2 to km2

Endnotes

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1. The act that formed the National Park Service (NPS) is called the Organic Act. The critical passage that describes the mission of the NPS obligates it to “conserve the scenery and the natural and historic objects and the wild life [in the National Parks], and to provide for the enjoyment of the same in such manner and by such means as will leave them unimpaired for the enjoyment of future generations. The key word that relates to Yellowstone’s geysers is unimpaired. Return to Slide 9.