Uncertainty Practice Problems
This module is undergoing classroom implementation with the Math Your Earth Science Majors Need project. The module is available for public use, but it will likely be revised after classroom testing.
Reported mass uncertainty
Powdered rock samples weighed on a digital scale.
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Provenance: Graham Baird, University of Northern Colorado
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Problem 1. Geochemists may need to know the total mass of a powdered sample before determining the percent concentration of certain elements. The manufacturer has extensively tested the balance and reports that the uncertainty in any measurement is 0.3%. The electronic display reads 3.67 g. Provide the measurement with the uncertainty value and the uncertainty range.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty has been provided as 0.3%, which can also be written as a decimal 0.003.
Step 2. Determine the uncertainty value for the reported measurement.
Multiply the sample mass by the percent in the decimal representation of the percent uncertainty: 3.67 g x 0.003 = 0.011 g is the uncertainty value.
Step 3. Report the measurement with the uncertainty value.
The measurement will be reported, followed by ± and the uncertainty value: 3.67 ± 0.011 g
Step 4. Report the measurement as a range.
Both add and subtract the uncertainty value to the measurement to define the lower and upper bounds of the range, then report the range: 3.67 g - 0.011 g = 3.659 g; 3.67 g + 0.011 g = 3.681 g. The range could be reported either as 3.659 to 3.681 g. or as 3.659 - 3.681 g.
Reported digital caliper uncertainty
Calipers measuring the length of a rock cylinder.
Provenance: Graham Baird, University of Northern Colorado
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Problem 2. Geophysicist measuring the seismic velocity of samples in a laboratory need to know the total length of a sample precisely before determining P-wave and S-wave velocities through the samples. The manufacturer has extensively tested the digital caliper and reports that the uncertainty in any measurement is 0.2%. The electronic display reads 706.38 mm. Provide the measurement with the uncertainty value and the uncertainty range.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty has been provided as 0.2%, which can also be written as a decimal 0.002.
Step 2. Determine the uncertainty value for the reported measurement.
Multiply the sample length by the percent in the decimal representation of the percent uncertainty: 706.38 mm x 0.002 = 1.41 mm is the uncertainty value.
Step 3. Report the measurement with the uncertainty value.
The measurement will be reported, followed by ± and the uncertainty value: 706.38 ± 1.41 mm.
Step 4. Report the measurement as a range.
Both add and subtract the uncertainty value to the measurement to define the lower and upper bounds of the range, then report the range: 706.38 mm - 1.41 mm = 704.97 mm; 706.38 mm + 1.41 mm = 707.79 mm. The range could be reported either as 704.97 mm to 707.79 mm. or as 704.97 - 707.79 mm.
Interpreting slip rate uncertainty from a graph
Geodetically-inferred and geologic slip rates for fault segments in Southern California.
Provenance: Modified from Figure 3 of Ray Y. Chuang, Kaj M. Johnson; Reconciling geologic and geodetic model fault slip-rate discrepancies in Southern California: Consideration of nonsteady mantle flow and lower crustal fault creep. Geology 2011;; 39 (7): 627–630. doi:10.1130/G32120.1
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Graphs are common ways to display uncertainty, and the ability to accurately interpret such visual representations is an important Earth science skill that can take practice to develop.
Problem 3: Examine the following graph to determine the uncertainty in the geologic fault slip rate (mm/yr) on the San Andreas Fault Mojave section (SAF-Mo).
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty is provided, but must be interpreted from the graph.
Step 2 Determine the uncertainty value for the reported measurement.
Locate the confidence interval of interest on the graph. In the upper right hand side, find the SAF-Mo label. Then, locate the axis of interest on the graph. The horizontal axis is labeled 'geologic slip rate,' so the horizontal lines extending from the symbol represent the geologic slip rate uncertainty. Each line extends horizontally 10 mm/yr from the data point symbol. This is the uncertainty for the reported measurement.
Step 3 Report the measurement with the uncertainty value.
The SAF-Mo data point symbol indicates a geologic slip rate of 30 mm/yr. That is the mean geologic slip rate for the San Andreas Fault Mojave section.
Therefore the confidence interval is 30 `+-` 10 mm/yr.
Estimating instrument uncertainty
Microfossil of a mollusk at Red Rock Canyon National Conservation Area.
Provenance: ZooFari, CC BY 3.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Microfossils.JPG.
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Earth scientists use a variety of instruments to make measurements, including many that involve complex electronics and report values to many decimal places. The instructions that come with the instrument (or that can often be found online) may have a list of specifications related to accuracy, precision, drift, etc.
Problem 4: An electronic balance (scale) reads the mass of a microfossil as 0.1059 g. You are reliably getting this value again and again and don't have the instruction book that might tell you more about uncertainty. Report the uncertainty in this measurement.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty in your scale must be estimated.
Step 2. Determine the finest gradation of your measurement and then divide in half.
There are four decimal places given, so the finest gradation is 0.0001 g and 0.0001/2 is 0.00005 g.
Step 3. Report the uncertainty
Estimating uncertainty of an angle measured by a field compass
Field compass showing the angle to the landmark as indicated by the north end of the compass needle (white).
Provenance: Graham Baird, University of Northern Colorado
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Problem 5. Your field compass has a marking every degree. Report the bearing to a landmark with the uncertainty value and as an uncertainty range.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty in your compass must be estimated.
Step 2. Determine the finest gradation of your measurement and then divide in half.
The compass has 1 degree as the minimum gradation. Therefore, the estimated error is 0.5 degrees.
Step 3. Take a measurement. Then report the measurement with the uncertainty value.
The bearing to the landmark in this case reads as 27 degrees. The measurement will be reported, followed by ± and the uncertainty value: 27 ± 0.5 degrees
Step 4. Report the measurement as a range.
Both subtract and add the uncertainty value to the measurement to define the lower and upper bounds of the range, then report the range: 27 degrees - 0.5 degrees = 26.5 degrees; 27 degrees + 0.5 degrees = 27.5 degrees. The range could be reported either as 26.5 to 27.5 degrees or as 26.5 - 26.5 degrees.
Determining uncertainty of rock composition from data
In geochemistry and mining, the concentration of something, such as gold in rocks, needs to be reported with an appropriate confidence interval. Accurately reporting uncertainty is important to be able to later compare multiple values or whether a site is likely to be economically viable for mining or not.
A granodiorite specimen with trace amounts of gold.
Provenance: Graham Baird, University of Northern Colorado
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Problem 6: Samples of granodiorite collected in the Sierra Nevada batholith contain some gold, but how much? How do you know how to interpret lab results that are returned to you? Give the uncertainty associated with your answer (95% confidence interval). Use the data set of
gold concentration in parts per billion (ppb) data (Comma Separated Values 288bytes Jun5 24) to provide the average concentration with uncertainty, assuming a normal distribution of data.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty must be calculated.
Step 2. Get the data into a spreadsheet.
Either:
- download the Gold concentration (ppb) data (Comma Separated Values 288bytes Jun5 24) and open from a spreadsheet application such as Google Sheets or Excel, or
- copy and paste the data from this page into a blank spreadsheet such as Google Sheets or Excel.
Gold_Gottfried1972 - Sheet1.csv| Gold concentration (ppb) | |
|---|
| Sample ID | Concentration of gold (ppb) |
|---|
| TP-5 | 0.3 |
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| TP-9 | 2.8 |
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| TP25 | 0.4 |
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| TP-32 | 3.8 |
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| TP-66 | 0.8 |
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| TP-72 | 0.6 |
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| TP-74 | 4.2 |
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| TP-76 | 1.4 |
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| TP-81 | 0.5 |
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| TP-112A | 0.7 |
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| TP-152 | 0.5 |
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| TP-171 | 0.9 |
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| TP-201 | 0.9 |
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| TP-211 | 0.6 |
|---|
| |
|---|
| Data source: Gottfried et al. 1972, Table 22 granodiorites | |
|---|
Download data (288bytes)
(last updated 2024-06-05 14:50:43)
Step 3. Use the spreadsheet functions average, stdev and count to determine mean, standard deviation and number.
- Calculate values for the mean, standard deviation, and number of data points.
`barx`, the mean is calculated using a spreadsheet by entering "=
average(
B3:B16)" into a cell.
B3:B16 indicates the cells on the spreadsheet that contains the data.
- The standard deviation is calculated using a spreadsheet by entering "=stdev(B3:B16)" into a cell.
- The number of data points is calculated using a spreadsheet by entering "=count(B3:B16)" into a cell.
Step 4. Plug values into the formula for the confidence interval
- `barx+-1.96*(s/sqrt(n))`
- Entering values found in Step 2: `1.3+-1.96*(1.3/sqrt(14))`
(Note: the units included at the end)
Calculating the confidence interval for ice thickness from data
Sea ice floating in front of an ice sheet
Provenance: From https://pixabay.com/photos/iceberg-ocean-winter-cold-snow-7994536/
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Knowing the confidence interval for sea ice thickness is necessary for determining changes in sea ice due to climate change.
Problem 7: Multiple measurements of sea ice thickness have been made in a given area to attempt to represent the average. Give the average ice thickness in centimeters (cm) and the uncertainty associated with your answer (95% confidence interval). Use the data set of ice thickness (cm) data (Comma Separated Values 168bytes Jun5 24) assuming a normal distribution of data.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty must be calculated.
Step 2. Get the data into a spreadsheet.
Either:
- download the ice thickness (cm) data (Comma Separated Values 168bytes Jun5 24) and open from a spreadsheet application such as Google Sheets or Excel, or
- copy and paste the data from this page into a blank spreadsheet such as Google Sheets or Excel.
icethickness_cm - Sheet1.csv| location | ice thickness (cm) |
|---|
| 1 | 101 |
|---|
| 2 | 205 |
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| 3 | 157 |
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| 4 | 308 |
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| 5 | 420 |
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| 6 | 192 |
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| 7 | 334 |
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| 8 | 250 |
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| 9 | 225 |
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| 10 | 95 |
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| 11 | 440 |
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| 12 | 209 |
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| 13 | 292 |
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| 14 | 123 |
|---|
| 15 | 221 |
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| 16 | 91 |
|---|
| 17 | 448 |
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| 18 | 294 |
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| 19 | 287 |
|---|
Download data (168bytes)
(last updated 2024-06-05 15:48:51)
Step 3. Use the spreadsheet functions average, stdev and count to determine mean, standard deviation and number.
- `barx`, the mean is calculated using a spreadsheet by entering "=average(B2:B20)" into a cell. B2:B20 indicates the cells on the spreadsheet that contains the data.
- The standard deviation is calculated using a spreadsheet by entering "=stdev(B2:B20)" into a cell.
- The number of data points is calculated using a spreadsheet by entering "=count(B2:B20)" into a cell.
Step 4 Plug values into the formula for the confidence interval
- `barx+-1.96*(s/sqrt(n))`
- Entering values found in Step 2: `247+-1.96*(112/sqrt(19))`
(Note: the units included at the end)
Calculating the confidence interval for percentage from data
Quadrat in a marsh
![[reuse info]](/images/information_16.png)
Provenance: Photo by Cathy Stubbs.
Reuse: https://www.flickr.com/photos/dougbeckers/6885246111
In science, it is often not possible to count everything, but instead we estimate the percent of something such as the percent cover of certain marsh grass species within a specific area.
Problem 8: A randomly selected set of quadrats (squares) have been measured for percent of the species D. spicata. Report the average and the uncertainty associated with your answer (95% confidence interval). Use the data set of D. spicata (%) data (Comma Separated Values 160bytes Jun5 24) assuming a normal distribution of data.
Step 1. Decide if the uncertainty is provided, must be estimated, or needs to be calculated.
In this case, the uncertainty must be calculated.
Step 2. Get the data into a spreadsheet.
Either:
- download the D. spicata (%) data (Comma Separated Values 160bytes Jun5 24) and open from a spreadsheet application such as Google Sheets or Excel, or
- copy and paste the data from this page into a blank spreadsheet such as Google Sheets or Excel.
Distichlis spicata percentage from Barn Island Marsh, CT| Quadrat number | D. spicata (%) |
|---|
| 1 | 50 |
|---|
| 2 | 60 |
|---|
| 3 | 75 |
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| 4 | 25 |
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| 5 | 10 |
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| 6 | 3 |
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| 7 | 1 |
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| 8 | 60 |
|---|
| 9 | 100 |
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| 10 | 75 |
|---|
| 11 | 10 |
|---|
| 12 | 75 |
|---|
| 13 | 100 |
|---|
| 14 | 90 |
|---|
| 15 | 35 |
|---|
| 16 | 75 |
|---|
| 17 | 60 |
|---|
| 18 | 80 |
|---|
| 19 | 70 |
|---|
| 20 | 70 |
|---|
Download data (160bytes)
(last updated 2024-06-05 22:39:15)
Step 3. Use the spreadsheet functions average, stdev and count to determine mean, standard deviation and number.
- `barx`, the mean is calculated using a spreadsheet by entering "=average(B2:B21)" into a cell. B2:B21 indicates the cells on the spreadsheet that contains the data.
- The standard deviation is calculated using a spreadsheet by entering "=stdev(B2:B21)" into a cell.
- The number of data points is calculated using a spreadsheet by entering "=count(B2:B21)" into a cell.
Step 4 Plug values into the formula for the confidence interval
- `barx+-1.96*(s/sqrt(n))`
- Entering values found in Step 2: `56+-1.96*(32/sqrt(20))`
(Note: the unit in this case is "%")
Next steps
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