Quantitative Skills > Issues and Discussion > Quantitative Preparation of Earch Science Graduate Students > Quantitative Competencies
Session Report: 1:00—5:30 PM, Tuesday, Feb. 21, 2006
(Download this report. (Acrobat (PDF) 82kB Apr6 06))

Math mechanics - can you crank out a correct answer?

This is a list of mathematical concepts that incoming graduate students should be able to use or at least define. Many students will have had exposure to these concepts prior to arriving at college, but maybe not to the extent that they can actually use them. The list was developed to encompass math concepts that we use in our own research, and hence would want our students to be able to use as well. This list should be considered a work in progress, and feedback and suggestions are encouraged. Items in bold may be important to include as well.


  1. graphing, graph interpretation
  2. unit conversion
  3. dimensional analysis
  4. back-of-the-envelope, order of magnitude calculations
  5. substitution of variables
  6. solving systems of equations (link to linear algebra)


  1. Dependent vs. independent variables
  2. Separation of variables
  3. Types of functions (linear, power, exponential, logarithmic)
  4. Periodic (trig) functions
  5. Time series analysis

Multi-variable Analysis

  1. trigonometry
  2. vectors
  3. directional derivatives
  4. gradient, slope
  5. matrices
  6. linear algebra
  7. sensitivity analysis
  8. eigenvalues

Statistics and Probability

  1. Error analysis (mean, median, std deviation, confidence interval)
  2. regression analysis (R2, χ2)
  3. Conditional probability
  4. Accuracy analysis
  5. Probability distributions
  6. Signal processing/pattern recognition
  7. Bayesian stats


  1. Ratio
  2. Rate
  3. Sum and integral
  4. Derivative and Partial derivative


Students should appreciate that science is intrinsically uncertain. It is important that they be comfortable with the notion that a deterministic description of a state or a process does not exist, and this does not render the knowledge useless. Sources of uncertainty are numerous, and students should be able to identify and quantify sources of error such as experimental design, measurement error, propagation of error, model inadequacy, etc. Students should be able to quantify uncertainty and demonstrate an ability to use the uncertainty estimates to assess the quality of a solution.

Logical analysis

Students should be able to take a problem, devise strategies for addressing the problem, implement strategies with appropriate tools and skills, and work toward resolution of the problem and understanding the results. These are some of the skills that could be used for various parts of logical analysis.

An example: A leaking underground storage tank, perhaps containing toxins.

Ability to learn independently

Students need to gain the confidence to understand that resources are available that enable them to solve problems. Students should practice using outside resources (e.g, books, web, journals) to gain the tools and methods to solve problems. Students need to gain confidence that they can expand their mathematical skill sets on their own. Transferring skills to new problems underlies the whole research process. It can involve solving quantitative problems with mathematical tools that are not traditionally used. We would like students to pick up tools and techniques from textbooks and modify or combine them appropriately to solve a new problem.

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