Lec 4

Value proposition for a University: what value do you get and how does that fit into this course?

Learn critical/analytical thinking.

Learn content; how to construct a professional argument that data support a scientific model.

Character development.

Review of an example: Value Proposition A critical thinking

Pencil problem review

There are two ways to fill the box:

-take one random sample. Make 100 identical cylinders from this sample and line them up. This will give a distribution L= 100 x (1.00 +- 0.2) mm = 100 +-20 mm

-take 100 random samples and line them up. This will give a distribution L= 100 (1.00 +- 0.2/Sqrt[100]) = 100 +- 2 mm where the mean and std dev of that mean is used as the value that is multiplied by 100. Now when you fill the box, the 100 cylinders are randomly drawn from the population and are NOT identical. In this case the distribution you get is the distribution of means for a random sample of 100 cylinders placed in a line. The mean diameter for these 100 is 1.00 + 0.2/Sqrt[100]. Now when you line up 100 of cylinders with this mean and standard deviation you get a distribution of total lengths given by 100 x( 1.0 +0.2/Sqrt[100]) = 102 mm for 1 standard deviation and 104 mm for 2 standard deviations.

Remember the derivation of error in an average value using derivatives? delVmean=Sqrt[(delV)^2/n] = delV/Sqrt[n]. This is an error in the average value NOT the error in the sum value. The sum value is the total length which is about 100 mm while the average value of 100 readings has a mean of 1 mm. Label the plots of the distributions for individual lead diameters (mean 1.0 mm), the length of 100 next to each other (mean 100 mm), and avg. values (mean 1.0 mm).

Plot with labels three distributions: PDF for single cylinders (mean and std dev of 1.0 +0.2 mm), a line of 100 cylinders (100.0 +100 0.2/Sqrt[100] mm), and the cylinder diameter for 100 cylinders (100.0 +100 0.2/Sqrt[100])/100 = 1.0 + 0.2/Sqrt[100] mm). Note that you have to divide by 100 for the average value which then agrees with the error determined by propagation of errors for a mean value.

Why is this critical thinking? Not plug and chug. You have to think critically about what the formulas mean.

Review of an example: Value Proposition A learning to think analytically

Population pyramid http://www.randalolson.com/2015/07/14/rethinking-the-population-pyramid/ What is the error in the population data? How would you find it? Repeat the measurement. But the data is digital. There are only integer numbers of people. Since you can’t ask them all you take a sample of the population. However, some people might not respond to a survey because they were on vacation, some don’t know their age, etc. There is sampling error in a survey, a topic that has been carefully studied. http://www.nss.gov.au/nss/home.NSF/NSS/4354A8928428F834CA2571AB002479CE?opendocument

Sample size is important! You make 10 measurements of the voltage in the e/m lab. By averaging these you get a point in the distribution of mean values which has a stnd dev smaller than the distribution of individual values by 1/sqrt[10]. If you sample a larger number of voltages then the distribution of mean values gets smaller and you know the voltage more accurately. This is then used in the propagation of error formula for the error in e/m.

If you had a population of 1000000 “identical” USA countries and sampled the number of males between ages 50 and 55 in 100 such countries you would get a distribution. You can’t do this. You have to estimate the error differently.

One might argue that in general a distribution of the sum has a smaller percentage error for large numbers N. Take the box of N pencil cylinders as an example. The sum is N x( davg + sigma/Sqrt[N]) = N x( davg + Sqrt[N] sigma). The standard deviation increases as Sqrt[N] but the percent error decreases as Sqrt[N] sigma/N or sigma/Sqrt[N]. So a graph of percent error would have less spread with increasing N.

Why would sampling more people lead to less percent error? One error is that the people are not in the right age bin. Some are born in Jan and others in Dec so these average out in an estimate of the number of people of a certain age. If you had a couple of people in each year bin then a random sample might have people in one bin born in Dec matched with a few in Jan of the next year yet they are about the same age. This would make your age variable unreliable since a different sample of 10 could yield a very different result.

Another error is in determining sex. Is sex binary. If it is not then a sum would even out such an estimate.

Conclusion: often the larger the sample the less percent error. Note that percent error is what is presented in the last in the population pyramid link. Look how smooth the percent error graph is. Do you think the person who came up with this graph thinks it is good (does the artist know when the work is really good)?

The error should be presented with the data, as in the population pyramid graph, but often isn’t because it is not known. I’m taking a class on public works from the city of Golden. We learned about water law which requires the difference between the volume of water taken out of clear creek minus that put back in via sewage be a certain amount. There are instruments to measure both flows (the latter has large fluctuations due to things like football games). I asked the engineer what is the error in the flow gauges? He had to think about that. Yet such errors could lead to a violation of water law.

Why is this learning content?

Example: Value Proposition A critical thinking

A student asked me: What do I do when I go into the lab? I have a model to test, collect data on it, and see how repetitive it is. I often repeat it for different days to understand possible systematic errors (temperature variations, drift in accuracy of instruments, etc.). I think critically about the data and try to figure out how I can reduce the error (i.e. to get more accurate and precise RC data). When the error is minimized AND the data repeat, I then collect lots of data and then do a statistical analysis. In these labs I’d like you to learn the first part where you see how repetitive the data is and then do a simple calculation to see how well the data support the model. If you ignore this step you will perform statistics on lousy data. Garbage data in yield garbage statistics out. The first step is think critically about your data.

If you are a theorist (doing homework problems) you need to think critically about the result of your calculation. You should pretty much know the answer before the calculation because the calculation can easily yield results which are wrong. Think of how many mistakes could be made in the final result for a calculation requiring 10 steps.

Why is this critical thinking?

What is critical thinking? You can’t make an instrument to measure it if you don’t operationally define it.

Use Oxford and Cambridge links. Discuss some of the problems on the test, particularly the one on testing assumptions. Relate this to the RC assumptions. https://www.oxford-royale.co.uk/articles/critical-thinking-skills-university.html

“Despite variation in definitions of critical thinking, there is significant agreement on its core components. The American Philosophical Association’s (1990) definition, which reflects the consensus of 200 policy makers, employers, and professors, describes critical thinking as: “purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference as well as explanation of the evidential, conceptual and methodological considerations on which a judgment is based” (p. 2). Along these lines, Pascarella and Terenzini (2005) offer an operational definition of critical thinking largely based on the work of Erwin (2000):

“Most attempts to define and measure critical thinking operationally focus on an individual’s capability to do some or all of the following: identify central issues and assumptions in an argument, recognize important relationships, make correct references from the data, deduce conclusions from information or data provided, interpret whether conclusions are warranted based on given data, evaluate evidence of authority, make self-corrections, and solve problems (p. 156).”

Bok’s (2006) definition of critical thinking captures similar qualities: “The ability to think critically—ask pertinent questions, recognize and define problems, identify arguments on all sides of an issue, search for and use relevant data and arrive in the end at carefully reasoned judgments—is the indispensable means of making effective use of information and know- ledge (p. 109).”

From the Oxford critical thinking paper “Thinking skills test assessment (TSA) test specifications” at the bottom of page 10, critical thinking skills involve: drawing and summarising conclusions, applying principles, identifying assumptions and reasoning errors, and assessing the impact of additional evidence.”

Review the section “The importance of critical thinking” in the link above.

What are the pertinent questions based on? The model, heuristic, or pattern that is being discussed. What do you think is the biggest hindrance to critical thinking?

You don’t care because the things being taught are not important to you. You don’t have the time to critically evaluate the material, due to some deadline. You then rely on System 1 thinking. In business it is often the case that the most important thing is meeting a deadline rather than the quality of the work. Mines is good at training students to get the work in on time with less of an emphasis on truly understanding it, in my opinion.

Example of Critical thinking in the RC lab:

Applying principles: conservation of energy in the form of Kirchhoff’s laws

Identifying assumptions and reasoning errors: -R and C are constant. -The battery does not affect the circuit (as shown in the circuit diagrams in textbooks). -The scope does not affect the circuit (its input impedance add at least a resistance in parallel). -Errors in C (number of sig figs printed on device) and R (depends slightly on temperature and therefore current through it)

Discuss how there are two ways to deal with the model. First does it decay exponentially and second how do we determine the effects of errors in R and C? To deal with the first plot data on semi-log scale. Expect straight line. Show data indicating that RC decay is not exponential. Mention that the error is in the A/D resolution of 8 bits which shows up on a log scale dramatically. Say the max voltage is 10. The 8 bit (.04 V resolution) will only register a max of 2.3 for 10 V (and for the next bit, 2.298 for 9.96 V). At the minimum, it registers -5 for 4 mV (and for the next bit -4.8 for 8 mV). Also note that no straight line can go through all data points within the 8 bit error. At the minimum voltage there is a fluctuation about the least significant bit which shows up as the jitter in the steps.

Drawing and summarising conclusions (the data do not support the model) -how do you determine the error in the data? -repeat the measurement! Trigger the voltmeter to acquire when the switch is opened. Add data for 10 runs at a given time delay to get mean and standard deviation. Does straight line go through all data within error?

Show ImportFitData.nb

How do you convince someone of your conclusions? Emphasis is on supporting your conclusion with evidence.

Assessing the impact of additional evidence -try different values of R and C -REPEAT the measurement to get an idea of the error. -use a voltage standard or reference with a voltage divider to test the meter. -look at how C is measured (LC resonant circuit which specifies C at different frequencies).