# Apple-Google Wage Fixing and Systems Thinking

It seems that some of the most successful people in the world, at some of the world’s largest and most respected companies, engaged in an apparently wide-spread and illegal effort to limit employees’ job opportunities and wages.

This was stupid. It was stupid for the obvious reason: it was illegal. It was stupid for a more insidious reason, though: it will backfire on the company. We can explore why if we use a couple of system archetypes to think about the situation. These archetypes will come in handy in a wide range of situations.

The problem was retaining talent, who could be easily enticed away by more attractive compensation packages or to work at more exciting companies. Either it was easier to get an attractive compensation packages at a competitor, or the work did not stay sufficiently interesting or engaging. Simply put: employees were not happy enough.

The fix was to limit recruitment among competing companies.

In systems thinking, this is a classic “shifting the burden” dynamic. In shifting the burden, pictured below, you have two types of solutions to the symptoms of a problem: the fundamental solution—a corrective action for the root cause—and the symptomatic solution. The symptomatic solution reduces the symptom, but also creates a side effect that has a negative impact on the fundamental problem. Symptomatic solutions only have a temporary benefit before things get worse.

The classic shifting the burden system archetype describes wage-fixing practices as solutions to employee turnover.

The green links, labelled “+,” indicate that the two conditions at either end of the tail increase and decrease together. The application of more symptomatic solution causes an increase in the side effect; reducing the use of symptomatic solutions causes a decrease in the side effect. The red links, labelled “–,” indicate that the two conditions at either end work in opposite directions. An increase in the side effect causes a decrease in the effectiveness of the fundamental solution.

There are three cycles, or loops, in this diagram. Two of them are “balancing loops;” over time, one factor tends to balance out the other, and the situation stabilizes. The third loop is a “self-reinforcing loop;” such loops will “snow-ball” or continue increasing over time:

1. Applying a symptomatic solution
2. increases the side effect
3. which reduces the effectiveness of the fundamental solution
4. which increases the symptom
5. which drives more application of the symptomatic solution
6. and increases the side effect.

The solution is to focus on fundamental solutions—get at the root cause—and avoid or limit reliance on symptomatic solutions. Symptomatic solutions are always temporary and usually make things worse in the long term; fundamental solutions are permanent and don’t have negative side effects.

Why employees leave is also explained, in broad strokes, by another system archetype: the limits to growth.

The classic limits to growth system archetype describes why it’s hard to keep employees engaged and hold down turnover.

Here we have a self-reinforcing dynamic created by successes and interesting work with good compensation. Over time, employees should generate more success and have more interesting work and better compensation. However, this is coupled to a balancing loop that becomes stronger over time, and slows down the reinforcing loop, like the brakes in a car. Employees burn out, or generally stop producing as much. This balancing loop is driven by some limiting condition, which makes the slowing action—burnout or disengagement—stronger over time. While this simple version looks like it will lead to a steady state, more realistic versions often result in a crash, where the results not only level off, but actually decrease.

The solution to a limits to growth system is to attack the limiting condition. If employees get bored doing the same thing over time, then you have to find a way for them to be engaged with enough new, interesting work. If they can earn substantially better compensation packages at competitors, then you have to (approximately) match those packages.

If you don’t fix the limiting condition, you might see a temporary improvement via the dynamics of shifting the burden (or the closely-related archetype, fixes that fail), but in the long term the problem will only get worse.

# A Simple Introduction to the Graphing Philosophy of ggplot2

“The emphasis in ggplot2 is reducing the amount of thinking time by making it easier to go from the plot in your brain to the plot on the page.” (Wickham, 2012)

“Base graphics are good for drawing pictures; ggplot2 graphics are good for understanding the data.” (Wickham, 2012)

I’m not ggplot2’s creator, Hadley Wickham, but I do find myself in discussions trying to explain how to build graphs in ggplot2. It’s a very elegant system, but also very different from other graphing systems. Once you understand the organizing philosophy, ggplot2 becomes very easy to work with.

### The grammar of ggplot2 graphics

There is a basic grammar to all graphics production. In R‘s base graphics or in Excel, you feed ranges of data to a plot as x and y elements, then manipulate colors, scale dimensions and other parts of the graph as graphical elements or options.

ggplot2’s grammar makes a clear distinction between your data and what gets displayed on the screen or page. You feed ggplot2 your data, then apply a series of mappings and transformations to create a visual representation of that data. Even with base graphics or Excel we never really plot the data itself, we only create a representation; ggplot2 makes this distinction explicit. In addition, ggplot2’s structure makes it very easy to tweak a graph to look the way you want by adding mappings.

A ggplot2 graph is built up from a few basic elements:

 1 Data The raw data that you want to plot 2 Geometries geom_ The geometric shapes that will represent the data. 3 Aethetics aes() Aesthetics of the geometric and statistical objects, such as color, size, shape and position. 4 Scales scale_ Maps between the data and the aesthetic dimensions, such as data range to plot width or factor values to colors

Putting it together, the code to build a ggplot2 graph looks something like:

data
+ geometry to represent the data,
+ aesthetic mappings of data to plot coordinates like position, color and size
+ scaling of ranges of the data to ranges of the aesthetics


A real example shows off how this all fits together.

library(ggplot2)
# Create some data for our example
some.data <- data.frame(timer = 1:12,
countdown = 12:1,
category = factor(letters[1:3]))
# Generate the plot
some.plot <- ggplot(data = some.data, aes(x = timer, y = countdown)) +
geom_point(aes(colour = category)) +
scale_x_continuous(limits = c(0, 15)) +
scale_colour_brewer(palette = "Dark2") +
coord_fixed(ratio=1)
# Display the plot
some.plot


Demonstration of the key concepts in the grammar of graphics: data, geometries, aesthetic mappings and scale mappings.

Here you can see that the data is passed to ggplot(), aesthetic mappings between the data and the plot coordinates, a geometry to represent the data and a couple of scales to map between the data range and the plot ranges.

### More advanced parts of the ggplot2 grammar

The above will get you a basic graph, but ggplot2 includes a few more parts of the grammar that you’ll want to be aware of as you try to visualize more complex data:

 5 Statistical transformations stat_ Statistical summaries of the data that can be plotted, such as quantiles, fitted curves (loess, linear models, etc.), sums and so o. 6 Coordinate systems coord_ The transformation used for mapping data coordinates into the plane of the data rectangle. 7 Facets facet_ The arrangement of the data into a grid of plots (also known as latticing, trellising or creating small multiples). 8 Visual Themes theme The overall visual defaults of a plot: background, grids, axe, default typeface, sizes, colors, etc.

Hadley Wickham describes various pieces of this grammar in recorded presentations on Vimeo and YouTube and the online documentation to ggplot2. The most complete explanation is in his book ggplot2: Elegant Graphics for Data Analysis (Use R!) (Wickham, 2009).

### References

Wickham, Hadley. ggplot2: Elegant Graphics for Data Analysis. Dordrecht, Heibelberg, London, New York: Springer, 2009. Print.
Wickham, Hadley. A Backstage Tour of ggplot2 with Hadley Wickham. 2012. Video. YouTube. Web. 21 Mar 2014. . Contributed by REvolutionAnalytics.

# Normality and Testing for Normality

Many of our statistical tests make assumptions about the distribution of the underlying population. Many of the most common—ImR (XmR) and XbarR control charts, ANOVA, t-tests—assume normal distributions in the underlying population (or normal distributions in the residuals, in the case of ANOVA), and we’re often told that we must carefully check the assumptions.

At the same time, there’s a lot of conflicting advice about how to test for normality. There are the statistical tests for normality, such as Shapiro-Wilk or Anderson-Darling. There’s the “fat pencil” test, where we just eye-ball the distribution and use our best judgement. We could even use control charts, as they’re designed to detect deviations from the expected distribution. We are discouraged from using the “fat pencil” because it will result in a lot of variation from person to person. We’re often told not to rely too heavily on the statistical tests because they are not sensitive with small sample sizes and too sensitive to the tails. In industrial settings, our data is often messy, and the tails are likely to be the least reliable portion of our data.

I’d like to explore what the above objections really look like. I’ll use R to generate some fake data based on the normal distribution and the t distribution, and compare the frequency of p-values obtained from the Shapiro-Wilk test for normality.

### A Function to test normality many times

First, we need to load our libraries

library(ggplot2)
library(reshape2)


To make this easy to run, I’ll create a function to perform a large number of normality tests (Shapiro-Wilk) for sample sizes n = 5, 10 and 1000, all drawn from the same data:

#' @name assign_vector
#' @param data A vector of data to perform the t-test on.
#' @param n An integer indicating the number of t-tests to perform. Default is 1000
#' @return A data frame in "tall" format
assign_vector <- function(data, n = 1000) {
# replicate the call to shapiro.test n times to build up a vector of p-values
p.5 <- replicate(n=n, expr=shapiro.test(sample(my.data, 5, replace=TRUE))$p.value) p.10 <- replicate(n=n, expr=shapiro.test(sample(my.data, 10, replace=TRUE))$p.value)
p.1000 <- replicate(n=n, expr=shapiro.test(sample(my.data, 1000, replace=TRUE))\$p.value)
#' Combine the data into a data frame,
#' one column for each number of samples tested.
p.df <- cbind(p.5, p.10, p.1000)
p.df <- as.data.frame(p.df)
colnames(p.df) <- c("5 samples","10 samples","1000 samples")
#' Put the data in "tall" format, one column for number of samples
#' and one column for the p-value.
p.df.m <- melt(p.df)
#' Make sure the levels are sorted correctly.
p.df.m <- transform(p.df.m, variable = factor(variable, levels = c("5 samples","10 samples","1000 samples")))
return(p.df.m)
}


### Clean, random data

I want to simulate real-word conditions, where we have an underlying population from which we sample a limited number of times. To start, I’ll generate 100000 values from a normal distribution. To keep runtimes low I’ll have assign_vector() sample from that distribution when performing the test for normality.

n.rand <- 100000
n.test <- 10000
my.data <- rnorm(n.rand)
p.df.m <- assign_vector(my.data, n = n.test)


We would expect that normally distributed random data will have an equal probability of any given p-value. i.e. 5% of the time we’ll see p-value ≤ 0.05, 5% of the time we’ll see p-value > 0.05 and ≤ 0.10, and so on through > 0.95 and ≤ 1.00. Let’s graph that and see what we get for each sample size:

ggplot(p.df.m, aes(x = value)) +
geom_histogram(binwidth = 1/10) +
facet_grid(facets=variable ~ ., scales="free_y") +
xlim(0,1) +
ylab("Count of p-values") +
xlab("p-values") +
theme(text = element_text(size = 16))


Histogram of p-values for the normal distribution, for sample sizes 5, 10 and 1000.

This is, indeed, what we expected.

Now let’s compare the normal distribution to a t distribution. The t distribution would pass the “fat pencil” test—it looks normal to the eye:

ggplot(NULL, aes(x=x, colour = distribution)) +
stat_function(fun=dnorm, data = data.frame(x = c(-6,6), distribution = factor(1)), size = 1) +
stat_function(fun=dt, args = list( df = 20), data = data.frame(x = c(-6,6), distribution = factor(2)), linetype = "dashed", size = 1) +
scale_colour_manual(values = c("blue","red"), labels = c("Normal","T-Distribution")) +
theme(text = element_text(size = 12),
legend.position = c(0.85, 0.75)) +
xlim(-4, 4) +
xlab(NULL) +
ylab(NULL)

Starting with random data generated from the t-distribution:

my.data <- rt(n.rand, df = 20)


Histogram of p-values for the t distribution, for sample sizes 5, 10 and 1000.

The tests for normality are not very sensitive for small sample sizes, and are much more sensitive for large sample sizes. Even with a sample size of 1000, the data from a t distribution only fails the test for normality about 50% of the time (add up the frequencies for p-value > 0.05 to see this).

### Testing the tails

Since the t distribution is narrower in the middle range and has longer tails than the normal distribution, the normality test might be failing because the entire distribution doesn’t look quite normal; we haven’t learned anything specifically about the tails.

To test the tails, we can construct a data set that uses the t distribution for the middle 99% of the data, and the normal distribution for the tails.

my.data <- rt(n.rand, df = 20)
my.data.2 <- rnorm(n.rand)
# Trim off the tails
my.data <- my.data[which(my.data < 3 & my.data > -3)]
# Add in tails from the other distribution
my.data <- c(my.data, my.data.2[which(my.data.2 < -3 | my.data.2 > 3)])


Histogram of p-values for sample sizes 5, 10 and 1000, from a data set constructed from the t distribution in the range -3 to +3 sigmas, with tails from the normal distribution below -3 and above +3.

Despite 99% of the data being from the t distribution, this is almost identical to our test with data from just the normal distribution. It looks like the tails may be having a larger impact on the normality test than rest of the data

Now let’s flip this around: data that is 99% normally-distributed, but using the t distribution in the extreme tails.

my.data <- rnorm(n.rand)
my.data.2 <- rt(n.rand, df = 20)
# Trim off the tails
my.data <- my.data[which(my.data < 3 & my.data > -3)]
# Add in tails from the other distribution
my.data <- c(my.data, my.data.2[which(my.data.2 < -3 | my.data.2 > 3)])


Histogram of p-values for sample sizes 5, 10 and 1000, from a data set constructed from the normal distribution in the range -3 to +3 sigmas, with tails from the t-distribution below -3 and above +3.

Here, 99% of the data is from the normal distribution, yet the normality test looks almost the same as the normality test for just the t-distribution. If you check the y-axis scales carefully, you’ll see that the chance of getting p-value ≤ 0.05 is a bit lower here than for the t distribution.

To make the point further, suppose we have highly skewed data:

my.data <- rlnorm(n.rand, 0, 0.4)


This looks like:

For small sample sizes, even this is likely to pass a test for normality:

### What have we learned?

• With small sample sizes, everything looks normal.
• The normality tests are, indeed, very sensitive to what goes on in the extreme tails.

In other words, if we have enough data to fail a normality test, we always will because our real-world data won’t be clean enough. If we don’t have enough data to reliably fail a normality test, then there’s no point in performing the test, and we have to rely on the fat pencil test or our own understanding of the underlying processes.

Don’t get too hung up on whether your data is normally distributed or not. When evaluating and summarizing data, rely mainly on your brain and use the statistics only to catch really big errors in judgement. When attempting to make predictions about future performance, e.g. calculating Cpk or simulating a process, recognize the opportunities for errors in judgment and explicitly state you assumptions.

# Rewriting plot.qcc using ggplot2 and grid

The free and open-source R statistics package is a great tool for data analysis. The free add-on package qcc provides a wide array of statistical process control charts and other quality tools, which can be used for monitoring and controlling industrial processes, business processes or data collection processes. It’s a great package and highly customizable, but the one feature I wanted was the ability to manipulate the control charts within the grid graphics system, and that turned out to be not so easy.

I went all-in and completely rewrote qcc’s plot.qcc() function to use Hadley Wickham’s ggplot2 package, which itself is built on top of grid graphics. I have tested the new code against all the examples provided on the qcc help page, and the new ggplot2 version works for all the plots, including X-bar and R, p- and u- and c-charts.

In qcc, an individuals and moving range (XmR or ImR) chart can be created simply:

library(qcc)
my.xmr.raw <- c(5045,4350,4350,3975,4290,4430,4485,4285,3980,3925,3645,3760,3300,3685,3463,5200)
x <- qcc(my.xmr.raw, type = "xbar.one", title = "Individuals Chart\nfor Wheeler sample data")
x <- qcc(matrix(cbind(my.xmr.raw[1:length(my.xmr.raw)-1], my.xmr.raw[2:length(my.xmr.raw)]), ncol = 2), type = "R", title = "Moving Range Chart\nfor Wheeler sample data")


This both generates the plot and creates a qcc object, assigning it to the variable x. You can generate another copy of the plot with plot(x).

To use my new plot function, you will need to have the packages ggplot2, gtable, qcc and grid installed. Download my code from the qcc_ggplot project on Github, load qcc in R and then run source("qcc.plot.R"). The ggplot2-based version of the plotting function will be used whenever a qcc object is plotted.

library(qcc)
source("qcc.plot.R")
my.xmr.raw <- c(5045,4350,4350,3975,4290,4430,4485,4285,3980,3925,3645,3760,3300,3685,3463,5200)
x <- qcc(my.xmr.raw, type = "xbar.one", title = "Individuals Chart\nfor Wheeler sample data")
x <- qcc(matrix(cbind(my.xmr.raw[1:length(my.xmr.raw)-1], my.xmr.raw[2:length(my.xmr.raw)]), ncol = 2), type = "R", title = "Moving Range Chart\nfor Wheeler sample data")


Below, you can compare the individuals and moving range charts generated by qcc and by my new implementation of plot.qcc():

The qcc individuals chart as implemented in the qcc package.

The qcc individuals chart as implemented using ggplot2 and grid graphics.

The qcc moving range chart as implemented in the qcc package.

The qcc moving range chart as implemented using ggplot2 and grid graphics.

### New features

In addition to the standard features in qcc plots, I’ve added a few new options.

size or cex
Set the size of the points used in the plot. This is passed directly to geom_point().
font.size
Sets the size of text elements. Passed directly to ggplot() and grid’s viewport().
title = element_blank()
Eliminate the main graph title completely, and expand the data region to fill the empty space. As with qcc, with the default title = NULL a title will be created, or a user-defined text string may be passed to title.
new.plot
If TRUE, creates a new graph (grid.newpage()). Otherwise, will write into the existing device and viewport. Intended to simplify the creation of multi-panel or composite charts.
digits
The argument digits is provided by the qcc package to control the number of digits printed on the graph, where it either uses the default option set for R or a user-supplied value. I have tried to add some intelligence to calculating a default value under the assumption that we can tell something about the measurement from the data supplied. You can see the results in the sample graphs above.

### Lessons Learned

This little project turned out to be somewhat more difficult than I had envisioned, and there are several lessons-learned, particularly in the use of ggplot2.

First, ggplot2 really needs data frames when plotting. Passing discrete values or variables not connected to a data frame will often result in errors or just incorrect results. This is different than either base graphics or grid graphics, and while Hadley Wickham has mentioned this before, I hadn’t fully appreciated it. For instance, this doesn’t work very well:

my.test.data <- data.frame(x = seq(1:10), y = round(runif(10, 100, 300)))
my.test.gplot <- ggplot(my.test.data, aes(x = x, y = y)) +
geom_point(shape = 20)
index.1 <- c(5, 6, 7)
my.test.gplot <- my.test.gplot +
geom_point(aes(x = x[index.1], y = y[index.1]), col = "red")
my.test.gplot


Different variations of this sometimes worked, or sometimes only plotted some of the points that are supposed to be colored red.

However, if I wrap that index.1 into a data frame, it works perfectly:

my.test.data <- data.frame(x = seq(1:10), y = round(runif(10, 100, 300)))
my.test.gplot <- ggplot(my.test.data, aes(x = x, y = y)) +
geom_point(shape = 20)
index.1 <- c(5, 6, 7)
my.test.subdata <- my.test.data[index.1,]
my.test.gplot <- my.test.gplot +
geom_point(data = my.test.subdata, aes(x = x, y = y), col = "red")
my.test.gplot


Another nice lesson was that aes() doesn’t always work properly when ggplot2 is called from within a function. In this case, aes_string() usually works. There’s less documentation than I would like on this, but you can search the ggplot2 Google Group or Stack Overflow for more information.

One of the bigger surprises was discovering that aes() searches for data frames in the global environment. When ggplot() is used from within a function, though, any variables created within that function are not accessible in the global environment. The work-around is to tell ggplot which environment to search in, and a simple addition of environment = environment() within the ggplot() call seems to do the trick. This is captured in a stack overflow post and the ggplot2 issue log.

my.test.data <- data.frame(x = seq(1:10), y = round(runif(10, 100, 300)))
my.test.gplot <- ggplot(my.test.data, environment = environment(), aes(x = x, y = y)) +
geom_point(shape = 20)
index.1 <- c(5, 6, 7)
my.test.subdata <- my.test.data[index.1,]
my.test.gplot <- my.test.gplot +
geom_point(data = my.test.subdata, aes(x = x, y = y), col = "blue")
my.test.gplot


Finally, it is possible to completely and seamlessly replace a function created in a package and loaded in that package’s namespace. When I set out, I wanted to end up with a complete replacement for qcc’s internal plot.qcc() function, but wasn’t quite sure this would be possible. Luckily, the below code, called after the function declaration, worked. One thing I found was that I needed to name my function the same as the one in the qcc package in order for the replacement to work in all cases. If I used a different name for my function, it would work when I called plot() with a qcc object, but qcc’s base graphics version would be used when calling qcc() with the parameter plot = TRUE.

unlockBinding(sym="plot.qcc", env=getNamespace("qcc"));
assignInNamespace(x="plot.qcc", value=plot.qcc, ns=asNamespace("qcc"), envir=getNamespace("qcc"));
assign("plot.qcc", plot.qcc, envir=getNamespace("qcc"));
lockBinding(sym="plot.qcc", env=getNamespace("qcc"));


### Outlook

For now, the code suits my immediate needs, and I hope that you will find it useful. I have some ideas for additional features that I may implement in the future. There are some parts of the code that can and should be further cleaned up, and I’ll tweak the code as needed. I am certainly interested in any bug reports and in seeing any forks; good ideas are always welcome.

### References

• R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.
• Scrucca, L. (2004). qcc: an R package for quality control charting and statistical process control. R News 4/1, 11-17.
• H. Wickham. ggplot2: elegant graphics for data analysis. Springer New York, 2009.
• Wheeler, Donald. “Individual Charts Done Right and Wrong.” Quality Digest. 2 Feb 20102 Feb 2010. Print. <http://www.spcpress.com/pdf/DJW206.pdf>.

# Individuals and Moving Range Charts in R

Individuals and moving range charts, abbreviated as ImR or XmR charts, are an important tool for keeping a wide range of business and industrial processes in the zone of economic production, where a process produces the maximum value at the minimum costs.

While there are many commercial applications that will produce such charts, one of my favorites is the free and open-source software package R. The freely available add-on package qcc will do all the heavy-lifting. There is little documentation on how to create a moving range chart, but the code is actually quite simple, as shown below.

The individuals chart requires a simple vector of data. The moving range chart needs a two-column matrix arranged so that qcc() can calculate the moving range from each row.

library(qcc)
my.xmr.raw <- c(5045,4350,4350,3975,4290,4430,4485,4285,3980,3925,3645,3760,3300,3685,3463,5200)
#' Create the individuals chart and qcc object
my.xmr.x <- qcc(my.xmr.raw, type = "xbar.one", plot = TRUE)
#' Create the moving range chart and qcc object. qcc takes a two-column matrix
#' that is used to calculate the moving range.
my.xmr.raw.r <- matrix(cbind(my.xmr.raw[1:length(my.xmr.raw)-1], my.xmr.raw[2:length(my.xmr.raw)]), ncol=2)
my.xmr.mr <- qcc(my.xmr.raw.r, type="R", plot = TRUE)


This produces the individuals chart:

The qcc individuals chart.

and the moving range chart:

The qcc moving range chart.

The code is also available as a gist.

### References

• R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.
• Scrucca, L. (2004). qcc: an R package for quality control charting and statistical process control. R News 4/1, 11-17.
• Wheeler, Donald. “Individual Charts Done Right and Wrong.” Quality Digest. 2 Feb 20102 Feb 2010. Print. <http://www.spcpress.com/pdf/DJW206.pdf>.

# Scientific Consensus

I have seen a number of comments lately that “consensus” has no place in science, and that claims that there is a “scientific consensus” are just thinly veiled political double-speak. I have to take issue with such criticisms. In fact, consensus is one of the cornerstones of science.

This is not quite the same consensus used in politics or everyday life. Consensus means “general agreement” and is usually achieved through some form of discussion and negotiation. Consensus is therefore agreement over opinions, and is often agreement over a course of action despite differing opinions.

In science, consensus is derived from data and independent replication of experiments. It is the consensus that an idea—specifically, a testable hypothesis—is correct. It is the expression of scientists that a hypothesis is (a) scientifically testable and falsifiable, (b) that it has not been falsified and (c) that it explains the universe better than competing hypotheses. It is a consensus derived from the replication of observations or tests by other researchers.

Part of the problem, here, is that science has a habit of taking everyday words and developing very specific meanings around them. This happens because scientists need to communicate clearly and exactly at times, and language is messy and full of fuzzy concepts. The same thing happens in a lot of occupations. For example, accountants also develop specific meanings for everyday words.

Scientists no longer argue over the validity of Newton’s hypothesis on the gravitational force because there is broad consensus that the hypothesis is correct (as far as it goes). Objects attract each other according to their mass and the distance between them, and there have been plenty of independent experiments confirming the specific relationship, F = -G M1m2 / r2. The consensus is so strong that it’s referred to as Newton’s law of gravity. Likewise, scientists no longer argue over the geocentric model of the universe because there is broad consensus—derived from data collected over centuries by many independent researchers—that the Earth is not at the center of the solar system, let alone the universe.

Conversely, there is very little consensus when it comes to the accelerating expansion of the universe. Cosmologists agree that the universe is expanding faster than can be explained by our current understanding of the universe, but there are many, conflicting hypotheses about the causes. There is consensus over the fact of the accelerating expansion but the data does not yet support consensus on the underlying physical processes or mechanism causing it, and so there is no consensus about the physical process.

In one sense, scientific consensus is stronger, or more robust, than we are used to thinking about in politics and everyday life, precisely because it is based on observation and careful analysis by independent groups. It’s not just consensus based on what we think might be true, or what we want to be true, but based on what careful observation tells us must be true. Scientific consensus is not subject to whim.

From another perspective, though, scientific consensus is much weaker than we are used to. In science, there is no downside to abandoning or overturning a consensus when the data points in a different direction. In fact, there is significant benefit to being the person who can overturn a previous consensus; we remember Galileo, Newton, Darwin, Einstein and others precisely because their work, collecting and analyzing data, was so pivotal in altering the scientific consensus. In everyday life, if you back out of a consensus agreement, it’s likely that others party to the agreement will feel betrayed. There can be a significant social cost to pay for backing out of a consensus, even when you are convinced that you are right. Scientists may sometimes feel this same social pressure, but the scientific method provides clear guidance for adopting or abandoning consensus, and it doesn’t focus on the people involved but rather on external, objective observations of how the universe works. Scientific consensus is, perhaps, more readily changed than conventional consensus.

So consensus does exist is science and it plays an important role in science. We should be careful to distinguish, though, between consensus based on independent replication of results and consensus based on preconceptions and social negotiation.

# And You Thought Physics Was *YAWN*

Part of my day job involves monitoring the renewable energy market, and particularly keeping abreast of storage technologies. It’s like combining my hobby with my job.

A new wind turbine design was recently announced, the SeaTwirl. It’s an off-shore turbine design using vertical blades. The key technological advance here is that it includes a method to store energy, so that it can continue to produce electricity when the wind stops blowing. This ability to deliver a constant output is important, because, as you may have heard, wind energy is intermittent; you only get electricity when the wind blows, and only to the degree that it’s blowing. Demand, unfortunately, doesn’t follow wind’s intermittancy—nobody stops to check that the wind is blowing before they turn on their lights—and the utilities, transmission system operators and distribution system operators all have to supply electricity to meet demand.

The SeaTwirl stores energy by integrating an unusual wind turbine design with a pumped hydro system. It has the turbine blades on a large circular ring, which rotates parallel to the water’s surface kind of like a hula-hoop. This ring is hollow (more or less), and is filled up with water when the wind blows. When the wind stops blowing, the momentum of the water keeps the tube spinning, generating electricity, and the water is allowed to drain back out, spinning a hydro turbine to generate electricity.

The nice thing about this is that the storage can always be “recharged” and it’s “free.” Or at least it seems to be. If you’ve ever held a bucket while spinning around, you know that spinning with an empty bucket takes a lot less effort than spinning around with a full bucket. In part, this is because of a property known as the moment of inertia. The heavier or larger a spinning object gets, the more it resists changes to its rate of rotation (or rpm).

If the SeaTwirl is filling up this horizontal “hula-hoop” with water, then the weight of the tube is increasing and so is the moment of inertia. As the moment of inertia increases, the energy needed to reach a given rpm increases. Wind turbines normally generate electricity in proportion to the wind speed, because the rpm of the blades is proportional to the wind speed. Increase the moment of inertia and you decrease the rpm, which means you generate less electricity for a given wind speed.

Now comes the physics. For something shaped like a hula-hoop, the moment of inertia, I, is calculated from the mass, M, and the radius of the hoop, R, according to:

$I = M R^{2}$

The energy, E, in a spinning object is equal to the moment of inertia times the speed of rotation, ω, according to

$E=\frac{1}{2} I \omega ^{2}$

If we know the energy (because we know the wind speed), then we can calculate the speed of rotation, ω, by rearranging that equation to get ω on the left-hand side:

$\omega = \sqrt{\frac{2E}{I}}$

We can then replace I with mass and radius from the first equation to get

$\omega = \sqrt{\frac{2E}{MR^{2}}}$

So we can see that, if we don’t change the energy E (or don’t change the wind speed), and don’t change the radius R of the spinning hoop, then increasing the mass M results in a slower rate of rotation.

From SeaTwirl’s website and press releases, we can estimate how big the SeaTwirl is, which will let us estimate how much slower a full SeaTwirl will spin than an empty one, and therefore how much less electricity must be generated. We can calculate this by taking the ratio of ω full to ω empty, so that the parts that we don’t have to know E and R.

$\frac{\omega_{full}}{\omega_{empty}} = \frac{\sqrt{\frac{2E}{M_{full}R^{2}}}}{\sqrt{\frac{2E}{M_{empty}R^{2}}}} = \sqrt{\frac{\frac{2E}{R^{2}}}{\frac{2E}{R^{2}}}}\sqrt{\frac{M_{empty}}{M_{full}}}=\sqrt{\frac{M_{empty}}{M_{full}}}$

For the SeaTwirl, we now have to find out what R is, and estimate M for both filled and empty

The whole turbine assembly is made of composite materials, which probably have a density, $\rho_{c}$, of around 2500 kilograms per cubic meter (similar to fiberglass). Water has a density, $\rho_{w}$, of near 1000 kilograms per cubit meter (depending on temperature). The diameter of the turbine will be near 180 meters, so the radius, R, of our “hula-hoop” is half that, or 90 meters. From the pictures, it looks like the thickness of that hula-hoop is a few percent of the total diameter of the turbine, so we can figure an outside diameter of the “hula-hoop” of about 2 meters, for a radius, r, of 1 meter. Figure that at least ten percent of this is composite, and the rest is the hollow, water-filled portion.

To estimate the weight of the water in the “hula-hoop,” we can approximate the water as being a cylinder of radius $r_{w} = 0.9r$ and length equal to the circumference of the “hula-hoop,” $l = 2\pi R$. The volume of such a cylinder is equal to the cross-sectional area of the water column, $A_{w}=\pi r_{w}^{2}$ times the length of the column, l. The total mass of the water, $m_{w}$ is the density times this volume.

$m_{w} = \rho_{w}\pi r_{w}^{2}2\pi R = \rho_{w}3\pi (0.9r)^{2} R$

Plugging in our estimates for the above values gives us

$m_{w} = 1000 \cdot 3\pi (0.9)^{2} 90 = 687000$ kg

That’s a lot of water.

Now for the empty “hula-hoop.” We can treat it in the same way: a cylinder of material of radius r, length $l = 2\pi R$. However, we don’t want to calculate for a solid cylinder of composite; we have to subtract out the hollow part with radius $r_{w}$. So the mass of the composite is

$m_{c} = \rho_{c} 2 \pi R ( \pi r^{2} - \pi r^{2}_{w} )$

$m_{c} = 2500 \cdot 3 \pi 90 (1^{2} - 0.9^{2} ) = 403000$ kg

So the water more than doubles the weight of the hoop.

From the picture, you can see that there’s another hoop at the top, and the two hoops are connected by the turbines, which combined are probably worth at least another hoop in weight, so we can further assume that the mass of this bottom hoop, empty, is roughly one-third of the total mass of the movable parts of turbine.

The mass of the turbine, empty, is therefore about 1200000 kg, or 1200 tons. Filled with water, this goes up to about 1900000 kg, or 1900 tons. Empty, that’s maybe twice the largest off-shore turbine currently in existence, but this thing is easily twice as big as any current turbine, so our estimate appears to be in the right neighborhood.

Now we go back to our equation for the ratio of the rotational velocities, ω, and plug in these weights:

$\frac{\omega_{full}}{\omega_{empty}} = \sqrt{\frac{M_{empty}}{M_{full}}} =\sqrt{\frac{1200}{1900}} = 0.8$

So we get about 80% as much electricity from a water-filled turbine as from an empty one, when the wind blows. This is a direct efficiency loss due to the storage of energy in the spinning-water-hoop.

In addition, there’s the efficiency losses in loading water into the hoop, or “charging” the hoop, and the efficiency losses of “discharging” the hoop, running the water back out through a hydro turbine. Pumped hydro is usually about 72% efficient, or less, in each direction, so the total round-trip efficiency of storage + discharge is about 50% efficient. There are a lot of other storage technologies that do at least this good, if not better.

These two figures, the 80% efficiency loss of just operating the turbine and the 50% storage-discharge efficiency, can be used to directly compare SeaTwist with other wind turbine + storage technology solutions. Any storage technology that has at least a 50% round-trip efficiency and increases the total system cost by less than 20% over the system’s operating lifetime will outperform SeaTwist in terms of return on investment.