Understanding Data

When analyzing data, I have often found it useful to think of the data as being one of four main types, according to the typology proposed by Stevens.[1] Different types of data have certain characteristics; understanding what type of data you have helps with selecting the analysis to perform perform while preventing basic mistakes.

The types, or “scales of measurement,” are:

Nominal
Data identifying unique classifications or objects where the order of values is not meaningful. Examples include zip codes, gender, nationality, sports teams and multiple choice answers on a test.
Ordinal
Data where the order is important but the difference or distance between items is not important or not measured. Examples include team rankings in sport (team A is better than team B, but how much better is open to debate), scales such as health (e.g. “healthy” to “sick”), ranges of opinion (e.g. “strongly agree” to “strongly disagree” or “on a scale of 1 to 10”) and Intelligence Quotient.
Interval
Numeric data identified by values where the degree of difference between items is significant and meaningful, but their ratio is not. Common examples are dates—we can say 2000 CE is 1000CE + 1000 years, but 1000 CE is not half of 2000 CE in any meaningful way—and temperatures on the Celsius and Fahrenheit scales, where a difference of 10° is meaningful, but 10° is not twice as hot as 5°.
Ratio
Numeric data where the ratio between numbers is meaningful. Usually, such scales have a meaningful “0.” Examples include length, mass, velocity, acceleration, voltage, power, duration, energy and Kelvin-scale temperature.

The generally-appropriate statistics and mathematical operations for each type are summarized in table 1.

Table 1: Scales of measurement and allowed statistical and mathematical operations.
Scale Type Statistics Operations
Nominal mode, frequency, chi-squared, cluster analysis =, ≠
Ordinal above, plus: median, non-parametric tests, Kruskal-Wallis, rank-correlation =, ≠, >, <
Interval plus: arithmetic mean, some parametric tests, correlation, regression, ANOVA (sometimes), factor analysis =, ≠, >, <, +, –
Ratio plus: geometric and harmonic mean, ANOVA, regression, correlation coefficient =, ≠, >, <, +, -, ×, ÷

While this is a useful typology for most use, and certainly for initial consideration, there are valid criticisms of Stevens’ typology. For example, percentages and count data have some characteristics of ratio-scale data, but with additional constraints. e.g. the average of the counts $\overline{(2, 2, 1)} = 1.66\ldots$ may not be meaningful. This typology is a useful thinking tool, but it is essential to understand the statistical methods being applied and their sensitivity to departures from underlying assumptions.

Types of data in R

R[2] recognizes at least fifteen different types of data. Several of these are related to identifying functions and other objects—most users don’t need to worry about most of them. The main types that industrial engineers and scientists will need to use are:

numeric
Real numbers. Also known as double, real and single (note that R stores all real numbers in double-precision). May be used for all scales of measurement, but is particularly suited to ratio scale measurements.
complex
Imaginary real numbers can be manipulated directly as a data type using

x <- 1 + i2

or

x <- complex(real=1, imaginary=2)

Like type numeric, may be used for all scales of measurement.

integer
Stores integers only, without any decimal point. Can be used mainly for ordinal or interval data, but may be used as ratio data—such as counts—with some caution.
logical
Stores Boolean values of TRUE or FALSE, typically used as nominal data.
character
Stores text strings and can be used as nominal or ordinal data.

Types of variables in R

The above types of data can be stored in several types, or structures, of variables. The equivalent to a variable in Excel would be rows, columns or tables of data. The main ones that we will use are:

vector
Contains one or many elements, and behaves like a column or row of data. Vectors can contain any of the above types of data but each vector is stored, or encoded, as a single type. The vector

c(1, 2, 1, 3, 4)
## [1] 1 2 1 3 4

is, by default, a numeric vector of type double, but

c(1, 2, 1, 3, 4, "name")
## [1] "1" "2" "1" "3" "4" "name"

will be a character vector, or a vector where all data is stored as type character, and the numbers will be stored as characters rather than numbers. It will not be possible to perform mathematical operations on these numbers-stored-as-characters without first converting them to type numeric.

factor
A special type of character vector, where the text strings signify factor levels and are encoded internally as integer counts of the occurrence of each factor. Factors can be treated as nominal data when the order does not matter, or as ordinal data when the order does matter.
factor(c("a", "b", "c", "a"), levels=c("a","b","c","d"))
## [1] a b c a
## Levels: a b c d
array
A generalization of vectors from one dimension to two or more dimensions. Array dimensions must be pre-defined and can have any number of dimensions. Like vectors, all elements of an array must be of the same data type. (Note that the letters object used in the example below is a variable supplied by R that contains the letters a through z.)

# letters a - c in 2x4 array
array(data=letters[1:3], dim=c(2,4))
##      [,1] [,2] [,3] [,4]
## [1,] "a"  "c"  "b"  "a"
## [2,] "b"  "a"  "c"  "b"

# numbers 1 - 3 in 2x4 array
array(data=1:3, dim=c(2,4))
##      [,1] [,2] [,3] [,4]
## [1,]    1    3    2    1
## [2,]    2    1    3    2
matrix
A special type of array with the properties of a mathematical matrix. It may only be two-dimensional, having rows and columns, where all columns must have the same type of data and every column must have the same number of rows. R provides several functions specific to manipulating matrices, such as taking the transpose, performing matrix multiplication and calculation eigenvectors and eigenvalues.

matrix(data = rep(1:3, times=2), nrow=2, ncol=3)
##      [,1] [,2] [,3]
## [1,]    1    3    2
## [2,]    2    1    3
list
Vectors whose elements are other R objects, where each object of the list can be of a different data type, and each object can be of different length and dimension than the other objects. Lists can therefore store all other data types, including other lists.

list("text", "more", 2, c(1,2,3,2))
## [[1]]
## [1] "text"
##
## [[2]]
## [1] "more"
##
## [[3]]
## [1] 2
##
## [[4]]
## [1] 1 2 3 2
data.frame
For most industrial and data scientists, data frames are the most widely useful type of variable. A data.frame is the list analog to the matrix: it is an $m \times n$ list where all columns must be vectors of the same number of rows (determined with NROW()). However, unlike matrices, different columns can contain different types of data and each row and column must have a name. If not named explicitly, R names rows by their row number and columns according to the data assigned assigned to the column. Data frames are typically used to store the sort of data that industrial engineers and scientists most often work with, and is the closest analog in R to an Excel spreadsheet. Usually data frames are made up of one or more columns of factors and one or more columns of numeric data.

data.frame(rnorm(5), rnorm(5), rnorm(5))
##     rnorm.5.  rnorm.5..1  rnorm.5..2
## 1  0.2939566  1.28985202 -0.01669957
## 2  0.3672161 -0.01663912 -1.02064116
## 3  1.0871615  1.13855476  0.78573775
## 4 -0.8501263 -0.17928722  1.03848796
## 5 -1.6409403 -0.34025455 -0.62113545

More generally, in R all variables are objects, and R distinguishes between objects by their internal storage type and by their class declaration, which are accessible via the typeof() and class() functions. Functions in R are also objects, and the users can define new objects to control the output from functions like summary() and print(). For more on objects, types and classes, see section 2 of the R Language Definition.

Table 2 summarizes the internal storage and R classes of the main data and variable types.

Table 2: Table of R data and variable types.
Variable type Storage type Class Measurement Scale
vector of decimals double numeric ratio
vector of integers integer integer ratio or interval
vector of complex complex complex ratio
vector of characters character character nominal
factor vector integer factor nominal or ordinal
matrix of decimals double matrix ratio
data frame list data.frame mixed
list list list mixed

References

1. Stevens, S. S. “On the Theory of Scales of Measurement.” Science. 103.2684 (1946): 677-680. Print.
2. R Core Team (2017). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

Introduction to R for Excel Users

As the saying goes, when all you have is a hammer, everything looks like a nail. Excel was designed to do simple financial analyses and to craft financial statements. Though its capabilities have been expanded over the years, it was never designed to perform the sort of data analysis that industry scientists, engineers and Six Sigma belts need to perform on a daily basis.

Most data analyses performed in Excel look more like simple financial spreadsheets rather than actual data analysis, and this quality of work translates into bad—or at least sub-optimal—business decisions. There are alternatives to Excel, and the free, open-source data analysis platform R is one of them.

Unfortunately, R has a steep learning curve. I’m offering, for free, a short primer on R [PDF] where I’ve sought to make that learning curve a little less painful for engineers and scientists who normally work in Excel.

Background

A couple of years ago, I was developing a short course to teach R to scientists and engineers in industry who normally used Excel. The goal was to help them transition to a more capable tool. My course design notes morphed into a handout, and when plans for the course fell through, that handout grew into a self-study guide, which I later adapted into this seventy-page, stand-alone introduction for Excel users.

Organization

The primer walks the reader through the basics of R, starting with a brief overview of capabilities, then diving into installation, basic operations, graphical analysis and basic statistics. I believe that a picture is worth a thousand words, so it’s light on text and heavy on examples and visuals.

The end of the book rounds out with a look at some of the most useful add-ons, the briefest of introductions to writing your own, custom functions in R, and a cross-reference of common Excel functions with their equivalents in R.

The text is broken up into chapters and fully indexed so that it can be used either as a walk-through tutorial or as a quick reference.

Sample Size Matters: Design and Cost

References

• R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.
• H. Wickham. ggplot2: elegant graphics for data analysis. Springer New York, 2009.

Sample Size Matters: Design and Experiments

Previously, I introduced the idea that samples do not look exactly like the populations that they are drawn from, and had a closer look at what impact sample size has on our ability to estimate population statistics like mean, proportion or Cpk from samples. Here, I will have a closer look at how this uncertainty impacts our engineering process. In the next post, I will tie in the engineering impacts and decisions to the business value and costs.

Difference to detect

When we are testing, we’re either testing to determine that the new product or process performs better than the old or, for cost reduction projects, that the cheaper product or process is at least as good as the existing one.

This means that we need to detect a difference between old and new values, such as a difference in the mean weight between the new and old parts. The larger the sample size, $n$ the smaller the difference, $\Delta$ that we can detect. The error, $\epsilon$, in our estimate of the differences gets smaller as sample size increases:

$\epsilon_{\Delta} \propto \frac{\sigma}{\sqrt{n}}$

Given the uncertainties in our estimate of $\mu$ and $\sigma$, illustrated above, it should be clear now that with small sample sizes we can only detect large differences of many multiples of the sample standard deviation, $S$.

Mean

When trying to determine if a new product or process is better than an old one, we are usually interested in shifting the mean. We want a product to be lighter, provide more power, or a process to work faster. In such cases, we need to estimate the difference of the means, $\Delta = \mu_2 - \mu_1$ and ensure that it is different than 0 (or some other pre-determined value). The minimum difference that we can reliable detect is plotted below for different sample sizes.

Standard deviation

In many Six Sigma projects, and any time we want to shift the mean closer to a specification limit, we need to compare the new population standard deviation with the old. The simplest way of making this comparison is by taking the ratio $F = \sigma_{2}^{2} / \sigma_{1}^{2}$, where $\sigma_{2}^{2}$ is the larger of the two variances. The dependence on sample size is illustrated below.

You can see from the inset plot, which includes sample sizes of 2 and 3, that small sample sizes really hurt comparisons of variance, and that interesting differences in variance can’t be detected until we have more than 10 samples.

Proportions

Proportions, such as fraction of defective parts between a new and old design, can be compared by looking at the difference between the two proportions, $\Delta = \left| p_1 - p_0 \right|$.

You can see from this that proportions data provides much less information than variable data; we need much larger sample sizes to achieve usefully small $\Delta$.

Summary and look forward

When designing experiments, the goal is to detect some difference between two populations. The uncertainty in our measurements and the variation in the parts has a big impact on how many parts we need to test, or greatly limits what we can learn from an experiment.

Next time, I’ll show how these calculations of sample size and uncertainty impact the busines.

Sample Size Matters: Uncertainty in Measurement

In my previous post, I gave a brief introduction to populations and samples, and stated that sample size impacts our ability to know what a population really looks like. In this post, I want to show this relationship in more detail. In future posts, I will look at how sample size considerations impact our engineering process and what impacts this has on the business.

Mean and sample size

The error in our estimate of the mean, $E$, is proportional to the standard deviation of the sample, $S$, and the sample size, $n$.

$E \propto \frac{S}{\sqrt{n}}$

We can visualize this easily enough by plotting the 95% confidence interval. When we sample and calculate the sample mean ($\overline{X}$), the true population mean, $\mu$, (what we really want to know) is likely to be anywhere in the shaded region of the graph below.

This graph shows the 95% confidence region for the true population mean, $\mu$; there’s a 95% chance that the true population mean is within this band. The “0” line on the y axis is our estimate of the mean, $\overline{X}$. We can’t know what the true population mean is, but it’s clear that if we use more samples, we can be sure that our estimate is closer to the true mean.

Standard deviation and sample size

Likewise, when we calculate the sample standard deviation, $S$, the true standard deviation, $\sigma$ has a 95% chance of being within the confidence band below. For small sample sizes (roughly less than 10), the measured standard deviation can be off from the true standard deviation by several times. Even for ten samples, the potential error is nearly $\pm 1$ standard deviation.

Proportion and sample size

For proportions, the situation is similar: there is a 95% chance that the true sample proportion, $p$, is within the shaded band based on the measured sample proportion $\hat{p}$. Since this confidence interval depends on $\hat{p}$ and cannot be standardized the way $\mu$ and $\sigma$ can be, confidence intervals for two different proportions are plotted.

For small $n$, proportions data tells us very little.

Process capability and production costs

The cost of poor quality in product or process design can be characterized by the Cpk:

$Cpk = \mathrm{minimum} \begin{cases}\frac{USL - \mu}{3\sigma} \\\frac{\mu - LSL}{3\sigma}\end{cases}$

Where USL is the upper specification limit (also called the upper tolerance) and LSL is the lower specification limit (or lower tolerance).

We can estimate the defect rate (defects per opportunity, or DPO) from the Cpk:

$DPO = 1 - \Pr\left(X < 3 \times Cpk - 1.5\right)$

That probability function is calculated in R with pnorm(3 * Cpk - 1.5) and in Excel with NORMSDIST(3 * Cpk - 1.5). The 1.5 is a typical value used to account for uncorrected or undetected process drift.

Since we don’t know $\mu$ and $\sigma$, we have to substitute $\overline{X}$ and $S$. The uncertainty in these estimates of the population $\mu$ and $\sigma$ mean that we have uncertainty in what the true process Cpk (or defect rates) will be once we’re in production. When our sample testing tells us that the Cpk should be 1.67 (the blue line), the true process Cpk will actually turn out to be somewhere in the shaded band:

Below the blue line, our product or process is failing to meet customer expectations, and will result in lost customers or higher warranty costs. Above the blue line, we’ve added more cost to the production of the product than we need to, reducing our gross profit margin. Since that gray band doesn’t completely disappear, even at 100 samples, we can never eliminate these risks; we have to find a way to manage them effectively.

The impact of this may be more evident when we convert from Cpk to defect rates (ppm):

Summary and a look forward

With a fair sampling process, samples will look similar to—and statistically indistinguishable from—the population that they were drawn from. How much they look like the population depends critically on how many samples are tested. The uncertainties, or errors in our estimates, resulting from sample size decisions have impacts all through our design analysis and production planning.

In the next post, I will explore in more detail how these uncertainties impact our experiment designs.

Sample Size Matters

I find that Six Sigma and Design for Six Sigma courses are often eye-opening experiences for participants. There is an experience of discovering that there are tools available to answer problems that have vexed them, and learning that good engineering and science decisions can lead directly to good business outcomes through logical steps.

One of the most remarkable such moments is when students realize the importance of sample size. In the best cases, there is a forehead-slapping moment where the student realizes that much of the testing they’ve done in the past has probably been a complete waste of time; that while they thought they were seeing interesting differences and making good decisions, they were in fact only fooling themselves by comparing too-small data sets.

I want to show in the next few blog posts why sample size matters, both from a technical perspective and from a business perspective.

Design example

Throughout the next few posts, I’ll use the example of a manufactured product which the customer requires weigh at least 100 kg, sells for about $140 and that costs$120 to manufacture and convert to a sale (the cost of goods sold, or COGS, is $120). Amount Sales 140 COGS 120 Material 60 Labor and Overhead 60 Gross Profit 20 We want to develop a new version of the product, using a modified design and a new process that, by design, will reduce the cost of material by 10%. The old cost of material was 50% of COGS, or$60. To achieve the material cost reduction of 10%, we have to remove $6 in material costs, improving gross profit to$26.

We believe that the current design masses 120 kg, so we estimate that our new part mass should be $120 - 0.1 \times 120 = 108$ kg.

Current Design New Design Target
Part Weight 120 108

Seems like we might be done at this point, and I’ve seen plenty of engineering projects that stop here. Unfortunately, this isn’t the whole story. Manufacturing will be unable to produce parts of exactly 108 kg, so they’ll need a tolerance range to check parts against. We have that customer requirement for at least 100 kg, so any variation has to stay above that. We also want to save money relative to the current design, so we don’t want many parts to weigh much more than this, especially since the customer isn’t really willing to pay us for the “extra” material beyond 100 kg.

Population versus sample statistics

Most of process or product improvement is concerned with reducing the standard deviation, $\sigma$, shifting the mean (a.k.a. average), $\mu$, or reducing a proportion, $p$, of a process or product characteristic. These summary statistics refer to the population characteristics—the mean, standard deviation or proportion of all parts of a certain design that will ever be produced, or all times that a production step will ever be completed in the intended manner.

Since we can’t measure the whole population up front—we will be producing parts for a long time—we have to draw a sample from the population, and use the statistics of that sample to gain insight into the total population. We can visualize this, somewhat crudely, with the following:

We can imagine that the blue circles are conforming parts, and the orange octagons are non-conforming parts. If the sampling process is fair, then the sample proportion $\hat{p}$ will be close to—and statistically indistinguishable from—the true population proportion $p$. In the population we have 44 parts total, 8 defective parts and 36 conforming parts. In the sample that we drew, we have 10 parts total, 9 conforming and 1 defective. While $(p = 8/36 = 1/4 \ne \hat{p} = 1/9$, statistically we have

matrix(c(1, 8, 10-1, 44-8), ncol=2) %>%
chisq.test(simulate.p.value = TRUE)

##
##  Pearson's Chi-squared test with simulated p-value (based on 2000
##  replicates)
##
## data:  matrix(c(1, 8, 10 - 1, 44 - 8), ncol = 2)
## X-squared = 0.3927, df = NA, p-value = 0.6692


With such a high p-value (0.67), we fail to reject the null hypothesis that $\hat{p} = p$; in more colloquial terms, we conclude that the apparent difference between 8/36 and 1/9 is only due to random errors in sampling. (For larger counts of successes and failures, prop.test() would also work and would be more informative.)

From our perspective, of course, we don’t know what the population looks like. We don’t have any way of knowing with certainty—or accessing data about—future performance, so there is no way for us to know what the total population looks like. In lieu of population data, we develop a sampling process that allows us to fairly draw a sample from that population.

While we want to know the true population mean, $\mu$, the true population standard deviation, $\sigma$, or the true population proportion $p$, we can only calculate the sample mean, $\overline{X}$, the sample standard deviation, $S$, or the sample proportion $\hat{p}$.

From the known sample, we then reason backward to what the true population looks like. This is where statistics comes into play; statistics allows us to place rigorous boundaries on what the population may look like, without fooling ourselves. Sample size is critical to controlling the uncertainty in these boundaries.

Summary and a look forward

Testing in product development—and usually in production—involves sampling a product or process. Samples never look exactly like the population that we are concerned about, but if the sampling process is fair then the samples will be statistically indistinguishable from the population. With due awareness of the statistical uncertainties, we can use samples to make decisions about the population.

In the next post, I will look at how sample size impacts the uncertainty in our estimation of population statistics like the mean and standard deviation. In a later post, I will look at how this uncertainty impacts the business.

A short aside on statistical tests for proportions

The usual way to compare two proportions would be a proportions test (prop.test() in R), but because we have so few samples to compare, the results may be unreliable and prop.test() generates an appropriate warning. fisher.test() provides an exact estimate of the p-value, but the assumptions are violated with data like this, where we are sampling a fixed number of parts (i.e. row sums are fixed, but column sums are not controlled). This leaves us with using a chi-squared test (chisq.test() in R) which is less informative but does the job. Either the Barnard test or Bayesian estimation based on Monte Carlo simulation would be more informative and possibly more robust.

The Most Useful Data Plot You’ve Never Used

Those of us working in industry with Excel are familiar with scatter plots, line graphs, bar charts, pie charts and maybe a couple of other graph types. Some of us have occasionally used the Analysis Pack to create histograms that don’t update when our data changes (though there is a way to make dynamic histograms in Excel; perhaps I’ll cover this in another blog post).

One of the most important steps in data analysis is to just look at the data. What does the data look like? When we have time-dependent data, we can lay it out as a time-series or, better still, as a control chart (a.k.a. “natural process behavior chart”). Sometimes we just want to see how the data looks as a group. Maybe we want to look at the product weight or the cycle time across production shifts.

Unless you have Minitab, R or another good data analysis tool at your disposal, you have probably never used—maybe never heard of—boxplots. That’s unfortunate, because boxplots should be one of the “go-to” tools in your data analysis tool belt. It’s a real oversight that Excel doesn’t provide a good way to create them.

For the purpose of demonstration, let’s start with creating some randomly generated data:

head(df)

##   variable   value
## 1   group1 -1.5609
## 2   group1 -0.3708
## 3   group1  1.4242
## 4   group1  1.3375
## 5   group1  0.3007
## 6   group1  1.9717

tail(df)

##     variable   value
## 395   group1  1.4591
## 396   group1 -1.5895
## 397   group1 -0.4692
## 398   group1  0.1450
## 399   group1 -0.3332
## 400   group1 -2.3644


If we don’t have much data, we can just plot the points:

library(ggplot2)

ggplot(data = df[1:10,]) +
geom_point(aes(x = variable, y = value)) +
coord_flip() +
theme_bw()


But if we have lots of data, it becomes hard to see the distribution due to overplotting:

ggplot(data = df) +
geom_point(aes(x = variable, y = value)) +
coord_flip() +
theme_bw()


We can try to fix this by changing some parameters, like adding semi-transparency (alpha blending) and using an open plot symbol, but for the most part this just makes the data points harder to see; the distribution is largely lost:

ggplot(data = df) +
geom_point(aes(x = variable, y = value), alpha = 0.3, shape = 1) +
coord_flip() +
theme_bw()


The natural solution is to use histograms, another “go-to” data analysis tool that Excel doesn’t provide in a convenient way:

ggplot(data = df) +
geom_histogram(aes(x = value), binwidth = 1) +
theme_bw()


But histograms don’t scale well when you want to compare multiple groups; the histograms get too short (or too narrow) to really provide useful information. Here I’ve broken the data into eight groups:

head(df)

##   variable   value
## 1   group1 -1.5609
## 2   group1 -0.3708
## 3   group1  1.4242
## 4   group1  1.3375
## 5   group1  0.3007
## 6   group1  1.9717

tail(df)

##     variable   value
## 395   group8 -0.6384
## 396   group8 -3.0245
## 397   group8  1.5866
## 398   group8  1.9747
## 399   group8  0.2377
## 400   group8 -0.3468

ggplot(data = df) +
geom_histogram(aes(x = value), binwidth = 1) +
facet_grid(variable ~ .) +
theme_bw()


Either the histograms need to be taller, making the stack too tall to fit on a page, or we need a better solution.

The solution is the box plot:

ggplot() +
geom_boxplot(data = df, aes(y = value, x = variable)) +
coord_flip() +
theme_bw()


The boxplot provides a nice, compact representation of the distribution of a set of data, and makes it easy to compare across a large number of groups.

There’s a lot of information packed into that graph, so let’s unpack it:

Median
A measure of the central tendency of the data that is a little more robust than the mean (or arithmetic average). Half (50%) of the data falls below this mark. The other half falls above it.
First quartile (25th percentile) hinge
Twenty-five percent (25%) of the data falls below this mark.
Third quartile (75th percentile) hinge
Seventy-five percent (75%) of the data falls below this mark.
Inter-Quartile Range (IQR)
The middle half (50%) of the data falls within this band, drawn between the 25th percentile and 75th percentile hinges.
Lower whisker
The lower whisker connects the first quartile hinge to the lowest data point within 1.5 * IQR of the hinge.
Upper whisker
The upper whisker connects the third quartile hinge to the highest data point within 1.5 * IQR of the hinge.
Outliers
Any data points below 1.5 * IQR of the first quartile hinge, or above 1.5 * IQR of the third quartile hinge, are marked individually as outliers.

We can add additional values to these plots. For instance, it’s sometimes useful to add the mean (average) when the distributions are heavily skewed:

ggplot(data = df, aes(y = value, x = variable)) +
geom_boxplot() +
stat_summary(fun.y = mean, geom="point", shape = 10, size = 3, colour = "blue") +
coord_flip() +
theme_bw()


Graphs created in the R programming language using the ggplot2 and gridExtra packages.

References

1. R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical
Computing, Vienna, Austria. URL http://www.R-project.org/.
2. H. Wickham. ggplot2: elegant graphics for data analysis. Springer New York, 2009.
3. Baptiste Auguie (2012). gridExtra: functions in Grid graphics. R package version 0.9.1.
http://CRAN.R-project.org/package=gridExtra