شرط بندی فوتبال : Don’t Compare Averages

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Imagine you’re an executive, and you’re asked to
decide which of your sales leaders to give a big award/promotion/bonus to.
Your company is a tooth-and-claw capitalist company that only considers
revenue to be important, so the key element in your decision is who has got
the most revenue growth this year. (Given it’s 2020, maybe we sell

Here’s the all-important numbers.

nameaverage revenue increase (%)

And a colorful graph

Based on this, the decision looks easy. Bob, at just under 8%, has a
notably better revenue increase than his rivals who languish at 5%.

But lets dig deeper, and look at the individual accounts for each of our

nameaccount revenue increases (%)

This account-level data tells a different story. Bob’s high performance
is due to one account yielding a huge 80% revenue increase. All his other
accounts shrank. With Bob’s performance based on just one account,
is he really the best salespeep for the bonus?

Bob’s tale is a classic example of one the biggest problems with
comparing any group of data points by looking at the average. The usual
average, technically the mean, is very prone to one outlier swinging the
whole value. Remember the average net worth of a hundred homeless people is
$1B once Bill Gates enters the room.

The detailed account data reveals another difference. Although Alice and
Clara both have the same average, their account data tells two very
different stories. Alice is either very successful (~10%) or mediocre (~2%), while Clara
is consistently mildly successful (~5%). Just looking at the average hides this
important difference.

By this point, anyone who’s studied statistics or data visualization is
rolling their eyes at me being Captain Obvious. But this knowledge isn’t
getting transmitted to folks in the corporate world. I see bar charts
comparing averages all the time in business presentations. So I decided to
write this article, to show a range of visualizations that you can use to
explore this kind of information, gaining insights that the average alone
cannot provide. In doing this I hope I can persuade some people to stop only
using averages, and to question averages when they see others doing that.
After all there’s no point eagerly collecting the data you need to be a
data-driven enterprise unless you know how to examine that data

A strip chart shows all the individual numbers

So the rule is don’t compare averages when you don’t know what the
actual distribution of the data looks like. How can you get a good picture
of the data?

I’ll start with the case above, when we don’t have very many
data points. Often the best way to go for this is a strip chart, which
will show every data point in the different populations.

show code
ggplot(sales, aes(name, d_revenue, color=name)) +
  geom_jitter(width=0.15, alpha = 0.4, size=5, show.legend=FALSE) +
  ylab(label = "revenue increase (%)") +
  geom_hline(yintercept = 0) +

With this chart we can now clearly see the lone high point for Bob,
that most of his results are similar to Alice’s worst results, and that Clara
is far more consistent. This tells us far more than the earlier bar chart,
but isn’t really any harder to interpret.

You may then ask, how to plot this nice strip chart? Most people who
want to plot some quick graphs use Excel, or some other spreadsheet. I
don’t know how easy it is to plot a strip chart in the average
spreadsheet, as I’m not much of a spreadsheet user. Based on what I see in
management presentations, it may be impossible, as I hardly ever see one.
For my plotting I use R, which a frighteningly powerful statistics
package, used by people who are familiar with phrases like “Kendall rank
correlation coefficient” and “Mann-Whitney U test”. Despite this fearsome
armory, however, it’s pretty easy to dabble with the R system for simple
data manipulation and graph plotting. It’s developed by academics as
open-source software, so you can download and use it without worrying about
license costs and procurement bureaucracy. Unusually for the open-source
world, it has excellent documentation and tutorials to learn how to use
it. (If you’re a Pythonista, there’s also a fine range of Python libraries
to do all these things, although I’ve not delved much into that
territory.) If you’re curious about R, I have a
in the appendix of how I’ve learned what I know about it.

If you’re interested in how I generate the various charts I show
here, I’ve included a “show-code” disclosure after each chart which shows
the commands to plot the chart. The sales dataframes
used have two columns: name, and d_revenue.

What if we have a larger number of data points to consider? Imagine our
trio are now rather more important, each handling a couple of hundred
accounts. Their distributions still, however show the same basic characteristics, and we
can see that from a new strip chart.

show code
ggplot(large_sales, aes(name, value, color=name)) +
  geom_jitter(width=0.15, alpha = 0.4, size=2, show.legend=FALSE) +
  ylab(label = "revenue increase (%)") +
  geom_hline(yintercept = 0) +

One problem with the strip chart, however, is that we can’t see the
average. So we can’t tell whether Bob’s high values are enough to
compensate for this general lower points. I can deal with this by plotting
the mean point on the graph, in this case as a black diamond.

show code
ggplot(large_sales, aes(name, value, color=name)) +
  geom_jitter(width=0.15, alpha = 0.4, size=2, show.legend=FALSE) +
  ylab(label = "revenue increase (%)") +
  geom_hline(yintercept = 0) +
  stat_summary(fun = "mean", size = 5, geom = "point", shape=18, color = 'black') +

So in this case Bob’s mean is a bit less than the other two.

This shows that, even though I often disparage those who use means to
compare groups, I don’t think means are useless. My disdain is for those
who only use means, or use them without examining the overall
distribution. Some kind of average is often a useful element of a
comparison, but more often than not, the median is actually the better
central point to use since it holds up better to big outliers like Bob’s.
Whenever you see an “average”, you should always consider which is better:
median or mean?

Often the reason median is such an under-used function is because our tooling
doesn’t encourage use to use it. SQL, the dominant database query
language, comes with a built-in AVG function
that computes the mean. If you want the median, however, you’re usually
doomed to googling some rather ugly algorithms, unless your database has
the ability to load extension functions. If
some day I become supreme leader, I will decree that no platform can have
a mean function unless they also supply a median.

Using histograms to see the shape of a distribution

While using a strip chart is a good way to get an immediate sense of
what the data looks like, other charts can help us compare them in
different ways. One thing I notice is that many people want to use The One
Chart to show a particular set of data. But every kind of chart
illuminates different features of a dataset, and it’s wise to use several
to get a sense of what the data may be telling us. Certainly this is true
when I’m exploring data, trying to get a sense of what it’s telling me.
But even when it comes to communicating data, I’ll use several charts so
my readers can see different aspects of what the data is saying.

The histogram is a classic way of looking at a distribution. Here are
histograms for the large dataset.

show code
ggplot(large_sales, aes(value, fill=name)) +
  geom_histogram(binwidth = 1, boundary=0, show.legend=FALSE) +
  xlab(label = "revenue increase (%)") +
  scale_y_continuous(breaks = c(50,100)) +
  geom_vline(xintercept = 0) +
  theme_grey(base_size=30) +
 facet_wrap(~ name,ncol=1)

Histograms work really well at showing the shape of a single
distribution. So it’s easy to see that Alice’s deals clump into two
distinct blocks, while Clara’s have a single block. Those shapes are
somewhat easy to see from the strip chart too, but the histogram clarifies
the shape.

A histogram shows only one group, but here I’ve shown several together
to do the comparison. R has a special feature for this, which it refers to
as faceted plots. These kind of “small multiples” (a term coined by
Edward Tufte) can be very handy for comparisons. Fortunately R makes them
easy to plot.

Another way to visualize the shapes of the distributions is a density
plot, which I think of as a smooth curve of a histogram.

show code
ggplot(large_sales, aes(value, color=name)) +
  geom_density(show.legend=FALSE) +
  geom_vline(xintercept = 0) +
  xlab(label = "revenue increase (%)") +
  scale_y_continuous(breaks = c(0.1)) +
  theme_grey(base_size=30) +
  facet_wrap(~ name,ncol=1)

The density scale on the y axis isn’t very meaningful to
me, so I tend to remove that scale from the plot – after all the key
element of these are shapes of the distributions. In addition, since the
density plot is easy to render as a line, I can plot all of them on a
single graph.

show code
ggplot(large_sales, aes(value, color=name)) +
  geom_density(size=2) +
  scale_y_continuous(breaks = NULL) +
  xlab(label = "revenue increase (%)") +
  geom_vline(xintercept = 0) +

Histograms and density plots are more effective when there are more
data points, they aren’t so helpful when there’s only a handful (as with
the first example). A bar chart of counts is useful when there are only a few
values, such as the 5-star ratings on review sites. A few years ago Amazon
added such a chart for its reviews, which show the distribution in
addition to the average score.

Boxplots work well with many comparisons

Histograms and density plots are a good way to compare different
shapes of distributions, but once I get beyond a handful of graphs then
they become difficult to compare. It’s also useful to get a sense of
commonly defined ranges and positions within the distribution. This is
where the boxplot comes in handy.

show code
ggplot(large_sales, aes(name, value, color=name)) +
  geom_boxplot(show.legend=FALSE) +
  ylab(label = "revenue increase (%)") +
  geom_hline(yintercept = 0) +

The box plot focuses our attention on the middle range of the data, so
that half the data points are within the box. Looking at the graph we can
see more than half of Bob’s accounts shrank and that his upper quartile is
below Clara’s lower quartile. We also see his cluster of hot accounts at
the upper end of the graph.

The box plot works nicely with a couple of dozen items to compare,
providing a good summary of what the underlying data looks like. Here’s an
example of this. I moved to London in 1983 and moved to Boston a decade
later. Being British, I naturally think about how the weather compares in
the two cities. So here is a chart showing comparing their daily high
temperatures each month since 1983.

show code
ggplot(temps, aes(month, high_temp, color=factor(city))) +
  ylab(label = "daily high temp (°C)") +
  theme_grey(base_size=20) +
  scale_x_discrete(labels=month.abb) +
  labs(color = NULL) +
  theme(legend.position = "bottom") +

This is an impressive chart, since it summarizes over 27,000 data
points. I can see how the median temperatures are warmer in London during
the winter, but cooler in the summer. But I can also see how the
variations in each month compare. I can see that over a quarter of the
time, Boston doesn’t get over freezing in January. Boston’s upper quartile
is barely over London’s lower quartile, clearly indicating how much colder
it is in my new home. But I can also see there are occasions when Boston
can be warmer in January than London ever is during that winter month.

The box plot does have a weakness, however, in that we can’t see the
exact shape of the data, just the commonly defined aggregate points. This
may be an issue when comparing Alice and Clara, since we don’t see the
double-peak in Alice’s distribution in the way that we do with histogram
and density chart.

There are a couple of ways around this. One is that I can easily
combine the box plot with the strip chart.

show code
ggplot(large_sales, aes(name, value, color=name)) +
  geom_boxplot(show.legend=FALSE, outlier.shape = NA) +
  geom_jitter(width=0.15, alpha = 0.4, size=1, show.legend=FALSE) +
  ylab(label = "revenue increase (%)") +
  stat_summary(fun = "mean", size = 5, geom = "point", shape=18, color = 'black') +
  geom_hline(yintercept = 0) +

This allows me to show both the underlying data, and the important
aggregate values. In this plot I also included the black diamond that I
used before to show the position of the mean. This is a good way to
highlight cases like Bob where the mean and median are quite different.

Another approach is the violin plot, which draws a density plot into
the sides of the boxes.

show code
ggplot(large_sales, aes(name, value, color=name, fill=name)) +
  geom_violin(show.legend=FALSE, alpha = 0.5) +
  ylab(label = "revenue increase (%)") +
  geom_hline(yintercept = 0) +

This has the advantage of showing the shape of the distribution
clearly, so the double peak of Alice’s performance stands right out. As
with density plots, they only become effective with a larger number of
points. For the sales example, I think I’d rather see the points in the
box, but the trade-off changes if we have 27,000 temperature measurements.

show code
ggplot(temps, aes(month, high_temp, fill=factor(city))) +
  ylab(label = "daily high temp (°C)") +
  theme_grey(base_size=20) +
  labs(fill = NULL) +
  scale_x_discrete(labels=month.abb) +
  theme(legend.position = "bottom") +
  geom_violin(color = NA) 

Here we can see that the violins do a great job of showing the shapes
of the data for each month. But overall I find the box chart of this data
more useful. It’s often easier to compare by using significant signposts
in the data, such as the medians and quartiles. This is another case where
multiple plots play a role, at least while exploring the data. The box
plot is usually the most useful, but it’s worth at least a glance at a
violin plot, just to see if reveals some quirky shape.

Summing Up

  • Don’t use just an average to compare groups unless you understand
    the underlying distribution.
  • If someone shows you data with just an average ask: “what does the
    distribution look like?”
  • If you’re exploring how groups compare, use several different plots
    to explore their shape and how best to compare them.
  • If asked for an “average”, check whether a mean or median is better.
  • When presenting differences between groups, consider at least the
    charts I’ve shown here, don’t be afraid to use more than one, and pick
    those that best illustrate the important features.
  • above all: plot the distribution!


Domeniconi, David Colls, David Johnston, James Gregory, John Kordyback, Julie Woods-Moss, Kevin Yeung, Mackenzie Kordyback, Marco Valtas, Ned Letcher, Pat Sarnacke, Saravanakumar Saminathan, Tiago Griffo, and Xiao Guo

commented on drafts of this article on internal mailing lists.

My experience learning R

I first came across R about 15 years ago, when I did a little work with
a colleague on a statistical problem. Although I did a lot of maths in
school, I shied away from statistics. While I was very interested in the
insights it offers, I was deterred by the amount of calculation it
required. I have this odd characteristic that I was good at maths but not
good at arithmetic.

I liked R, particularly since it supported charts that were hardly
available elsewhere (and I’ve never much liked using spreadsheets). But R
is a platform with neighborhoods dodgy enough to make JavaScript seem
safe. In recent years, however, working with R has become much easier due
to the work of Hadley Whickham – the Baron Haussmann of R. He’s led the
development of the “tidyverse”: a series of libraries that make R very
easy to work with. All the plots in this article use his ggplot2 library.

In recent years I’ve used R more and more for creating any reports that
make use of quantitative data, using R as much for the calculations as for
the plots. Here the tidyverse dplyr library plays a big role. Essentially
it allows me to form pipelines of operations on tabular data. At one level
it’s a collection pipeline on the
rows of the table, with functions to map and filter the rows. It then goes
further by supporting table-oriented operations such as joins and pivots.

If writing such excellent software isn’t enough, he’s also co-written
an excellent book to learn to use R: R for Data
. I’ve found this to be a great tutorial on data analytics,
an introduction to the tidyverse, and a frequent reference. If you’re
at all interested in manipulating and visualizing data, and like to get
hands-on with a serious tool for the job, then this book is a great way to
go. The R community has done a great job with this and other books that help
explain both the concepts and tools of data science. The tidyverse
community has also built an first-rate open-source editing and development
environment called R Studio. I shall say no
more that when working with R, I usually use it over Emacs.

R certainly isn’t perfect. As a programming language it’s shockingly
quirky, and I’ve dared not stray from the tree-lined boulevards of simple
dplyr/ggplot2 pipelines. If I wanted to do serious programming in a
data-rich environment, I’d seriously consider switching to Python. But for
the kinds of data work I do, R’s tidyverse has proven to be an excellent

Tricks for a good strip chart

There’s a couple of useful tricks that I often reach for when I use a
strip chart. Often
you have data points with similar, or even the same values. If I plot
them naively, I end up with a strip chart like this.

show code
ggplot(sales, aes(name, d_revenue, color=name)) +
  geom_point(size=5, show.legend=FALSE) +
  ylab(label = "revenue increase (%)") +
  geom_hline(yintercept = 0) +

This plot is still better than that first bar chart, as it clearly
indicates how Bob’s outlier is different to his usual performance. But
with Clara having so many similar values, they all clump on top of each
other, so you can’t see how many there are.

The first of my tricks I use is to add some jitter. This adds some
random horizontal movement to the points of the strip chart, which allows
them to spread out and be distinguished. My second is to make the points
partly transparent, so we can see when they plot on top of each other.
With these two tricks, we can properly appreciate the number and position
of the data points.

Exploring the bin width for histograms

A histogram works by putting the data into bins. So if I have a bin
width of 1%, then all accounts whose revenue increase is between 0 and 1%
are put into the same bin, and the graph plots how many are in
each bin. Consequently the width (or amount) of bins makes a big
difference to what we see. If I make larger bins for this dataset, I get
this plot.

show code
ggplot(large_sales, aes(value, fill=name)) +
  geom_histogram(binwidth = 5, boundary=0,show.legend=FALSE) +
  scale_y_continuous(breaks = c(50,100)) +
  xlab(label = "revenue increase (%)") +
  geom_vline(xintercept = 0) +
  theme_grey(base_size=30) +
  facet_wrap(~ name,ncol=1)

Here the bins are so wide that we can’t see the two peaks of Alice’s

The opposite problem is that if the bins are too narrow the plot
becomes noisy. So when I plot a histogram, I experiment with the bin
width, trying different values to see which ones help expose the
interesting features of the data.

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