## More on equal-area histograms

##### May 31, 2023

Today, I'm returning to those "equal-area histograms" that Andrew wrote about last month. I have two previous posts about this. The first post introduces the concept: in a traditional histogram, the columns have the same bin width while the column heights can represent a variety of metrics, such as counts, relative frequencies (i.e. proportion of the data) and densities; in the equal-area histogram, the columns have varying widths while the area of each column is constant, and determined by the number of bins (columns).

Here is a comparison of the two types of histograms.

In a second post, I explained the differences between using counts, frequencies and densities in the vertical axis. The underlying issue is that the histogram is not merely a column chart, in which the width of the columns is arbitrary and data-free - in the histogram, both the heights and widths of columns carry meaning. One feature of the histogram that almost everyone expects is that the area of the columns sum up to 1. This aligns with a desired interpretation of probabilities of data falling into specified ranges, as we'd like the amount of data in the entire range to add up to 100%. Unfortunately, the two items are usually incompatible with each other.

If the height of the columns represents the probability of data falling into the range as indicated by its width, then the sum of the column heights is 1, which implies that the sum of the column areas cannot be 1. On the other hand, if the column areas add up to 1, then the column heights will not add up to 1, and thus, in this scenario, we cannot interpret the column heights to be probabilities. As explained in the second post, the column heights in this situation are densities, which can be defined as the proportion of data divided by the bin width. Intuitively, it gives information on how dense or sparse the data are within the specified range.

***

Today's post start with a toy dataset, containing randomly generated values from a normal distribution (bell curve) centered at 4 and with standard deviation 1.

Here is the traditional histogram of the dataset, using 100 equal-width bin. (I generated 10,000 values)

Next, I created a panel of four equal-area histograms, with increasingly number of bins. Each is built from the same underlying dataset.

The first histogram divides the data into 4 bins; then 10 bins, 20 bins and 100 bins.

In the 4-bin case, each column contains 1/4 = 25% of the data. The middle two columns contain 50% of the data, and they have high densities, as the widths of these columns are low. It's a crude approximation of the familiar bell curve.

As we increase the number of bins, the columns in the middle of the distribution, where most of the data are concentrated, become narrower. In the sparse regions, the column width doesn't necessarily grow because each column must contain 1/n of the data, where n is the number of columns. As the number of columns increases, each column contains less of the data.

The bottom chart is the "percentogram", which is what Andrew's correspondent proposed. The number of bins is set to 100, so each column contains exactly 1 percent of the data. For a normal distribution, the columns in the middle are very tall and thin.

The reason why the middle of the percentogram looks faded is that I asked for a white border around each column. But when the columns are so thin, even if one sets the border width very small, what readers see is a mixture of orange and white.

With high number of bins, we notice a few things: a) the outline of the histogram becomes "ragged" (the more bins there are), b) the middle columns become razor-thin c) the width conceded by the middle columns is absorbed not by the columns at the edges but those between the peak and the edge.

I'm struggling a bit to justify this percentogram versus the typical, equal-width histogram.

Let me go down a different path.

***

In "principled" histograms, the column heights represent data densities, while the total area of the columns add up to 1. This leads us to a new understanding of the relationship between the equal-width histogram and the equal-area histogram.

We start with data density defined by (proportion of data) / (bin width). Those two values are not independent - one is fully determined by the other, given the underlying dataset. In a traditional equal-width histogram, the question is: how much of the data is found in a column of fixed width? In the new equal-area histogram, the question is: how wide is the bin that contains a fixed amount of data? In the former, the denominator is fixed while the numerator varies; the opposite occurs in the latter.

***

We also recognize that given the range of the data, there is a relationship between the the set of bin widths in the two types of histograms. In the traditional histogram, all bin widths have the same value, equal to the range of the data divided by the number of bins. Think of this as the average bin width. In an equal-area histogram, the set of bin widths varies: however, the sum of the bin widths must still add up to the range of the data. For two comparable histograms with the same number of bins, the average of the bin widths must be the same for both sets. (I'm ignoring any rounding situations in which the range of the histogram is larger than the range of the data.)

Now, consider the middle of the normal distribution where the data are dense. In the traditional histogram, the column in the middle still has width equal to the average bin width. In the equal-area histogram, the middle column has width much smaller than the average bin width. In other words, we can think of the column in the traditional histogram being broken up into many thin and slim columns in the equal-area histogram, each containing 1% of the data in the case of the percentogram.

The height of the column is the data density. In the traditional histogram, the middle column is the pooled sample of larger size; in the equal-area histogram, each of those thin and slim columns is a partition of the sample. This explains observation (a) above in which the outline of the equal-area histogram is more ragged - it's because each column contains fewer data from which to estimate the data density.

But this raggedness is artificial, sampling noise.

***

The sparse areas are more complicated still. It's also the reverse of the above. On the edges of the normal distribution, the columns of the new histogram are wider than those of the traditional histogram. So, we can think of breaking up the edge column of the new histogram into multiple columns of the traditional histogram.

The interpretation is more complicated because the data are sparse in this region. Obviously, the estimates of density on the traditional histogram in sparse regions are poor because not enough data reside in there. The density estimate on the new histogram is based on a larger sample size.

However.

Yes, however, whether the new histogram's density estimate is better depends on the shape of the tail of the distribution. A normal distribution has exponential tails, which means that the data density declines quite drastically the further we go into the tail. Therefore, the new histogram averages the data densities across a large part of the tail, wiping out the exponential shape while the traditional histogram preserves that shape - at the expense of greater sampling variability due to smaller sample sizes.

***

For what it's worth, let's look at some histograms for an exponential random variable.

The data are extremely dense on the left side while it has a long tail on the right side.

Here are the four equal-area histograms for 4, 10, 20 and 100 bins.

The four-bin version gives a nice summary of the shape. As the number of bins goes up, as before, the denser regions now have tall, thin spikes. Again, because of the white borders, the last histogram with 100 bins is faded where the data are densest. (So obviously, don't follow my lead, and eliminate borders if you want to use it.)

The 100-bin version looks almost the same as the traditional histogram.

***

At this stage of the exploration, I still haven't found a compelling reason to switch to equal-area hist0grams. In the denser regions, it's adding sampling noise. If I don't care about the sparser areas, specifically, the shape of the tails, maybe they provide a cleaner presentation.

## Visual story-telling: do you know or do you think?

##### May 22, 2023

One of the most important data questions of all time is: do you know? or do you think?

And one of the easiest traps to fall into is: I think, therefore I know.

***

Visual story-telling can be great but it can also mislead. Deception sometimes happens when readers are nudged to "fill in the blanks" with stuff they think they know, but they don't.

A Twitter reader asked me to look at the map in this Los Angeles Times (paywall) opinion column.

The column promptly announces its premise:

Years of widening economic inequality, compounded by the pandemic and political storm and stress, have given Americans the impression that the country is on the wrong track. Now there’s empirical data to show just how far the country has run off the rails: Life expectancies have been falling.

The writer creates the expectation that he will reveal evidence in the form of data to show that life expectancies have been driven down by economic inequality, pandemic, and politics. Does he succeed?

***

The map portrays average life expectancy (at birth) for some mysterious, presumably very recent, year for every county in the United States. From the color legend, we learn that the bottom-to-top range is about 20 years. There is a clear spatial pattern, with the worst results in the south (excepting south Florida).

The choice of colors is telling. Red and blue on a U.S. map has heavy baggage, as they signify the two main political parties in the country. Given that the author believes politics to be a key driver of health outcomes, the usage of red and blue here is deliberate. Throughout the article, the columnist connects the lower life expectancies in southern states to its politics.

For example, he said "these geographical disparities aren't artifacts of pure geography or demographics; they're the consequences of policy decisions at the state level... Of the 20 states with the worst life expectancies, eight are among the 12 that have not implemented Medicaid expansion under the Affordable Care Act..."

Casual readers may fall into a trap here. There is nothing on the map itself that draws the connection between politics and life expectancies; the idea is evoked purely through the red-blue color scheme. So, as readers, we are filling in the blanks with our own politics.

What could have been done instead? Let's look at the life expectancy map side by side with the map of the U.S. 2020 Presidential election.

Because of how close recent elections have been, we may think the political map has a nice balance of red and blue but it isn't. The Democrats' votes are heavily concentrated in densely-populated cities so most of the Presidential election map is red. When placed next to each other, it's obvious that politics don't explain the variance in life expectancy well. The Midwest is deep red and yet they have above average life expectancies. I have circled out various regions that contradict the claim that Republican politics drove life expectancies down.

It's not sufficient to point to the South, in which Republican votes and life expectancy are indeed inversely correlated. A good theory has to explain most of the country.

***

The columnist also suggests that poverty is the cause of low life expectancy. That too cannot be gleaned from the published map. Again, readers are nudged to use their wild imagination to fill in the blank.

Data come to the rescue. Here is a side-by-side comparison of the map of life expectancies and the map of median incomes.

A similar conundrum. While the story feels right in the South, it fails to explain the northwest, Florida, and various other parts of the country. Take a look again at the circled areas. Lower income brackets are also sometimes associated with high life expectancies.

***

The author supplies a third cause of lower life expectancies: Covid-19 response. Because Covid-19 was the "most obvious and convenient" explanation for the loss of life expectancy during the pandemic, this theory suggests that the red areas on the life expectancy map should correspond to the regions most ravaged by Covid-19.

Let's see the data.

The map on the right shows the number of confirmed cases until June 2021. As before, the correlation holds somewhat in the South but there are notable exceptions, e.g. the Midwest. We also have states with low Covid-19 cases but below-average life expectancy.

***

What caused the decline of life expectancy in the U.S. - which began before the pandemic, and has continued beyond - is highly complex, beyond what a single map or a pair of maps or a few pairs of maps could convey. Showing a red-blue map presents a trap for readers to fall into, in which they start thinking, without knowing.

## Graph workflow and defaults wreak havoc

##### May 12, 2023

For the past week or 10 days, every time I visited one news site, it insisted on showing me an article about precipitation in North Platte. It's baiting me to write a post about this lamentable bar chart (link):

***

This chart got problems, and the problems start with the tooling, which dictates a workflow.

I imagine what the chart designer had to deal with.

For a bar chart, the tool requires one data series to be numeric, and the other to be categorical. A four-digit year is a number, which can be treated either as numeric or categorical. In most cases, and by default, numbers are considered numeric. To make this chart, the user asked the tool to treat years as categorical.

Many tools treat categories as distinct entities ("nominal"), mapping each category to a distinct color. So they have 11 colors for 11 years, which is surely excessive.

This happens because the year data is not truly categorical. These eleven years were picked based on the amount of rainfall. There isn't a single year with two values, it's not even possible. The years are just irregularly spaced indices. Nevertheless, the tool misbehaves if the year data are regarded as numeric. (It automatically selects a time-series line chart, because someone's data visualization flowchart says so.) Mis-specification in order to trick the tool has consequences.

The designer's intention is to compare the current year 2023 to the driest years in history. This is obvious from the subtitle in which 2023 is isolated and its purple color is foregrounded.

How unfortunate then that among the 11 colors, this tool grabbed 4 variations of purple! I like to think that the designer wanted to keep 2023 purple, and turn the other bars gray -- but the tool thwarted this effort.

The tool does other offensive things. By default, it makes a legend for categorical data. I like the placement of the legend right beneath the title, a recognition that on most charts, the reader must look at the legend first to comprehend what's on the chart.

Not so in this case. The legend is entirely redundant. Removing the legend does not affect our cognition one bit. That's because the colors encode nothing.

Worse, the legend sows confusion because it presents the same set of years in chronological order while the bars below are sorted by amount of precipitation: thus, the order of colors in the legend differs from that in the bar chart.

I can imagine the frustration of the designer who finds out that the tool offers no option to delete the legend. (I don't know this particular tool but I have encountered tools that are rigid in this manner.)

***

Something else went wrong. What's the variable being plotted on the numeric (horizontal) axis?

The answer is inches of rainfall but the answer is actually not found anywhere on the chart. How is it possible that a graphing tool does not indicate the variables being plotted?

I imagine the workflow like this: the tool by default puts an axis label which uses the name of the column that holds the data. That column may have a name that is not reader-friendly, e.g. PRECIP. The designer edits the name to "Rainfall in inches". Being a fan of the Economist graphics style, they move the axis label to the chart title area.

The designer now works the chart title. The title is made to spell out the story, which is that North Platte is experiencing a historically dry year. Instead of mentioning rainfall, the new title emphasizes the lack thereof.

The individual steps of this workflow make a lot of sense. It's great that the title is informative, and tells the story. It's great that the axis label was fixed to describe rainfall in words not database-speak. But the end result is a confusing mess.

The reader must now infer that the values being plotted are inches of rainfall.

Further, the tool also imposes a default sorting of the bars. The bars run from longest to shortest, in this case, the longest bar has the most rainfall. After reading the title, our expectation is to find data on the Top 11 driest years, from the driest of the driest to the least dry of the driest. But what we encounter is the opposite order.

Most graphics software behaves like this as they are plotting the ranks of the categories with the driest being rank 1, counting up. Because the vertical axis moves upwards from zero, the top-ranked item ends up at the bottom of the chart.

***

Moving now from the V corner to the D corner of the Trifecta checkup (link), I can't end this post without pointing out that the comparisons shown on the chart don't work. It's the first few months of 2023 versus the full years of the others.

The fix is to plot the same number of months for all years. This can be done in two ways: find the partial year data for the historical years, or project the 2023 data for the full year.

(If the rainy season is already over, then the chart will look exactly the same at the end of 2023 as it is now. Then, I'd just add a note to explain this.)

***

Here is a version of the chart after doing away with unhelpful default settings:

## Deconstructing graphics as an analysis tool in dataviz

##### Apr 20, 2023

One of the useful exercises I like to do with charts is to "deconstruct" them. (This amounts to a deeper version of the self-sufficiency test.)

Here is a chart stripped down to just the main visual elements.

The game is to guess what is the structure of the data given these visual elements.

I guessed the following:

• The data has a top-level split into two groups
• Within each group, the data is further split into 3 parts, corresponding to the 3 columns
• With each part, there are a variable number of subparts, each of which is given a unique color
• The color legend suggests that each group's data are split into 7 subparts, so I'm guessing that the 7 subparts are aggregated into 3 parts
• The core chart form is a stacked column chart with absolute values so relative proportions within each column (part) is important
• Comparing across columns is not supported because each column has its own total value
• Comparing same-color blocks across the two groups is meaningful. It's easier to compare their absolute values but harder to compare the relative values (proportions of total)

If I knew that the two groups are time periods, I'd also guess that the group on the left is the earlier time period, and the one on the right is the later time period. In addition to the usual left-to-right convention for time series, the columns are getting taller going left to right. Many things (not all, obviously) grow over time.

The color choice is a bit confusing because if the subparts are what I think they are, then it makes more sense to use one color and different shades within each column.

***

The above guesses are a mixed bag. What one learns from the exercise is what cues readers are receiving from the visual structure.

Here is the same chart with key contextual information added back:

Now I see that the chart concerns revenues of a business over two years.

My guess on the direction of time was wrong. The more recent year is placed on the left, counter to convention. This entity therefore suffered a loss of revenues from 2017-8 to 2018-9.

The entity receives substantial government funding. In 2017-8, it has 1 dollar of government funds for every 2 dollars of revenues. In 2018-9, it's roughly 2 dollars of government funds per every 3 dollars of revenues. Thus, the ratio of government funding to revenues has increased.

On closer inspection, the 7 colors do not represent 7 components of this entity's funding. The categories listed in the color legend overlap.

It's rather confusing but I missed one very important feature of the chart in my first assessment: the three columns within each year group are nested. The second column breaks down revenues into 3 parts while the third column subdivides advertising revenues into two parts.

What we've found is that this design does not offer any visual cues to help readers understand how the three columns within a year-group relates to each other. Adding guiding lines or changing the color scheme helps.

***

Next, I add back the data labels:

The system of labeling can be described as: label everything that is not further broken down into parts on the chart.

Because of the nested structure, this means two of the column segments, which are the sums of subparts, are not labeled. This creates a very strange appearance: usually, the largest parts are split into subparts, so such a labeling system means the largest parts/subparts are not labeled while the smaller, less influential, subparts are labeled!

You may notice another oddity. The pink segment is well above \$1 billion but it is roughly the size of the third column, which represents \$250 million. Thus, these columns are not drawn to scale. What happened? Keep reading.

***

Here is the whole chart:

A twitter follower sent me this chart. Elon Musk has been feuding with the Canadian broadcaster CBC.

Notice the scale of the vertical axis. It has a discontinuity between \$700 million and \$1.7 billion. In other words, the two pink sections are artificially shortened. The erased section contains \$1 billion (!) Notice that the erased section is larger than the visible section.

The focus of Musk's feud with CBC is on what proportion of the company's funds come from the government. On this chart, the only way to figure that out is to copy out the data and divide. It's roughly 1.2/1.7 = 70% approx.

***

The exercise of deconstructing graphics helps us understand what parts are doing what, and it also reveals what cues certain parts send to readers.

In better dataviz, every part of the chart is doing something useful, it's free of redundant parts that take up processing time for no reason, and the cues to readers move them towards the intended message, not away from it.

***

I'm not sure why old data was cited because in the most recent accounting report, the proportion of government funding was around 65%.

Source of funding is not a useful measure of pro- or anti-government bias, especially in a democracy where different parties lead the government at different times. There are plenty of mouthpiece media that do not apparently receive government funding.

## Bivariate choropleths

##### Apr 03, 2023

A reader submitted a link to Joshua Stephen's post about bivariate choropleths, which is the technical term for the map that FiveThirtyEight printed on abortion bans, discussed here. Joshua advocates greater usage of maps with two-dimensional color scales.

As a reminder, the fundamental building block is expressed in this bivariate color legend:

Counties are classified into one of these nine groups, based on low/middle/high ratings on two dimensions, distance and congestion.

The nine groups are given nine colors, built from superimposing shades of green and pink. All nine colors are printed on the same map.

Without a doubt, using these nine related colors are better than nine arbitrary colors. But is this a good data visualization?

Specifically, is the above map better than the pair of maps below?

The split map is produced by Josh to explain that the bivariate choropleth is just the superposition of two univariate choropleths. I much prefer the split map to the superimposed one.

***

Superimposing the two univariate maps solves one problem: it removes the need to scan back and forth between two maps, looking for the same locations, something that is imprecise. (Unless, the map is interactive, and highlighting one county highlights the same county in the other map.)

For me, that's a small price to pay for quicker translation of color into information.

## Yet another off radar plot

##### Feb 23, 2023

Bloomberg compares people's lives in retirement in this interesting dataviz project (link, paywall). The "showcase" chart is a radar plot that looks like this:

The radar plot may count as the single chart type that has the most number of lives. I'm afraid this one does not go into the hall of fame, either.

The setup leading to this plot is excellent, though. The analytical framework is to divide the retirement period into two parts: healthy and not so healthy. The countries in the radar plot are in fact ordered by the duration of the "healthy retirement period", with France leading the pack. The reference levels used throughout the article is the OECD average. On average, the OECD resident retires at age 64, and dies at age 82, so they spend 18 years in retirement, and 13 of them while "healthy".

In the radar plot, the three key dates are plotted as yellow, green and purple dots. The yellow represents the retirement age, the green, the end of the healthy period, and the purple, the end of life.

Now, take 10, 20, 30 seconds, and try to come up with a message for the above chart.

Not easy at all.

***

Notice the control panel up top. The male and female data are plotted separately. I place the two segments next to each other:

It's again hard to find any insight - other than the most obvious, which is that female life expectancy is higher.

But note that the order for the countries is different for each chart, and so even the above statement takes a bit of time to verify.

***

There are many structural challenges to using radar charts. I'll cover one of these here - the amount of non data-ink baggage that comes with using this chart form.

In the Bloomberg example, the baggage includes radial gridlines for countries, concentric gridlines for the years dimension, the country labels around the circle, the age labels in the middle, the color legend, the set of arrows that map to the healthy retirement period, and the country ranks (and little arrow) that indicate the direction of reading. That's a lot of information to process.

In the next post, I'll try a different visual form.

## Area chart is not the solution

##### Feb 14, 2023

A reader left a link to a Wiki chart, which is ghastly:

This chart concerns the trend of relative proportions of House representatives in the U.S. Congress by state, and can be found at this Wikipedia entry. The U.S. House is composed of Representatives, and the number of representatives is roughly proportional to each state's population. This scheme actually gives small states disporportional representation, since the lowest number of representatives is 1 while the total number of representatives is fixed at 435.

We can do a quick calculation: 1/435 = 0.23% so any state that has less than 0.23% of the population is over-represented in the House. Alaska, Vermont and Wyoming are all close to that level. The primary way in which small states get larger representation is via the Senate, which sits two senators per state no matter the size. (If you've wondered about Nate Silver's website: 435 Representatives + 100 Senators + 3 for DC = 538 electoral votes for U.S. Presidental elections.)

***

So many things have gone wrong with this chart. There are 50 colors for 50 states. The legend arranges the states by the appropriate metric (good) but in ascending order (bad). This is a stacked area chart, which makes it very hard to figure out the values other than the few at the bottom of the chart.

A nice way to plot this data is a tile map with line charts. I found a nice example that my friend Xan put together in 2018:

A tile map is a conceptual representation of the U.S. map in which each state is represented by equal-sized squares. The coordinates of the states are distorted in order to line up the tiles. A tile map is a small-multiples setup in which each square contains a chart of the same design to faciliate inter-state comparisons.

In the above map, Xan also takes advantage of the foregrounding concept. Each chart actually contains all 50 lines for every state, all shown in gray while the line for the specific state is bolded and shown in red.

***

A chart with 50 lines looks very different from one with 50 areas stacked on each other. California, the most populous state, has 12% of the total population so the line chart has 50 lines that will look like spaghetti. Thus, the fore/backgrounding is important to make sure it's readable.

I suspect that the designer chose a stacked area chart because the line chart looked like spaghetti. But that's the wrong solution. While the lines no longer overlap each other, it is a real challenge to figure out the state-level trends - one has to focus on the heights of the areas, rather than the boundary lines.

[P.S. 2/27/2023] As we like to say, a picture is worth a thousand words. Twitter reader with the handle LHZGJG made the tile map I described above. It looks like this:

You can pick out the states with the key changes really fast. California, Texas, Florida on the upswing, and New York, Pennsylvania going down. I like the fact that the state names are spelled out. Little tweaks are possible but this is a great starting point. Thanks LHZGJG! ]

## If you blink, you'd miss this axis trick

##### Jan 31, 2023

When I set out to write this post, I was intending to make a quick point about the following chart found in the current issue of Harvard Magazine (link):

This chart concerns the "tectonic shift" of undergraduates to STEM majors at the expense of humanities in the last 10 years.

I like the chart. The dot plot is great for showing this data. They placed the long text horizontally. The use of color is crucial, allowing us to visually separate the STEM majors from the humanities majors.

My intended post is to suggest dividing the chart into four horizontal slices, each showing one of the general fields. It's a small change that makes the chart even more readable. (It has the added benefit of not needing a legend box.)

***

Then, the axis announced itself.

I was baffled, then disgusted.

Here is a magnified view of the axis:

It's not a linear scale, as one would have expected. What kind of transformation did they use? It's baffling.

Notice the following features of this transformed scale:

• It can't be a log scale because many of the growth values are negative.
• The interval for 0%-25% is longer than for 25%-50%. The interval for 0%-50% is also longer than for 50%-100%. On the positive side, the larger values are pulled in and the smaller values are pushed out.
• The interval for -20%-0% is the same length as that for 0%-25%. So, the transformation is not symmetric around 0

I have no idea what transformation was applied. I took the growth values, measured the locations of the dots, and asked Excel to fit a polynomial function, and it gave me a quadratic fit, R square > 99%.

This formula fits the values within the range extremely well. I hope this isn't the actual transformation. That would be disgusting. Regardless, they ought to have advised readers of their unusual scale.

***

Without having the fitted formula, there is no way to retrieve the actual growth values except for those that happen to fall on the vertical gridlines. Using the inverse of the quadratic formula, I deduced what the actual values were. The hardest one is for Computer Science, since the dot sits to the right of the last gridline. I checked that value against IPEDS data.

The growth values are not extreme, falling between -50% and 125%. There is nothing to be gained by transforming the scale.

The following chart undoes the transformation, and groups the majors by field as indicated above:

***

Yesterday, I published a version of this post at Andrew's blog. Several readers there figured out that the scale is the log of the relative ratio of the number of degrees granted. In the above notation, it is log10(100%+x), where x is the percent change in number of degrees between 2011 and 2021.

Here is a side-by-side view of the two scales:

The chart on the right spreads the negative growth values further apart while slightly compressing the large positive values. I still don't think there is much benefit to transforming this set of data.

P.S. [1/31/2023]

(1) A reader on Andrew's blog asked what's wrong with using the log relative ratio scale. What's wrong is exactly what this post is about. For any non-linear scale, the reader can't make out the values between gridlines. In the original chart, there are four points that exist between 0% and 25%. What values are those? That chart is even harder because now that we know what the transform is, we'd need to first think in terms of relative ratios, so 1.25 instead of 25%, then think in terms of log, if we want to know what those values are.

(2) The log scale used for change values is often said to have the advantage that equal distances on either side represent counterbalancing values. For example, (1.5) (0.66) = (3/2) (2/3)  = 1. But this is a very specific scenario that doesn't actually apply to our dataset.  Consider these scenarios:

History: # degrees went from 1000 to 666 i.e. Relative ratio = 2/3
Psychology: # degrees went from 2000 to 3000 i.e. Relative ratio = 3/2

The # of History degrees dropped by 334 while the number of Psychology degrees grew by 1000 (Psychology I think is the more popular major)

History: # degrees went from 1000 to 666 i.e. Relative ratio = 2/3
Psychology: from 1000 to 1500, i.e. Relative ratio = 3/2

The # of History degrees dropped by 334 while # of Psychology degrees grew by 500
(Assume same starting values)

History: # degrees went from 1000 to 666 i.e. Relative ratio = 2/3
Psychology: from 666 to 666*3/2 = 999 i.e. Relative ratio = 3/2

The # of History degrees dropped by 334 while # of Psychology degrees grew by 333
(Assume Psychology's starting value to be History's ending value)

Psychology: # degrees went from 1000 to 1500 i.e. Relative ratio = 3/2
History: # degrees went from 1500 to 1000 i.e. Relative ratio = 2/3

The # of Psychology degrees grew by 500 while the # of History degrees dropped by 500
(Assume History's starting value to be Psychology's ending value)

## Visual cues affect how data are perceived

##### Jan 24, 2023

Here's a recent NYT graphic showing California's water situation at different time scales (link to article).

It's a small multiples display, showing the spatial distribution of the precipitation amounts in California. The two panels show, respectively, the short-term view (past month) and the longer-term view (3 years). Precipitation is measured in relative terms,  so what is plotted is the relative ratio of precipitation in the reference period, with 100 being the 30-year average.

Green is much wetter than average while brown is much drier than average.

The key to making this chart work is a common color scheme across the two panels.

Also, the placement of major cities provides anchor points for our eyes to move back and forth between the two panels.

***

The NYT graphic is technically well executed. I'm a bit unhappy with the headline: "Recent rains haven't erased California's long-term drought".

At the surface, the conclusion seems sensible. Look, there is a lot of green, even deep green, on the left panel, which means the state got lots more rain than usual in the past month. Now, on the right panel, we find patches of brown, and very little green.

But pay attention to the scale. The light brown color, which covers the largest area, has value 70 to 90, thus, these regions have gotten 10-30% less precipitation than average in the past three years relative to the 30-year average.

Here's the question: what does it mean by "erasing California's long-term drought"? Does the 3-year average have to equal or exceed the 30-year average? Why should that be the case?

If we took all 3-year windows within those 30 years, we're definitely not going to find that each such 3-year average falls at or above the 30-year average. To illustrate this, I pulled annual rainfall data for San Francisco. Here is a histogram of 3-year averages for the 30-year period 1991-2020.

For example, the first value is the average rainfall for years 1989, 1990 and 1991, the next value is the average of 1990, 1991, and 1992, and so on. Each value is a relative value relative to the overall average in the 30-year window. There are two more values beyond 2020 that is not shown in the histogram. These are 57%, and 61%, so against the 30-year average, those two 3-year averages were drier than usual.

The above shows the underlying variability of the 3-year averages inside the reference time window. We have to first define "normal", and that might be a value between 70% and 130%.

In the same way, we can establish the "normal" range for the entire state of California. If it's also 70% to 130%, then the last 3 years as shown in the map above should be considered normal.

## Accounting app advertises that it doesn't understand fractions

##### Jan 17, 2023

I captured the following image of an ad at the airport at the wrong moment, so you can only see the dataviz but not the text that came with it. The dataviz is animated with blue section circling around and then coming to a halt.

The text read something like "75% of the people who saw this ad subsequently purchased something". I think the advertiser was TripActions. It is an app for accounting. Too bad their numbers people don't know 75% is three-quarters and their donut chart showed a number larger than 75%.

***

Browsing around the TripActions website, I also found this pie chart.

The radius of successive sectors is decreasing as the size of the proportions shrinks. As a result, the same two sectors labeled 12% at the bottom have differently-sized areas. The only way this dataviz can work is if the reader decodes the angles sustained at the center, and ignores the areas of the sectors. However, the visual cues all point readers to the areas rather than the angles.

In this sense, the weakness of this pie chart is the same as that of the racetrack chart, discussed recently here.

In addition, the color dimension is not used wisely. Color can be used to group the expenses into categories, or it can be used to group them by proportion of total (20%+, 10-19%, 5-9%, 1-4%, <1%).