Conjecture regarding Percent

I was recently at a conference where two talks back-to-back included bar graphs showing the percentage of population P that was component X. In both graphs, one bar was exceptionally large when P was particularly small. The reason for this was that X dropped, but all the other components making up P dropped much further, so X became a disproportionately large component of P.

Ultimately, this issue is due to the fact that X and all other components of P are not very closely related, such that the whole population P is not really indicative of much of anything, while the individual components are indicative of their own little areas. This is like saying that apples make up X percent of fruit. What harms apples probably doesn't effect all fruit, and vice versa, so apples could remain the same while total fruit plummets when the oranges all rot. Apples become disproportionately represented in fruit, but not because of anything they did.

However, as I saw this, it occurred to me that percentage was not a very good metric, since people were drawn to a few anomolously high points. So the solution that occurred to me was that one could draw a bar graph such that the height represented percent, while the width of each bar indicated the total population P. In this scenario, the area of the bar, rather than the height, becomes the key feature.

Since the percent (using the names from the first paragraph) is calculated as:
(X/P)*100
then the area of the bar becomes:
X(X/P)*100.
Or
100X^2/P.

However, this is a squared unit, so the metric to show would be the square root of this or:

10X/sqrt(P).

This seems to have some really interesting properties, but it also is so obvious that it must have been named and thoroughly diagnosed. My questions for this forum are: What is the name of this metric? and What are the issues with this metric?

The perception you got from those changing percentages shows a common misunderstanding of Percentage.

A percentage never gives any hint to the total amount, and it never shows a chnage in total amount.

But in any graphical display of a percentage, espacially when using bars, the common (false)perception is more like total amount than really percentage.
If you are now look at more sample (with changing total amount) you will get a false perception. One solution could be not to use bars for display.

Speeking about your solution, I'd make up an example (several samples) and show it differnet people to test their perception.

You'd be more likely to use a pie chart to show percentages.

This gives a better idea of what each portion actually represents of the 100%.

Just my $0.02, correct me if that's wrong :p

s.

A pie chart could also mislead, but the real killer about using pie charts is that sceintists never use them. This is probably because there are generally error bars on bar graphs, and pie charts can't show that well. However, in this case, a pie chart could have been a more effective display if the pie was sized to the total population.

i don't understand... percentages should not be used for a bar graph... of course that is misleading. you could have two identical bar graphs displaying how much salt there was proportionally to water in a solution of salt and water - and then boil the water and have a new graph. it would appear that there was actually more salt - which is not the case at all! when people see bar graphs, they are expecting it to show amounts, neevr percentages. that is, in my opinion, what bar graphs are for - pie graphs are perfectly acceptable for percentages, and for percent change it is probably preferred. i had no idea that scientists didn't use pie graphs... and what is this about the size of the pie graph? there is no rule about how big the graph would have to be... the bigger you made it, the easier it would be to create accurately, but 10 percent of a nickel and 10 percent of a dinner plate are the same percents... personally, i believe that it would be better to keep the graphs separate, or just stick with one or the other.

It is misleading to use bar graphs, but you can't put error bars on a pie chart (easily). On the other hand, that is not a relevant objection in this case, since there were no error bars in the presentations I saw. The purpose of the figure was to show the percent of a salmon run that were made up of stock X for each of several years. Either a bar or a pie chart would normally work fairly well, since the underlying thought (not really a hypothesis) was that the percentage of the run that was X was always about the same. This was true for all years except when the total run was very small. In those years, X dropped less than the other components, and thus became a greater percentage of the total. That made the bar for the low run year look really large when in fact the total component X was only 3 fish.

Using the bar graph didn't show quite what was intended, but neither would a pie chart. The reason was that percentage was of interest only when the sample size was sufficiently large. Sample size wouldn't be a good graphic either, since it showed something completely different (relative survival) than what was wanted (run composition). The idea behind the post was that combining the two would actually give a better representation of the data than either metric alone.