Don't compare percentages

tags: percentages 

I wrote this for in 2006, but I keep looking for it at the beginning of each year as numeric illiterates try to do the same flawed analysis to prove something about programming languages in the last year (this year it’s Andrew Binstock at DDJ). I’m adding it to in case (now read only) suddenly disappears.

sigzero recently asked if the Perl community was atrophying, someone else pointed me to Tim O’Reilly’s State of the Computer Book Market, and there’s a Wikipedia revert war over Tiobe’s Programming Programming Index. All of these suffer from a misunderstanding of percentages and their value in making decisions and reaching conclusions. This is especially egregious when people divorce the percentages from the absolute numbers (either because the original data are restricted or the people know that they tell a different story).

I start with six topics I’ll call { A, B, C, D, E, F }, and I’ll arbitrarily assign them a number. It doesn’t matter what this number actually is. It could be hits on a web search engine, book sales on the topic, number of people moving to a town, or anything else. I’m just making all of this up to show that when people talk of trends in growth using percentages, they get the wrong answer because percentages remove information. The real answer involves some simple calculus (although without the fancy symbology) and a realization about what I should actually measure.

Let’s get to it then. I make up two sets of numbers. I’ll say one was from last month, and one was from this month. Since the universe is growing, the total results for this month are not the same as the that from last month. This is very important, because this is where percentages fail and how people use percentages incorrectly. Notice that in every case except one, the number for each topic increased. In the other case, the number stays the same.

Table 1: Size of topic, in absolute numbers,
for two consecutive months

    A         17,000        34,000
    B         50,000        70,000
    C          5,000         6,000
    D          8,000        16,000
    E         10,000        12,000
    F         10,000        10,000

    TOTAL    100,000       148,000

I’ll change these numbers to percentages of the totals now. I round off to the nearest integer, but that won’t matter much. This is the first place where I’ll lose some information. With percentages, the difference in the totals between the two months disappear, but I’ll compare the percentages as if the totals were the same, and I’ll end up with the wrong answer.

Table 2: Size of topic, in percentage of total number,
for two consecutive months

    A    17    23
    B    50    47
    C     5     4
    D     8    11
    E    10     8
    F    10     7

First, notice in Table 1 that B had the greatest overall change (20,000), but actually dropped by 3% in Table 2, while A had a smaller change (17,000) in Table 1, but shot up 6% in Table 2. A has a relative percentages growth of 9%, although it’s growing more slowly. All topics except F had an increase in absolute numbers, but four actually dropped in percentages. Both C and E increased by a fifth of their absolute number, which is remarkable growth in a month, but their percentages dropped. That is, although they actually trend upward, the percentages trend in the opposite direction of their growth. B, which contributed the most absolute growth, dropped as much as F in percentages, although F had absolutely no growth. According to the relative numbers, B is losing in relative size as fast as F, although it added double the total number of F just in growth in one period.

I’ll do this for another month, adding the same absolute numbers as before to each topic. Each topic has the same absolute growth from month to month. Each is steadily increasing in absolute numbers except for F, which stays the same.

Table 3: Size of topic, in absolute numbers,
for three consecutive months

    A         17,000        34,000       51,000
    B         50,000        70,000       90,000
    C          5,000         6,000        7,000
    D          8,000        16,000       24,000
    E         10,000        12,000       14,000
    F         10,000        10,000       10,000

    TOTAL    100,000       148,000      196,000

What happens to the percentages in steady state growth? I see the same relative downward trends in most cases, despite the fact that all but F are growing in actual size.

Table 4: Size of topic, in percentage of total number,
for three consecutive months
    A    17    23    26
    B    50    47    46
    C     5     4     4
    D     8    11    12
    E    10     8     7
    F    10     7     5

Now, in Table 4, the percentages are even worse for B. Again, the total increased by 48,000 hits, and again, B provided 20,000 new hits. But, by the percentages, B falls 1%. B is responsible for most of the growth, but also looks as if it is atrophying, while A is just getting up to the level where B started. Although all but one topic increased their absolute numbers, four fell in relative percentages.

Let’s go a year later than the last column, adding 12 times the same number of absolute growth to each result. Every topic continues a steady state growth.

Table 5: Size of topic, in absolute numbers, for three
consecutive months, and one year later (covering
15 total months)

    A         17,000        34,000       51,000       255,000
    B         50,000        70,000       90,000       330,000
    C          5,000         6,000        7,000        19,000
    D          8,000        16,000       24,000       120,000
    E         10,000        12,000       14,000        38,000
    F         10,000        10,000       10,000        10,000

    TOTAL    100,000       148,000      196,000       772,000

What happens to the percentages a year later, shown as the last column in Table 6?

Table 6: Size of topic, in percentage of total number,
for three consecutive months, then one year later (covering
15 total months)

    A    17    23    26    33
    B    50    47    46    43
    C     5     4     4     2
    D     8    11    12    16
    E    10     8     7     5
    F    10     7     5     1

It’s the same decline. Although B has over six times the absolute numbers it did 15 months ago, it lost 7% in relative numbers. It’s still the largest and the fastest growing topic by absolute numbers, but the relative numbers make it look like the fastest declining topic.

The relative numbers are based on the total absolute number, and the problem comes in when I compare the percentages from month to month since the total absolute number changes. Percentages remove information to make things easier to see, but they aren’t there to be compared to percentages from another measurement. If the total number stays the same, the trend in percentages might have meaning, but if I don’t report the total size, the percentages don’t enough information for me to compare them across measurements.

If I used this analysis to figure out which books to publish, I’d miss the biggest and fastest growing market. If I used them to decide which city to do business in, I’d miss the one with the most customers and the one that’s adding the most customers. Analysts used to like to use this sort of thing to show why Apple Computer was a poor bet since it only had a minuscule percentage of the market, but that was clearly wrong. If I’d bought APPL 3 years ago at $11, I would have gained over 400% in value, compared to MSFT, selling at $26, losing 14% in the same time period. It’s because relative numbers aren’t the important ones.

So how do I do it then? I want to see how things are growing and what I should pay attention to. I could plot the absolute numbers, but a first-derivative plot makes it easier to see the real change. Like percentages, it removes the total number, but it leaves in the information about the direction and magnitude of the change. That’s the number I really want to compare. The first derivative is simply the change in the number over the time period. It’s a bit of calculus, although I won’t take it to an infinitesimal time slice.

Table 7: First derivative table, showing the change over three consecutive months, and one year later.

    A   17,000    17,000    17,000    204,000
    B   20,000    20,000    20,000    240,000
    C    1,000     1,000     1,000     12,000
    D    8,000     8,000     8,000     96,000
    E    2,000     2,000     2,000     24,000
    F        0         0         0          0

Compare the utility of Table 7 to Table 6 for decision making. In Table 6, topics A and D look like they are the fastest growing, when in reality topic A grew more than either of them and continues to have a faster rate of change over the next year. The percentages have the least information, doing away with the absolute change and the absolute total size. If I’d bet my money on A because it looked as if it was growing fastest by percentages, at the end of the year I would have missed the top by 55,000, over a fifth of the absolute total for B.

Now, consider the business reason for looking at these sorts of numbers, and, for simplicity, imagine that the products for each topic have the same price. I don’t make more money by the higher rate of growth. Topic A doubled in its first month, but still sold less than Topic B. Betting on the fastest growing relative percentage (Topic A) in that case would have meant ignoring the greater number of sales I could get from Topic B. I make more money by making more sales, not more growth, and in this case, the worst-performer, percentage-wise, is the highest volume seller in every time period.

The first derivative plot shows that Topic B is the fastest growing market. I don’t care about the growth in terms of the size of the market, but the number of units I can sell. In business parlance, the percentage is the “market penetration”, but that’s not the interesting number to anyone other than those trying to drive someone out of a market (as in the case of the silly business between Microsoft and Apple arguments). Market penetration is the false idol of stock speculators. The totals and first-derivatives are the numbers that translate into real money and real value.

Further research