You are currently browsing the daily archive for 8 March 2011.

We need some logger rhythms. (We pause briefly while you ROFL.)

Generations of students have rejected logarithms because they’re just too hard. It used to be true, too — actually calculating a logarithm is extremely difficult, and has always been relegated to specialists who did little else. Those specialists created “log tables” that could be used to look up logarithms, but using those was an intricate task because the tables were necessarily made as short as possible, in order to keep from using up all the paper and ink in the world to print them, and finding the logarithm of the number you were looking at in the table took a good bit of error-prone ingenuity.

That’s no longer the case. Calculators abound, and in all but the cheapest getting the accurate logarithm is a matter of pushing a button. Call up the Windows Calculator, choose [View] –> Scientific, and what comes up has two buttons for logarithms, [log] and [ln][1]. Put in an ordinary number, press [log], and there’s the logarithm. Put in a logarithm and press [Inverse] [log], and there’s the ordinary number.

There’s a tsunami of numbers washing over us — budgets, test scores, polls, prices, and a thousand other things. The people who calculate and publish them are anxious to make points, and use all kinds of tricks to emphasize the differences they think are important. Logarithms can help you cut through the fog, because

Logarithms tell you if the difference is important.

The logarithms most people encounter in normal life are the Richter[2] scale for earthquakes and the “decibel” scale for sound (and a lot of other things). Earthquakes are reported directly as logarithms, decibels are logarithms multiplied by ten to avoid using the decimal point. In both cases, the important fact is that a difference in the second digit doesn’t matter much, where a difference in the first digit matters a lot — the difference between a 5.2 earthquake and a 5.3 is barely detectable to the folk who encounter it, but the difference between a 5.2 and a 6.2 is really significant, and 7.2 breaks things and kills people; the difference between 82 decibels (abbreviated dB) and 83 dB is just barely noticeable, 92 dB goes from “loud” to “really annoying”, and 102 dB is “possible hearing loss”.

Logarithmic scales have a starting point or “origin”, but unlike plain numbers (“linear scales”) the origin point doesn’t matter for purposes of comparison between two numbers. The decibel scale was created with exactly that in mind. The researcher tested a lot of people, figured out the minimum loudness difference they needed to say one sound is louder than the other, and assigned that as 1 dB. Remember that decibels are logarithms multiplied by ten, and go to your calculator; put in o.1 and press the [Inverse] and [log] buttons. The result is a long number, 1.2589 and a lot more digits. Round it off to 1.26; in order for most people to hear the difference in loudness, the actual difference in sound power has to be about 26%. The base point, or zero dB, was set at the softest sound most people could hear at all, but that’s arbitrary.[3]

The ratio in power between the minimum audible sound and permanent hearing damage is about 1,000,000,000,000 — one trillion. That’s a big number, cumbersome to use; what’s more important is that it can be misleading. A difference of 26% is 260,000,000,000 or 260 billion, which looks like a really big difference. Convert to dB, and it’s 120 dB (one trillion) versus 121 dB (one trillion 260 billion). Now the difference looks trivial, and it is. You’re already bleeding at the ears at 120 dB, and adding one more decibel is barely noticeable, either to you or to your doctor.

Now let’s apply that principle to the national debt and deficit cutting. The deficit is roughly one and a half trillion dollars. [log] gives 12.17, or about 122 dD (decidebts). The proposed budget cuts amount to about 60 billion; [log] gives 10.78 or about 108 dD. $60 billion sounds like a lot of money, and it is for you and me, but 122 – 108 = 14 dD, which doesn’t look like much — and it isn’t. An engineer would say that the smaller number is “down” or minus 14 dD from the larger one. Put -1.4 into your calculator, [inverse][log]: 0.0398, or less than 4% of the basic problem. If you’re bleeding to death, stanching the wound by 4% won’t help you much.

Similarly, let’s look at Iowahawk’s educational difference numbers as expanded by Michael Pollard, taking only one line cuz I’m lazy:

2009 4th Grade Math

White students: North Carolina 254, Texas 254, Virginia 251, Wisconsin 250, Georgia 247, South Carolina 245, (national average 248)

North Carolina 254, [log] 2.405, 24 dE (“deci-educations”)
Texas 254, [log] = 2.405, 24 dE
Virginia 251, [log] = 2.399, 24 dE
Wisconsin 250, [log] = 2.398, 24 dE
Georgia 247, [log] = 2.392, 24 dE
South Carolina 245, [log] = 2.389, 24 dE
Nat’l Average 248, [log] = 2.394, 24 dE

How remarkable[4]. There is no difference at any level that makes a difference.

Try it with some of the numbers you encounter. It’s a great way to cut through the fog and see what (if any) real differences exist.

[1] They’re both “logarithms”, but done to different “bases”. They give different numbers when pushed, and the difference is hyper-important to mathematicians, but both make the same comparisons; pick one and stick to it, and you’ll be fine.

[2] Now replaced by the “Moment Magnitude” scale. The difference is important to people in the business, but when reading the paper or a blog it doesn’t matter.

[3] Not really, but close enough to be useful. The Wikipedia article is much better if you care to wade through it, but it will just confuse a lot of people.

[4] The nice way of saying “Ain’t that some shit?”

Tip Jar

Donations (via PayPal)

Hit it, folks.
:fx:Calvin eyes:Puuleeeez?
You don't know many people who need it more.

When I Posted

March 2011