This will probably not rank up there with the weightiest columns ever posted in this space. After all, even the Babe couldn’t hit a home run every time. But just as in baseball, it’s good to go back to the basics when things have gotten muddled. So chalk this article up to an attempt along those lines.

Both the public and the media seem confused about monthly unemployment reports. Some of this might be because the data are confusing. True unemployment is underreported in the country. During bad times, the administration in power tends to favor reporting that makes things look better than they actually are. You can’t blame them, but it confuses people. Also, it’s unsporting to bend the scoring-rules to make your team look good.

Complete unemployment data are not available to government officials, so they report what they have. The two primary data elements reported can be called “A” and “B”: i.e.

A = Number of people drawing or newly applying for unemployment benefits.

B = Number of people currently working.

From these we obtain…

C = A + B (total current workforce)

and R = A/C (current unemployment rate)

Simple, eh? It recalls what one of my old teachers liked to say: “For every problem there’s a solution that is simple, elegant and wrong.” (Grandpa’s “lies, damned lies, and statistics” also comes to mind.)

Quantities A and B come readily to hand for government officials. Unfortunately, A is not a correct count of the total unemployed population, which means that C doesn’t really include all people who are working, could work and want to work. Economists say that at the national level, A understates the unemployed population significantly – perhaps by as much as 50%. When long-time unemployed workers use up their unemployment benefits, they are no longer included in A. Ditto for people who couldn’t apply for those benefits in the first place, because they were self-employed, etc. To show a correct unemployment rate, we need the true number of unemployed people: i.e.,

U = A + X

where X = Number of unemployed people outside the unemployment benefits system

and A = Number of people drawing or applying for unemployment benefits (as above).

Then we could calculate…

R_{T} = U/[B + A + X] (i.e., true unemployment rate)

Eyes glazed over now? Relax. The eyes of the government green-eyeshade corps are a little glazed-over too. It’s not their fault, as the aptly-named quantity “X” isn’t really available. It can only be estimated. That’s why government statisticians have stuck with the simpler (but incomplete) formulation, R = A/C.

All this would be OK if everyone agreed that the published statistics had limited accuracy and shouldn’t be micro-examined for trends. When politics gets involved, however, caveats about statistical limitations go out the window. This is happening now with statistic “R.” Because the economic news has been mostly bad for the past year, the current administration is desperate to see some encouraging trend in unemployment, which reached a high of 10.2% in November 2009. It dropped to 10% in December, and fell to 9.7% in January. This has been hailed as an indicator that the economy is finally “turning around.”

In both recent monthly reports, however, the number of people working (i.e., B, above) has actually declined – in January, by 20,000. This should have meant a higher unemployment rate. But the number of people on (or seeking) unemployment benefits *also declined*. This pushed down the unemployment rate, R, causing White House poobahs to shout hosannas over the “shrinking” jobless rate. It’s an arithmetic anomaly, however. (Does 20,000 fewer people at work really sound like better times?)

The result is confusion for the populace and reporters of the mainstream media. To some extent, what we might call “mathematical illiteracy” is also to blame. It’s a mathematical fact that reducing a fraction’s numerator and denominator by the same amount produces a fraction of lower value. That’s why we got lower unemployment rates for those two months.

For instance, take 50/100 (i.e., 0.500). Now subtract 10 from both numerator and denominator. The new fraction is 40/90 = 0.444… which is obviously less than 0.500.

This works for any fraction. But bigger numbers get some people confused. The rule is still true, but it’s not as obvious. Say 10,000,000 people are on (or applying for) unemployment benefits, and 90,000,000 are working. Then A=10,000,000, B=90,000,000; and the declared unemployment rate is:

R = (10,000,000)/(100,000,000) = 10%

So far, so good. But if next month 200,000 of those unemployed workers quit looking for work, or their benefits run out, the government says the number of unemployed workers is now 9,800,000, and the work-force totals 99,800,000. Thus, the unemployment rate is:

R = (9,800,000)/(99,800,000) = 9.82%

This looks encouraging, but othing has been gained. Not a single new person is working. Some bureaucrats might posture about the “recovery taking hold,” or some such nonsense, but it’s just an illusion. Government bookkeeping conventions have contrived to produce deceptive numbers.

(By the way, in case some readers suspect that the fraction-reduction principle isn’t always true – perhaps thinking that I contrived some special numbers here – please consult the algebraic proof in the footnote. [1] This isn’t rocket science. If you can understand the proof, thank your old algebra teacher. If you can’t, you’ve helped demonstrate another serious national problem.)

There is no substitute for good data. Incomplete data only furnish a distorted picture. I have seen estimates of the true unemployment rate, R_{T}, that range from 15% to as high as 22%. I have no idea if either figure is accurate because “X” is missing. The likelihood is strong, however, that the current value of R (9.7%) vastly underestimates reality. I know many unemployed people, personally. In the past I didn’t know anyone unemployed, so it’s empirically obvious that things are not good.

Are things getting better? I don’t know. But I do know that the country would be much better served with accurate data. All those high-priced federal employees ought to be smart enough to find X. If they did, then White House Press Secretary Robert Gibbs could stop playing Let’s Pretend with stories about “jobs saved,” etc. Americans need to know the real score. They’re getting impatient.

*******

[1] **Theorem:** Reducing a fraction’s numerator and denominator by the same amount always produces a fraction of lesser value.

**Conditions:** b > a > x > 0.

**Proof:**

- We postulate: a/b > (a-x)/(b-x)
- Then: b(b-x)a/b > b(b-x)(a-x)/(b-x)
- Cancelling terms gives: a(b-x) > b(a-x)
- Expanding gives: ab-ax > ab-bx
- Simplifying: -ax > -bx; or bx-ax > 0
- Or x(b-a) > 0. This inequality must be true because x > 0 and b > a.

QED.