<THE FOOLISH FOUR>

Fun with Statistics
by Ann Coleman
(TMF AnnC)

Reston, VA (September 4, 1998) -- You don't have to be a mathematical genius to understand and use statistics. Granted, it helps, but there are at least some simple statistical tests that are not that hard to grasp, and they can be very useful.

Disclaimer: I am not a statistician or even particularly well educated in statistics as a field. (Those of you who are... please be gentle.) The purpose of the following exercise is to demonstrate, in one very simple way, the validity of our advice about starting/renewing a Foolish Four portfolio in December/January.

We based our recommendation about the January start date on the average returns for portfolios starting in each month of the year. Quite simply, the average returns are highest for January, which suggests that there is something about that month. There are a number of ways to tell if that "January effect" is real, but the simplest and easiest to explain was suggested by a reader who asked, "How many times was January the best month?"

As I reported Wednesday, a January 2 start/renewal date provided the best returns in 5 of the last 25 years and was in the top three in 11 of those 25 years. That's not exactly impressive. Maybe that January thing was just a fluke?

Just to set the stage, let's think of dice -- the favorite tool of Statistics 201 teachers. Everyone knows that if you toss a die, it has an equally likely chance of coming up 1, 2, 3, 4, 5, or 6. If you toss it many, many times, you will roll each number an approximately equal number of times. Toss it 600 times and it will land on each number somewhere in the neighborhood of 100 times. Unless the die is loaded. When the die isn't loaded (when the results are totally random), each number has a 1 in 6 chance of turning up, or a probability of 16.7%.

Now, of course, nature doesn't follow those rules exactly. If you roll 60 times, you might get 9 ones, 9 twos, 15 threes, 12 fours, 11 fives and only 4 sixes. That's not exactly what one would "expect." So at what point do you start to suspect that the die is loaded?

That's where statistics can help.

A chi-squared test (Goodness of Fit test) compares expected results with actual results and tells you just how likely it is that the actual results occurred by chance. The results above, where we rolled the die 60 times, generated a chi-square value of 6.8. Even though the results differed from our expected results quite a bit, the chi-square significance tables for 6.8 at 5 degrees of freedom (degrees of freedom is the number of categories minus 1) tell us that we can expect that much of a deviation 10% of the time. To put it another way, the actual results will differ from the expected results to that degree at least one time out of 10. That's not enough to accuse me of using loaded dice. The chi-squared test is a hard one to pass.

In the case of our database, let's look at a 24 year period instead of 25. (The math is easier and the difference in statistical significance is minimal.) How many times would you expect to find any one month with the highest average returns? In any year, only one month will be the highest, one the next highest, etc. (Ties are unlikely and did not occur.) So the chance of being the highest month for any one year is 1 in 12. In a 12 year period you would expect to see January as the highest month once, and in a 24 year period, you would expect to see January come in first twice (and February twice and March twice, etc.) You would expect to see it place first, second, or third 3 times in a twelve year period and 6 times in a 24 year period.

But in reality, it placed first 5 times, instead of two, and in the top three 11 times, instead of 6. What is the probability of that happening "by chance?"

Well, we can't test for the probability of first place because our statistical sample isn't big enough. (You need an expected value of at least 5 in all categories, and our expected value is 2.) But the expected value for coming in first, second, or third is 6, so we can test that.

If the results were random, the chance of being in first, second, or third place would be 3 in 12, or 25%. The chance of being in 4th through 12th place would be 9 in 12, or 75%. This is a simpler test than the die test. Our chi-squared value is 5.5 with one degree of freedom (two categories). The probability of this distribution occurring by random chance is less than 5%.

Something is generally considered statistically significant if the probability of it occurring randomly is less than 5 per cent. "Statistically significant at the 95% (or .05) level" is how you will often hear it described. A looser standard will sometimes be evoked: "Statistically significant at the 90% (0.10) level," meaning the probability is less than 10% but greater than 5 percent that the phenomena will occur by chance. (Statisticians look at those results with some skepticism.) Our January effect passes this test. Something is probably loading the dice.

Fool on and prosper!

Current Dow Order | 1998 Dow Returns

What Happened to Robert Sheard?

09/04/98 Close

Stock Change Last -------------------- UK + 1/4 39.88 IP + 15/16 39.63 MO - 7/8 42.00 EK -1 81.00

Day Month Year FOOL-4 -0.20% 2.83% 4.18% DJIA -0.55% 1.34% -3.39% S&P 500 -0.85% 1.71% 0.36% NASDAQ -0.34% 4.49% -0.25% Rec'd # Security In At Now Change 12/31/97 206 Eastman Ko 60.56 81.00 33.75% 12/31/97 291 Union Carb 42.94 39.88 -7.13% 12/31/97 276 Philip Mor 45.25 42.00 -7.18% 12/31/97 289 Int'l Pape 43.13 39.63 -8.12% Rec'd # Security In At Value Change 12/31/97 206 Eastman Ko 12475.88 16686.00 $4210.13 12/31/97 291 Union Carb 12494.81 11603.63 -$891.19 12/31/97 276 Philip Mor 12489.00 11592.00 -$897.00 12/31/97 289 Int'l Pape 12463.13 11451.63 -$1011.50 CASH $754.73 TOTAL $52087.98