"The average long-term experience in investing is never surprising, but the short-term experience is always surprising," - Charles D. Ellis, *Winning the Loser’s Game*: *Timeless Strategies for Successful Investing*

The confusion of most investors is derived from their inability to look at large sets of data about stocks, times, managers or styles. Here is the reason for the confusion. With a small set of data, such as the 50 rolls of three dice shown in **Figure 3-8**, the assumption is that the chances of getting a six on the next roll was the best of all combinations. This poor representation of the long-term characteristics of the three dice is known as random drift, (in the casino they call it luck). This is similar to saying that an investor feels confident about a certain stock, time period, manager or style based on a recent short-term experience. In statistics, this is also known as a sampling error.

However, if one looks at the long term or a thousand rolls of three dice in **Figure 3-9** or 5 dice in **Figure 3-9a**, a far better representation of the risk and return characteristics is demonstrated, which reduces the confusion caused by sampling error, random drift, or luck. In Figure 3-9, it is evident that rolling a six is just as likely as rolling a 15 and a lot less likely than rolling a 10 or 11. The population characteristics for any large data set are best described by the average and the standard deviation, which represents the variance around the average. Investors who think they see a pattern or trend in monthly or quarterly returns are experiencing random drift, just like 50 or 60 rolls of the dice. They are being fooled by randomness.

**Figure 3-8**

**Figure 3-8**

The distributions of stock returns in Figures 3-10 to 3-14 look strikingly similar to the roll of the dice in Figures 3-9 above.

**Figure 3-10**

**Figure 3-11**

**Figure 3-12**

**Figure 3-13**

**Figure 3-14**

In the language of statistics, the distributions seen above are the result of the Central Limit Theorem. The central limit theorem is one of the most remarkable results of the theory of probability. In its simplest form, the theorem states that the sum of a large number of independent observations from the same distribution has, under certain general conditions, an approximate normal distribution. The approximation steadily improves as the number of observations increases. The theorem is considered the heart of probability theory, although a better name would be normal convergence theorem.

Suppose an ordinary coin is tossed 100 times and the number of heads is counted. This is equivalent to scoring one for a head and zero for a tail and computing the total score. The total number of heads is the sum of 100 independent, identically distributed random variables. The central limit theorem states the distribution of the total number of heads will be, to a very high degree of approximation, normal. This is illustrated graphically by repeating this experiment many times. The results of this experiment are displayed in a diagram. The percentage computed over the number of experiments is arranged along the vertical axis, and the total score or the number of heads is arranged along the horizontal axis. After a large number of repetitions, a curve appears that looks like the normal curve.

It has been empirically observed that various **natural phenomena**, such as the heights of individuals, daily returns of the S&P 500, the managers who fall in the top fifty percent of all managers, and the students who correctly guess the outcome of a coin flip, follow approximately a normal distribution, as seen below.

A suggested explanation is that these phenomena are **sums of a large number of independent random effects, **like the daily news that moves the market, and hence are approximately normally distributed by the central limit theorem.

From the transcript of the PBS Nova Special, The Trillion Dollar Bet, Boston University Professor of Economics, Zvi Bodie (Bodie research) put it this way: "In flipping a coin, if you flip it long enough, there may be a long run of heads, which doesn't at all imply that the person flipping it had the ability to make it come up heads. It could just be the luck of the toss."

More from The Trillion Dollar Bet:

"This strange view arose from an unexpected discovery. After the stock market crash of 1929, economists decided to find out whether traders really could predict how prices moved by looking at past patterns. They decided to run a series of experiments. In one of them they simply picked stocks at random. They threw darts at the Wall Street Journal while blindfolded. At the end of the year, this random choice outperformed the predictions of top traders. This was a revelation: prices must be moving totally at random, and although patterns came and went, they were there by chance alone and had no predictive value. **'The economists arrived at a devastating conclusion: it seemed just as plausible to attribute the success of top traders to sheer luck rather than skill.**'"

The answer was first given over 100 years ago, on March 29, 1900, by Louis Bachelier, in his landmark study on the **Theory of Speculation.** This has since been documented by hundreds of other researchers. Investors are either too lazy, uninterested in learning, or else they rely on some "stock market expert." They operate like gamblers in Vegas, hoping that their skill, which is really just luck, will lead them to market beating returns.

Instead, studies like the one below, have shown that the average investor only captures a small percentage of the market's return:

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## About the Author

Mark Hebner - Founder, Index Fund Advisors, Inc.

Founder and President of Index Fund Advisors, Inc., and author of Index Funds: The 12-Step Recovery Program for Active Investors. He is a Wealth Advisor, with an MBA from the University of California at Irvine and a BS in Pharmacy from the University of New Mexico with a specialization in Nuclear Pharmacy.