Average connotes mediocrity. Other synonyms include: ordinary, common, so-so, run-of-the-mill, and regular. Few engaged, self-directed investors strive to be average. They all aspire to beat the averages and excel. Many consider themselves to be above average and will deny the possibility that they might actually not belong in that category. However, I strive to only be average. What I don't want to be is the median investor.

According to Vanguard, at the end of 2010, the average balance in 401(k) accounts at Vanguard was roughly $79,000[i]. The median balance was roughly $27,000. Why is the average balance so much higher than the median? A small number of very large balances combine with a large quantity of small balances to produce such a distribution. Likewise, I would rather have the average salary of all U.S. wage earners, rather than the median. And I would much rather have the average amount of wealth than the median amount.

A brief review of a few basic statistics concepts may be in order. An average is the total value of all the samples divided by the number of samples. This is also referred to as the mean. The median is the numeric value that separates the higher half of a sample from the lower half. Many samples are normally distributed, such as people's heights or SAT scores. A normal distribution, also referred to a Gaussian function or informally as a bell curve, is symmetrical in relation to its mean or median. However, some samples, as described in the second paragraph above, are not normally distributed; and the median and mean values can differ quite significantly from each other.

For an extreme example of this phenomenon, consider a gathering of 40 of your closest friends and acquaintances. The median and average amount of wealth in this gathering is probably in the thousands or, if you are affluent, the millions. Now let's say that Warren Buffett joined this group. The average net wealth of everyone in the group is now in the billions. The median net wealth did not change appreciably.

If a sample is not normally distributed, the median of the sample and the average are typically not equal. If the distribution is positively skewed, as is the case in the examples cited above, then the average value is higher than the median. If the distribution is negatively skewed, then the average is lower than the median. The difference between the average and the median increases with the amount of "skewness." For more details about skewness, see the Wikipedia entry.

Is the distribution of returns on individual stocks positively skewed, negatively skewed, or normal? Over very short periods of time, the distribution is normal or close to it. However as the holding period or time horizon lengthens, the distribution becomes more positively skewed. A stock can decline up to 100% but no more. Gains are unlimited (in theory, anyway). In 2012, 16 stocks traded on either NYSE or NASDAQ gained more than 200% for the year. The best performing stock, IMPAC Mortgage Holdings (NYSEMKT:IMH), increased over 600%. If the time horizon is longer than a year (which is prudent for most investors), the divergence between the best performing stocks and the worst performing stocks increases and the distribution becomes even more skewed.

I did a quick test of this hypothesis with the 30 stocks of the Dow Jones Industrial Average. For 2012, the median return for the 30 stocks was 11.22%. The average return was 14.13%. For the four-year period from 2009 through 2012, the median return was 61.44% and the average return was 69.84%. For the 10-year period from 2003 through 2012, the median return was 106.57% and the average return was 134.66%. This is far from a conclusive examination, and any rigorous test of this hypothesis should include a much larger sample of stocks and many more periods. But this quick test does show that the average value exceeds the median value, and this disparity grows with time.

I then ask myself if I pick my own stocks, are the stocks I choose more likely to provide median or average returns? After perusing a list of 2012's best and worst performers, I quickly come to the conclusion that I am more likely to choose a group of median stocks rather than a group of average stocks. The type of stocks that appear on the list of best and worst performers for the year looks very similar to me. They are small, obscure stocks that have the potential for both extremely high rewards as well as the risk of losing most of your investment. Examples include Sarepta Therapeutics (NASDAQ:SRPT), Infinity Pharmaraceuticals (NASDAQ:INFI), Headwaters (NYSE:HW), Houston American Energy (NYSEMKT:HUSA), Jaguar Mining (NYSE:JAG), and Theratechnologies [THERF.PK]. All of these stocks either increased by more than 250% or declined by more than 88% during 2012.

If I chose stocks totally at random, I might hit one or more of the stocks on this list if I chose a large enough quantity. But if I exert any discretion about the types of stocks I buy, I would likely not buy anything so speculative. My choices are much more likely to include a subset from the middle of the distribution than from either edge. Because of this selective sampling, I will end up with a group that has a much closer resemblance to the median than the mean.

There is an easy way, however, to own the best performing stocks along with the worst performers and achieve the average performance. This is to simply buy a broad-based index fund, such as the Vanguard Total Stock Market Fund (VTSAX), Vanguard FTSE All-World ex-US Fund (VFWAX), or their ETF equivalents: VTI and VEU. These funds own virtually every stock in their particular universe. With this simple security selection strategy, I can be reasonably assured that I will achieve the average performance, which exceeds the median performance of all investors, taken as a group. Likewise, I see little point in paying someone else to invest on my behalf when they are more likely to achieve the median performance instead of the average performance, even before considering the performance drag caused by their expenses.

So I am satisfied with my mediocrity, content with the knowledge that I am outperforming the majority of investors by merely striving to be average.

[i] Steven P. Utkas, Jean A. Young, The great recession and 401(k) plan participant behavior, Vanguard Research, March 2011

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.