Portfolio risk versus returns: the statistics
Please expand this article. Some suggested sources are given hereafter. 
This is an advanced investing topic. Past performance does not predict future performance. 
Portfolio returns are expected to follow a normal distribution. Deviations from this pattern are a form of investment risk, described below.^{[1]}
Intro
Fund returns are constantly changing. Instead of showing these changes on a performance chart (over time), count the number of times a return falls on each value. Create a plot with returns on the horizontal scale (Xaxis), min to max; the vertical scale (Yaxis) with the number of times the return fell on that value. This format is called a distribution plot, as it shows how the values are distributed across the range of returns.
Statistics is the science of the collection, organization, and interpretation of data.^{[2]} Statistical methods use this plot to analyze past performance and to predict future returns. For example; since most returns fall near plot center, it seems likely that future returns will also be near the plot center.
Based on usage, this plot is known as a statistical distribution. For many types of fund returns, the outline of the plot has the shape of a bell (like the shape of the Liberty Bell) and is called a normal distribution.^{[3]}
 insert graphs here  Plot over time, then show distribution
 Fat Tails ... and how to get 'em (gummystuff)
Identify: normal distribution, left side, right side, fat tail, long tail.
The shape of the curve is described using 4 different terms (you need all 4)...
Taleb
 Understanding all the talk about "tails", forum discussion starting point about The Black Swan (Taleb book)
 White swan
 Gray swan
 Black swan
 How the Finance Gurus Get Risk all Wrong, forum discussion with link to Taleb's (and Mendelbrot's) paper of same title. Tutorial of a sort, emphasizing that use of the Bell curve for risk management can be misleading.
Rough outline
 Fat tails
 Long tails
 Left tale risk
 Right tale opportunity
The first four moments of a distribution:
 Expected (mean) return
 Standard deviation (volatility)
Investors define volatility to be the standard deviation. For example, William F. Sharpe defines the Return per Unit of Risk (Sharpe Ratio) as some Return divided by the Standard Deviation.^{[4]}
Statistics discussions
 Median of a distribution and how skewness affects the relationship between the mean and the median
 Correlations and confidence intervals
Lognormal distributions
 Why for annual returns the log normal distribution, rather than the normal distribution, should be used
The proper distribution for a normal model would be a lognormal distribution... The reason for the lognormal distribution is that changes are relative to other changes. The probability that the market drops by 20% in the second half of the year is relatively independent of what happened in the first half of the year, but it is a larger point loss if the first half of the year was a bull market.^{[5]}
See also
References
 ↑ Risk and Financial risk, from Wikipedia.
 ↑ Statistics, from Wikipedia
 ↑ Normal distribution, from Wikipedia
 ↑ Standard Deviation: some comments on Volatility and Risk and Risk
 ↑ Understanding all the talk about "tails", forum discussion, direct link to post.
External links
 Understanding all the talk about "tails", forum discussion.
 Normal distribution, from Wikipedia.
Gummystuff
Tutorials from Gummy(retired forum member): Gummy stuff. Hosted by the Bogleheads' forum sister Canadian site: Financial Wisdom
 Standard Deviation: some comments on Volatility and Risk and Risk
 Standard Deviation ... and the Square Root of Time
 Risk / Reward Ratios
 Risk: how to measure it?
 Fat Tails ... and how to get 'em
