# Statistical notation in plain English

I have a very limited understanding of statistics, and am having trouble translating an answer to this question from quant.SE, "Separating the wheat from the chaff", the answer being:

Larry Harris has a chapter on performance evaluation in Trading and Exchanges. He states that over a long period of time, a skilled asset manager will consistently have excess returns whereas a lucky one will be expected to have random and unpredictable returns. Thus, we start with the portfolio's market-adjusted return standard deviation:

\begin{equation} \sigma_{adj} = \sqrt{\sigma^2_{port} + \sigma^2_{mk} - 2\rho\sigma_{port}\sigma_{mk}} \end{equation}

where $\rho$ is the correlation between the market and portfolio returns.

For a sample size $n$ (generally number of years), the average excess returns, and the adjusted standard deviation from above, we have a t-statistic:

\begin{equation} t = \frac{\overline{R_{port}} - \overline{R_{mk}}}{\frac{\sigma_{adj}}{\sqrt{n}}} \end{equation}

Now we can simply determine the probability that the manager's excess returns were luck by plugging this t-statistic into the t-distribution's PDF with degrees-of-freedom $n - 1$. The lower the probability, the more we can believe the manager's excess returns were from skill.

At this point, I now understand the top formula, though this was based on the finding the book's the section on topic referenced in the answer above on Google Books.

What I don't get now is this formula:

\begin{equation} t = \frac{\overline{R_{port}} - \overline{R_{mk}}}{\frac{\sigma_{adj}}{\sqrt{n}}} \end{equation}

I understand it's somehow related to the t-distribution's "Probability Density Function" but I have no idea what the R or "overline" notation means, or for that matter, where the values are R-port and R-mk come from; "port" being portfolio, and "mk" being market. If it matters, since it took me a bit to figure this out "adj" just flags the sigma (standard deviation) as the "portfolio's market-adjusted return standard deviation."

Any suggestions?

• If you want someone on another site to explain their notation, you should ask them. There are conventions in finance, for example that won't be standard in all of statistics. I'll say a few things, though: $\bar x$ means "the sample mean of $x$'s". The poster shouldn't be using $\sigma$ if they're referring to a sample standard deviation (and if they do mean a population s.d. they don't have a $t$). – Glen_b Sep 1 '13 at 0:02
• +1 @Glen_b: Thanks for pointing out x¯ means "the sample mean of x's." Based on your feedback, it's my understanding that your suggestion is that I remove the question (which is fine, grateful for your help). Is it correct that removing the question is the best thing to do? Thanks. – blunders Sep 1 '13 at 0:16
• To the extent that your question is "explain this piece of statistical notation", such as $\bar x$, I think it's okay to stay. It's just that some parts of your confusion don't stem from issues with statistical convention, but from a poster not explaining his notation (most of us aren't in a better place to guess what $R$ or the subscripts might stand for than you are, though from the context, $R$ is 'return' and $R_{\text{port}}$ & $R_{\text{mk}}$ would be portfolio and market returns respectively). However, given the bad notation, it's hard to say when it's a population vs a sample quantity. – Glen_b Sep 1 '13 at 0:27
• Note as well that you don't "plug in" the statistic to the t-distribution's probability density function (pdf), but to its cumulative distribution function, & that 1 minus the resulting probability gives neither "the probability that the manager's excess returns were luck" nor "the probability that the fund manager would have earned those returns if he was unskilled", but the probability that the fund manager would have earned those returns or higher if he was unskilled (or whatever the null hypothesis is). – Scortchi - Reinstate Monica Sep 1 '13 at 0:44
• @Glen_b: So, think you've answered the question then unless I'm missing something. (Meaning you answered both what the likely meaning of what the R or "overline" notation means. Please post that as the answer and I'll mark it as such. The issues with the source answer such as PDF, CDF, etc. are likely beyond the scope of the question, and my current general understanding of stats. Again, thanks! – blunders Sep 1 '13 at 1:08

The notation "$\bar x$" means the sample mean of $x$'s, $\bar x = \frac{1}{n}(x_1+x_2+\ldots+x_n)$. So just take the (sample) average of whatever is under the bar.
$\sigma_{\text{something}}$ should represent a population standard deviation, not a sample standard deviation (for which instead one might use $s_{\text{something}}$).
From context, it appears that $R$ is a return, and the subscripts $\text{port}$ and $\text{mk}$ are "portfolio" and "market" respectively.