Alright ya'll it's about to get stuffy in here. Something I have strong interest in is global markets and the stock exchange (well, strong for a biochemist with no economics training past high school). Naturally, this sector is perfect for exploring the widespread use of statistics and the unique and powerful ways they can be applied. While disciples of Benjamin Graham will warn that the valuation of a company's stock is not 100% tied to potential for an upward trend, this undoubtedly plays some role in the type of trading done on the market today. More than 50% of trading on the NYSE is done via something called High-Frequency Trading, which uses complex algorithms to buy and sell securities on the millisecond time scale for an overwhelming addition of small differences, resulting in a large profit for the companies employing these buying and trading algorithms. I'm not nearly savvy enough to describe these algorithms with sufficient detail, but I do want to discuss some aspects of these algorithms and other probability/statistics related applications in stock trading.
The first technique (and arguably the most central) application of statistics in stock trading is in Asset Pricing Theory. This branch of investment theory uses the calculated effects (effect size!) of various macro-economic factors, or the behavior of theoretical indices (indices track many different stocks, or sectors and are like a mean value representing how a sector is doing. Common examples are the S&P 500 or the Dow Jones Industrial Average) to forecast the expected return of an asset. I realize that all sounds a bit vague, so let's focus on a specific example:
We are all familiar with the T-statistic (departure of a parameter from its notional value and its standard error) which we use in Student's T-Tests. In the case of investing, the utility of this statistic is almost exactly the same as when we would use it to compare means. In fact, one of the foundations of investment statistics is formation and testing of a null hypothesis. In Asset Pricing Theory, the null hypothesis would propose that "the expected return of the asset is not different from the risk-free rate of return". In other words, they compare the asset in question to the performance of risk-free investments like some bonds or savings accounts. Given the historical returns of an asset and the risk-free investment of choice, an investor may find a T-statistic describing the difference between their asset (our sample of interest) and the risk-free investment (background, WT, negative control, etc.). Investors will then use the T-statistic as an indicator of the probability of observing the asset's returns under the assumption of the null hypothesis. Another similarity is that, in this branch of economics, they set their statistically significant p-value as 0.05, and utilize confidence intervals to have a better understanding of how this asset is likely to behave. Similar to our experiments, these calculations also rely on a certain "N", where a single N could be individual transactions involving this asset (in this case, higher volume stocks would be advantaged, due to a greater number of values), but this is not always the case.
This type of statistical testing is vital to many stock analysts, who will use the likelihood that a stock will perform better than a risk-free investment as part of their assessment of how to score the stock (what they should advise their advisees or their firm to do with regard to the stock). It can also indicate when a stock may be "overpriced" or "on sale". In many trading circles, the direction that a stock is likely to go short-term is less important than the absolute value of the company and what a stock of that company should be "worth", hence "Asset Pricing Theory".