By Salil Mehta, statistician and blogger at (Statistical Ideas)



) -- There is a popular monthly chart of the

S&P 500

returns that is going around. It shows where the 'sell in May', and the 'Santa Claus rally' phrases come from. But let's look deeper at the probabilistic nature behind these monthly deviations, in the context of normal market gyrations.

Chart 1. This in the raw data for the S&P 500 since 1960. The total average monthly change among these returns is 0.5%. Could those monthly deviations on either side of 0.5%, therefore, have been the result of normal random distribution? In other words, could the ordering of months was just be by luck alone?

Chart 2. The variation in monthly returns, around each of the monthly averages above, also varies though by themselves these variations are not statistically significant.

Chart 3. We then show the original monthly S&P 500 return chart, alongside the 90% confidence interval associated with each of those months. We see 10% of the 12 months should not have the 0.5% total monthly average within their confidence interval. But we see that instead 3 months do (which is more than 10% of 12). They are shown below, in green. Alternatively, about 70% of the months should be within one standard deviation of the 0.5% total monthly average. This is roughly 8 of the 12 months, but empirically only four months conform on this basis.

So, on net, the results of the monthly average returns is not only interesting, but they do show some interesting statistical significance as well. And it points to some lift, barring any tapering execution, at year-end.

Written by Salil Mehta, creator of the Statistical Ideas blog.

At the time of publication, the author held no positions in any of the stocks mentioned, although positions may change at any time.

This commentary comes from an independent investor or market observer as part of TheStreet guest contributor program. The views expressed are those of the author and do not necessarily represent the views of TheStreet or its management.