Confidence levels

Confidence levels offer a more precise estimate of an average value by setting upper and lower limits for a data set’s central tendency. This range can suggest how precise a statistic is (mean, correlation, etc.) according to a certain probability. For example, 98-percent confidence levels suggest that the market is likely to have traded outside of this range just 2 percent of the time.

Assume the S&P 500’s average monthly move is 1.76 percent over the past 12 months, and we want to find the upper and lower confidence levels for this mean at the 95-percent confidence interval. (The higher the confidence interval, the wider this range is.)

Let's say the 12 monthly percentage returns are 1.2, 1.5, 1.7, 2.3, 4.5, 3.3, 6.7, -2.4, 1.1, 1.0, 2.1, and 1.9. The standard deviation is 2.46 percent, and for a 95-percent confidence level, you must go 1.96 standard deviations from the mean. The standard error of the mean is 0.71 percent (2.46 percent / (12)). The formula for upper and lower confidence levels are:

Lower limit = Mean - (1.96 standard deviations * 0.71 percent standard error) = 1.76 percent - (1.96)(0.71 percent) = 0.36 percent

Upper limit = Mean + (1.96 standard deviations * 0.71 percent standard error) = 1.76 percent + (1.96)(0.71 percent) = 3.15 percent

The site http://davidmlane.com/hyprstat/index.html offers relatively easy-to-digest definitions of this and other statistical terms.