Confidence Intervals

An important fact about the Normal Distribution is that 95% of its occurrences lie within a distance of 2 times Sigma either side of the average. For example, suppose you know that the average length of an adult male giraffe neck is 10 feet, and that Sigma has been computed to be 1 foot. Then, assuming that giraffe neck lengths are Normally Distributed (a pretty good assumption since physical traits tend to be averaged across many ancestors), you can be 95% confident that the next adult male giraffe you see will have a neck between 8 and 12 feet long (2 feet either side of 10).

If you don’t make book on giraffe neck lengths, here’s a more practical example. We’ll assume that you are away from your computer, or there would be a better way to do this. So imagine that you have just stepped dripping wet out of the shower when your spouse or significant other brings you the portable phone with the boss on the other end. “What’s profit going to be next year,” asks the boss. This is at least an enlightened boss who will accept a range of possibilities. You think fast. You know that profit has varied around $500,000 for the past 12 years with no apparent trend. And you are lucky enough to remember that the STDEV formula in Excel, which computes the Standard Deviation (another name for Sigma), yielded $50,000 when applied to the same 12 data points. So, as naked as the day you were born, you are able to say: “Boss, the odds are 19 to 1 that next year’s profit will be between $400,000 and $600,000.” Get it? Plus or minus $100,000 is the 95% confidence interval, and 95% is 19 times bigger than 5%.