# What conditions ensure that the standard deviation is a good measure of error?

While attempting to analyze error incurred in certain optics experiments, I am only able to make statements about the standard deviation of a suitable distance measure between the actual and expected outcomes.

I understand that the standard deviation is not always a good measure of error, for example in the case of a bi-modal distribution. Numerics show that the distribution of the expected outcomes is uni-modal, but is this 'knowledge' enough to ensure that the standard deviation is the right quantity to look at?

Is there a set of conditions that insure that the standard deviation a good measure of the 'error' of a distribution?

I would say the sd is a measure of 'spread' rather than 'error'.

One way of thinking of when the sd is appropriate is "If the mean is a good measure of central tendency, the sd is often a good measure of spread". Other things that can make the sd not so good:

Skewness. For a skewed distribution we tend to use the median; where we use the median, often the interquartile range is a good measure of spread.

Rates: The arithmetic mean is not a good way of averaging rates, because it also depends on the length of time at each rate. Here you want the harmonic mean. I am not sure what measure is best for spread of rates.

Different scales: If you are trying to combine numbers that are on different scales, the mean is not good. Consider a college admissions office that wants to combine math SAT score (0 to 800), verbal SAT (ditto), GPA (0-4) and a rating of the college essay (0-10). Then the arithmetic mean would weight SAT score and you want the geometric mean. Here, you could get a measure of spread by standardizing the variables.

• Thanks! That's useful language to think in terms of. So, for a distribution that is not skewed, and uni-modal, is it reasonable to state that the standard deviation is a good measure of the spread of the distribution?
– user30512
Oct 1 '13 at 11:14
• Right now I can't think of a case of a unimodal symmetric distribution where sd isn't at least reasonable. But there may be one. Oct 1 '13 at 11:15
• @PeterFlom There are an infinite number of symmetric unimodal examples where s.d. is not a good measure of spread. An obvious example would be a t distribution with two degrees of freedom. The mean exists, but standard deviation is pretty useless -- the population s.d. doesn't even exist. Oct 1 '13 at 14:06
• Thanks @Glen_b. Do you think is the standard deviation a reasonable measure of the spread of a distribution if the higher moments are bounded?
– user30512
Oct 14 '13 at 15:45
• @Curious If the population standard deviation exists and the sample standard deviation itself has a finite variance, then in many situations it can be a reasonable(/good) measure of spread - but then ultimately it really depends on what aspects of spread you're interested in measuring. Oct 14 '13 at 20:25