# Does it make sense to speak of whether the standard deviation is a "good measure" of some particular distribution?

I came across this question (picture below, the question where the drop-down bar is highlighted) on edulastic.com.

The answer, apparently, was that "The standard deviation is a good measure of the spread of the distribution for Freshmen."

What is the rationale for this? Or does this question simply not make sense? People do speak that way, and therefore it has some meaning, even if poorly and indirectly expressed.

If someone says "the standard deviation is a good measure of the spread of the distribution for Freshmen", I interpret that as roughly:

A normal distribution gives a good summary of the spread of the distribution for Freshmen.

I would prefer to talk this way instead, and I would prefer talk of medians and quartiles even more. Meanwhile I can accept the way that people talk now.

• By "people" do you mean teachers in US high schools? I attended school outside the US and have been at a US university for the past 10 years. This was the first time I've ever seen anyone saying (or writing) any such thing. (It was also one of the few times I was looking at US high school math material.)
– user378
Jun 15 '15 at 14:24
• @KennyLJ, try googling "standard deviation is a good": it turns up various examples of the phrase from universities and companies. The first hit for me is a post on the Stack Exchange for statistics, where even a moderator of that site uses the phrase. It's pretty common.
– user173
Jun 16 '15 at 2:00

The standard deviation is a measure of "dispersion". But "dispersion" needs to have a focal point, and this focal point in the case of the standard deviation is the mean.

So "standard deviation is a measure of dispersion around the mean", of how "far on average" are the actual observations from their mean value.

The mean itself is a useful summary measure of the distribution of the data, if it represents indeed a focal point, a concentration point, a point around which the actual occurrences tend to cluster. But this implies that the mean is useful when the distribution is a) unimodal and b) has occurences "to the left and to the right" of its mode (maximum). Then the mean gives us roughly the value around which occurences hover.

But when the data is not roughly reminiscent of a "symmetric unimodal" shape (as is for Freshmen), but has the appearance of the distribution of Seniors, then the mean itself stops being really useful as a measure of concentration: evidently, Seniors occurrences do not hover around the sample mean.

But if the focal point around which and in relation to which the standard deviation is defined, is in itself not that useful as a metric and it does not summarize important aspects of the distribution that the data follow, then the standard deviation itself stops being also a useful measure.

• For a distribution like that of the Seniors, I keep the following example in mind: Living in China, I often found that in walking around parks in the daytime, many of my fellow perambulators appeared to be grandparents with their grandchildren. As a result, in thinking about the average age of people walking around in the park, the mean would not be a great measure! Jun 20 '15 at 17:03
• @BenjaminDickman That is a very good example, of why the mean is not always mean-ingful Jun 20 '15 at 19:11

Take the data for the seniors. Calculate the median and standard deviation. These two number make more sense when talking about data that's scattered close to a normal distribution, a bell curve. A bi-modal distribution is valid, even interesting, but the words standard deviation don't quite apply the same way.

That said, it's a bad question. Better would be to talk about whether data fits a normal distribution, or the degree to which it does. But using words like "good" in math problems seems problematic to me.

• So "is a good measure" is out but "make more sense" is in?
– user173
Jun 14 '15 at 23:55
• "Does the data appear a normal distribution?" is how i'd approach it. Jun 15 '15 at 9:50