Wednesday, January 06, 2010

What is a Standard Deviation?

Entry for 31 December 2009:

An excerpt from Training the Trainers: A Research Training Manual, McLeod, Elliott & Wheeler, in press.

Commentary: I spent the end of November and the beginning of December working on revisions to the research training manual that John McLeod, Sue Wheeler and I have been working on for the past 2 years. John added a section on understanding common statistical concepts to the latest version, which I thought was a good idea, but as usual, we are struggling with how to explain in ordinary language what a standard deviation is. If I remember correctly, it was my attempt to explain this concept that lead one of our diploma students to make a t-shirt that read, “The last thing I understood was the baked potato I ate for lunch.” The following builds on John’s version and is my most recent attempt to get the idea across:

It is also important to know about the variability of scores - the amount of spread in the distribution of scores. The standard deviation statistic (abbreviated as s, s.d., sd or SD) is usually given in research reports as a measure of variability. In fact, the SD figure is the average variability of scores. That’s where the term “standard deviation” comes from: It is a kind of average or typical (hence “standard”) difference (hence “deviation”) between each person’s score and the mean score. It tells us how far off the average person is from the average, which also tells us how far we should trust the average as a way of representing the people in the sample: The bigger the SD, the farther people tend to be from average. Interesting facts about SDs:

• About two-thirds of people in a sample will have scores that are within one SD above or below the mean.

• About 95% of people will have scores that are within 2 SDs above or below the mean.

• On a standard 5-point rating scale, such as that used by the CORE Outcome Measure, many items will have SDs of about 1 scale point, so it’s a good idea to pay attention to items with SDs substantially above (1.5) or below (.5) that.

Further reflections. "Standard deviation" is another one of these oxymorons that we never think about: The idea of a standard/typical exception/deviation from the general case clashes internally; if we listen to it careful, we can hear its Bartokian dissonance as another kind of harmony. We tend simultaneously toward the general, the central tendency and to our own individuality and uniqueness. The Standard Deviation recognizes this and its own peculiarly quantitative way honours this individuality.


mark said...

So, if there are five people eating one apple each and only one person eats the core-pips and all- and all the other people except one other puts all their apple cores in the trash, except this one person, who eats their core but spits out the pips, then the SD score is the nature of the common variation (that they ate some of the core that the others didn't eat anything of) between the two who didn't put their cores in the trash?

Robert Elliott said...

Actually, there are three variables here, corresponding to what parts of the apple each person ate: the noncore part, the core w/ or w/o pips, and the pips. We can assign a value of 1 if that part was eaten and a value of 0 if that part was not eaten. This results in the following:
1. Everyone ate the noncore part of the apple, which means that there was a mean of 1 and and a standard deviation of 0. That makes this variable a constant, because it doesn't vary (i.e., it's not really a variable at all). It's all common; there is no variation.
2. Two out of 5 people ate the nonpip part of the core. This is a mean of .4 (or 40%), with an SD of .55. For a 0 - 1 scale and n = 5, that's a lot of variability, technically as much as it is possible to have.
3. One out of 5 people ate the pips also. This variable has a mean of .2 and an SD of .45.

On the other hand, it's probably more sensible to analyze this situation qualitatively, in which case we could conclude the following:
1. Eating the noncore part of the apple and avoiding the pips are each general themes, defined as occurring in all or all but one of the instances.
2. Eating the noncore part of the apple and throwing away all of the core is a typical theme, occurring in at least half of the examples.
3. Eating the core is a variant theme, defined as occurring in more than one example and less than half.
4. Eating the pips is a unique theme, defined as occurring in only one example.

mark said...

Thank you for explaining the standard deviation, here, Professor Elliot. My apple-eaters are obviously rather too few for you to be taxed by their variety!

Your original post seemed to point to a fundamental shift in our way of perceiving things ("Bartokian dissonances") if we are to analyze SD's in relation to more complex phenomena such as therapeutic outcomes?

If I have begun to understand the potential value of your research in terms of our capacity to shift our perceptual position: does it lie in the area of paying greater attention to the statistically smaller outcome, which nevertheless has within in it a constant? Or have I missed the point?

Robert Elliott said...

It's not so much looking at smaller effects, which tend to diminish toward being not very interesting, but rather embracing the variability pointed to by the standard deviation in order to see if we can understand it better. Some of this can be done quantitatively, but much of it then becomes territory for qualitative exploration. Under these circumstances, your pip-preferring apple-eater becomes more interesting, and we begin to wonder about his/her experience and whether the rest of us might be missing out on something nutritionally or otherwise.

mark said...

Thank you once again for taking the trouble to expand further on this.

So, hopefully without stretching your patience, if I have understood you correctly (and to use a different context): there is, say, research analysis of 50 therapeutic relationships of clients suffering from severe depression to determine effectiveness of outcomes.

20 of the clients require an average of 40 sessions to satisfactorily manage their depression, whereas 29 clients require an average of 30 sessions.

However there is only one client who is relieved of their depression in only one therapeutic session.

Upon analysis of this one session it is found that the therapist was almost totally silent and undemonstrative- and this differed significantly from the behaviour of the therapists in the other 49 samples.

As far as the Standard deviation is concerned, of the 50 therapeutic relationships scrutinized this is the least interesting because the variables in the other samples represent more frequently occurring phenomena (i.e. therapeutic interactions over a greater span of time than this one example, and more therapist activity)?

We don't learn as much, in other words, giving this one single-session example our scrutiny and analysis, as we do examining the clusters around the averages of the other sessions (assuming the deviations from the average aren't consistently extreme in the other samples).

Robert Elliott said...

The one-session therapy with a silent therapist is a classic example of what is called an "outlier" -- a case that is different enough from the rest of the cases to in fact represent a different population. Outliers are always interesting to look at, because they are most likely to surprise us, and being surprised means that we are learning something new. However, this case tells us nothing about the rest of the clients...

mark said...

Thanks again for all your input, and for being open and accessible- plenty for me to ponder here.

I think I get a sense of why it is difficult to explain in ordinary language!