# What is the relevance of standard deviation?

I do a humanities subject but have a statistics based analysis of a rating experiment and cannot get my head around standard deviation.

The rating test is 1 to 5

1) Mean: 1.7 Standard Deviation: 0.88 2) Mean: 4: Standard Deviation: 2

What on earth does SD represent here? I know that it is about dispersion but I don't get whether the SD is low or high here.

• Also, for 1) people mainly picked 1,2 and a few 3. For 2) people mainly picked 4, 5 and a few 3. From what I have read online, does the low score of SD represent the lack of deviation from the mean? – 123gwen Jan 25 '16 at 10:37
• To get a feeling for magnitude: what would be SD be if all ratings were 1? all 5? 50% 1s and 50% 5s? Try out numeric examples that you can see are extreme (ratings are constant; ratings vary as much as possible). – Nick Cox Jan 25 '16 at 10:46
• Standard deviation shows you a scale of your data, but it is not very informative in case if your distribution is skewed or have outliers which is almost always the case for this type of data. – German Demidov Jan 25 '16 at 10:55
• – Tim Jan 25 '16 at 11:01
• It is mathematically impossible for such data to have a mean of $4$ and SD of $2$. The SD cannot exceed $\sqrt{3}$, which occurs when one-quarter of the responses are "1" and three-quarters are "5". – whuber Jan 25 '16 at 16:13

## 1 Answer

Standard deviation is a kind of "typical distance from the mean", usually slightly larger than the average distance from the mean. (And so it's measured in the same units as the original observations.)

So yes, as you suggest in comments, a small SD indicates that most of the distribution is close to the mean.

If the standard deviation is in the ballpark of about 0.7-1, then a typical rating is about 1 point away from the mean.

If the standard deviation is 0 they're all the same rating. (e.g. if everyone picks 1, that will have a standard deviation of 0).

Generally speaking there's no absolute standard of "large" or "small" for standard deviations (it depends on what you're doing, what the values are measuring, and on a number of other things) -- but with ratings on 1 to some maximum (like 5) there is a "biggest possible" standard deviation, which is half the range*. Since the range is 4, a standard deviation of 2 is definitely "big", representing essentially complete (and even) polarization into 1 or 5 ratings.

* (times $\sqrt{\frac{n}{n-1}}$ for $n$ observations if we're using the Bessel-corrected standard deviation)

You might also compare to the SD for a completely even spread across all 5 ratings, which would be on the "spread out" side (i.e. that would be a relatively big SD). This is a standard deviation of a bit over 1.4 ($\sqrt{2}$ -- or rather, $\sqrt{2\frac{n}{n-1}}$ with the usual Bessel correction). So with ratings on 1 - 5 you might call 1.4 "biggish".

Here's a few examples to give some basis for comparison:

• Thanks so much that makes sense! However do I describe it as a high or small SD or is that not even how to describe a SD? Compared to the mean, do the SD figures given show that there is little dispersion? so for 1) Mean: 1.7 Standard Deviation: 0.88 - do I study the SD in terms of the mean or in terms of 1-5 (the range of results)? Just clarifying to make sure! – 123gwen Jan 25 '16 at 10:49
• One thing worth noting about the SD is that it's expressed in the units of the metric. In other words, it is not invariant to scale. To enable a direct comparison of the variability in your features, use the coefficient of variation calculated as the SD divided by the mean (for each metric). To Glen_b's point, this will never create an "absolute standard" or ground truth for comparison, but it will facilitate relative comparisons of "high" vs "low." – Mike Hunter Jan 25 '16 at 10:57
• @123gwen I wouldn't compare ratings on 1-5 to the mean. Note that if all the ratings are 5 (the only way to have a mean of 5), the SD must be 0. If all the ratings are 1 (the only way to have a mean of 1), the SD must be 0. If the mean is 3, the SD could be somewhere between 0 and 2. So for ratings like that, comparing SD to the mean really makes no sense. As I already said there's generally no absolute standard of large or small (you can say one standard deviation is larger than another). However, because there's a largest possible standard deviation (essentially 2 for a 1-5 rating), ... – Glen_b Jan 25 '16 at 11:12
• ... there's at least something to compare SD's to. – Glen_b Jan 25 '16 at 11:13
• I wouldn't calculate coefficient of variation (CV) here at all. The main use of CV = SD/mean is when SD is (roughly) proportional to the mean, so that the ratio is interesting and useful. That's not going to happen with ratings 1 to 5. As already pointed out by @Glen_b and hinted at by myself, a mean of 5 must mean a SD of 0. For this case, the fact that the SD has the same units as the ratings is a feature and in no sense a problem. – Nick Cox Jan 25 '16 at 11:31