According to Wikipedia (my emphasis),

In statistics, the standard deviation … is a measure that is used to

quantify the amount of variationor dispersion of a set of data values. A standard deviation close to 0 indicates that the data points tend to be very close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

The purpose of the standard deviation (SD), then, is to tell us **how varied** or uniform (SD → 0) the data is.

So, given a certain SD, how varied *is* the data? **What does the size of the standard deviation mean?**

Please explain the meaning of the SD by interpreting an *SD* = 1 (*M* = 0).

If you cannot interpret the size (quantity) of this SD, what other information would you need to be able to interpret it, and how would you interpret it, given that information? Please provide an example.

If, on the other hand, the quantity of the SD cannot be *qualified* in this manner, my argument is that it is essentially meaningless. If you disagree, please explain the meaning of the SD.

_{The following are earlier versions to give context to the answers. Unfortunately these didn't really convey what I wanted, and my attempt to ask it elsewhere was closed. (I don't need these versions answered now):}

_{First revision:}

_{What does the size of the standard deviation mean?}

_{For example, if I want to study human body size and I find that adult human body size has a standard deviation of 2 cm, I would probably infer that adult human body size is very uniform, while a 2 cm standard deviation in the size of mice would mean that mice differ surprisingly much in body size.}

_{Obviously the meaning of the standard deviation is its relation to the mean, and a standard deviation around a tenth of the mean is unremarkable (e.g. for IQ: SD = 0.15 * M).}

_{But what is considered "small" and what is "large", when it comes to the relation between standard deviation and mean? Are there guidelines similar to the ones that Cohen gives for correlations (a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small)?}

_{Original question:}

_{We always calculate and report means and standard deviations. But what does the size of the variance actually mean?}

_{For example, assume we are observing which seat people take in an empty room. If we observe that the majority of people sit close to the window with little variance, we can assume this to mean that people generally prefer siting near the window and getting a view or enough light is the main motivating factor in choosing a seat. If on the other hand we observe that while the largest proportion sit close to the window there is a large variance with other seats taken often also (e.g. many sit close to the door, others sit close to the water dispenser or the newspapers), we might assume that while many people prefer to sit close to the window, there seem to be more factors than light or view that influence choice of seating and differing preferences in different people.}

_{At what values can we say that the behavior we have observed is very varied (different people like to sit in different places)? And when can we infer that behavior is mostly uniform (everyone likes to sit at the window) and the little variation our data shows is mostly a result of random effects or confounding variables (dirt on one chair, the sun having moved and more shade in the back, etc.)?}

_{Are there guidelines for assessing the magnitude of variance in data, similar to Cohen's guidelines for interpreting effect size (a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small)?}

_{For example, if 90% (or only 30%) of observations fall within one standard deviation from the mean, is that uncommon or completely unremarkable?}

## Best Answer

Discussion of the new question:

It depends on what we're comparing to. What's the standard of comparison that makes that very uniform? If you compare it to the variability in bolt-lengths for a particular type of bolt that might be hugely variable.

By comparison to the same thing in your more-uniform humans example, certainly; when it comes to lengths of things, which can only be positive, it probably makes more sense to compare coefficient of variation (as I point out in my original answer), which is the same thing as comparing sd to mean you're suggesting here.

No, not always. In the case of

sizesof things oramountsof things (e.g. tonnage of coal, volume of money), that often makes sense, but in other contexts it doesn't make sense to compare to the mean.Even then, they're not necessarily comparable from one thing to another. There's no applies-to-all-things standard of how variable something is before it's variable.

Which things are we comparing here? Lengths to IQ's? Why does it make sense to compare one set of things to another? Note that the choice of mean 100 and sd 15 for one kind of IQ test is entirely arbitrary. They don't have units. It could as easily have been mean 0 sd 1 or mean 0.5 and sd 0.1.

Already covered in my original answer but more eloquently covered in whuber's comment -- there is no one standard, and there

can'tbe.Some of my points about Cohen there still apply to this case (sd relative to mean is at least unit-free); but even with something like say Cohen's d, a suitable standard in one context isn't necessarily suitable in another.

Answers to an earlier version

Well, maybe a lot of the time; I don't know that I

alwaysdo it. There's cases where it's not that relevant.The standard deviation is a kind of average* distance from the mean. The variance is the square of the standard deviation. Standard deviation is measured in the same units as the data; variance is in squared units.

*(RMS -- https://en.wikipedia.org/wiki/Root_mean_square)

They tell you something about how "spread out" the data are (or the distribution, in the case that you're calculating the sd or variance of a distribution).

That's not exactly a case of recording "which seat" but recording "distance from the window". (Knowing "the majority sit close to the window" doesn't necessarily tell you anything about the mean nor the variation about the mean. What it tells you is that the

mediandistance from the window must be small.)That the median is small doesn't of itself tell you that. You might infer it from other considerations, but there may be all manner of reasons for it that we can't in any way discern from the data.

Again, you're bringing in information outside the data; it might apply or it might not. For all we know the light is better far from the window, because the day is overcast or the blinds are drawn.

What makes a standard deviation large or small is not determined by some external standard but by subject matter considerations, and to some extent what you're doing with the data, and even personal factors.

However, with positive measurements, such as distances, it's sometimes relevant to consider standard deviation relative to the mean (the coefficient of variation); it's still arbitrary, but distributions with coefficients of variation much smaller than 1 (standard deviation much smaller than the mean) are "different" in some sense than ones where it's much greater than 1 (standard deviation much larger than the mean, which will often tend to be heavily right skew).

Be wary of using the word "uniform" in that sense, since it's easy to misinterpret your meaning (e.g. if I say that people are "uniformly seated about the room" that means almost the opposite of what you mean). More generally, when discussing statistics, generally avoid using jargon terms in their ordinary sense.

No, again, you're bringing in external information to the statistical quantity you're discussing. The variance doesn't tell you any such thing.

Not in general, no.

Cohen's discussion[1] of effect sizes is more nuanced and situational than you indicate; he gives a table of 8 different values of small medium and large depending on what kind of thing is being discussed. Those numbers you give apply to differences in independent means (Cohen's

d).Cohen's effect sizes are all scaled to be unitless quantities. Standard deviation and variance are not -- change the units and both will change.

Cohen's effect sizes are intended to apply in a particular application area (and even then I regard too much focus on those standards of what's small, medium and large as both somewhat arbitrary and somewhat more prescriptive than I'd like). They're more or less reasonable for their intended application area but may be entirely unsuitable in other areas (high energy physics, for example, frequently require effects that cover many standard errors, but equivalents of Cohens

effect sizesmay be many orders of magnitude more than what's attainable).Ah, note now that you have stopped discussing the size of standard deviation / variance, and started discussing the proportion of observations within one standard deviation of the mean, an entirely different concept. Very roughly speaking this is more related to the peakedness of the distribution.

For example, without changing the variance at all, I can change the proportion of a population within 1 sd of the mean quite readily. If the population has a $t_3$ distribution, about 94% of it lies within 1 sd of the mean, if it has a uniform distribution, about 58% lies within 1 sd of the mean; and with a beta($\frac18,\frac18$) distribution, it's about 29%; this can happen with all of them having the same standard deviations, or with any of them being larger or smaller without changing those percentages -- it's not really related to spread at all, because you defined the interval in terms of standard deviation.

[1]: Cohen J. (1992),

"A power primer,"

Psychol Bull.,112(1), Jul: 155-9.