# "The average is X, you can expect this to deviate to no more than Y" - what does this mean

I was googling on the subject of the average circumference of the human head and came across this Yahoo Answers post:

Wherein the respondent says “The average circumference of the human head is about 55.9cm for men, it is 54.47cm for women. You can expect this figure to deviate to no more than 2cm.”

Does this mean something other than I straightforwardly read it to mean? (that is, 55.9 plus or minus 2, but that’s all)

Or is this one of those terminology things where it doesn’t quite mean what the man on the street might think, but a statistician can understand it fully?

(for the curious, the reason I was exploring this topic: my bulbous head is 60cm in circumference, and I have trouble finding a hat to fit)

Seriously, dude, it’s Yahoo Answers. You shouldn’t expect precise terminology or accurate information.

Obviously it isn’t true that all men have heads 55.9 +/- 2 as yours is outside that range. Maybe the 2cm is the standard deviation, or (and this is more likely) the person is talking out of their ass.

I fully understand the unreliability of Yahoo Answers - I’m not asking about the reliability of this information - just whether that phraseology has meaning that is obscure to the layman.

Fair enough. Sorry if I was condescending or anything.

However, a source like Yahoo answers will also employ imprecise terminology. Thus, what the person MIGHT be referring to is standard deviation, though I wouldn’t hold my breath.

It’s either a mangling of a confidence interval, where some knucklehead is applying the confidence level (usually 95%) to the single interval (it applies to the entire family of intervals), or the numerical estimates are being treated as the true values of a population mean and standard error of some sort.

Problem - we generally do not know these true values, and normality can be suspect as an a priori assumption.

While people generally lean rather heavily on the asymptotic normality of families of sample means, it is not at all clear that one can assume the normality of the underlying population.

My money is on a rather loose interpretation of standard deviation, which is an average squared deviation from the mean. The standard deviation can be interpreted as the expected deviation from the mean.

But I agree, the answer was poorly phrased.

[nitpick]It’s the square root of the average squared deviation from the mean. Variance is the average squared deviation from the mean.
[/nitpick]

Even more nit-picky, the use of hard numbers suggests that these are sample-based estimates of means, standard deviations, yada yada, which do not support probabilistic statements about population behavior.

Here is a better answer to Big Head’s original thought about his abnormally large cranium. Not so big it seems.

:smack:

Not a nitpick by any stretch of the imagination, however. A nitpick would be arguing over using n vs. n-1 in the denominator of the sample estimate of the standard deviation, as both are asymptotically unbiased.

The square root operator is pretty crucial to the whole calculation. I thank you for pointing out my omission. Now I will go over to the corner and flagellate myself for making such a stupid error.

Actually, the sample standard deviation using the (n-1) denominator is unbiased for all n.