basically variance is the average squared deviation from the mean. it's just a random number that tells someone how much a particular data set varies. so by definition, Var(X) = E((X-u)^2), where u is the mean. we can use this to derive the other formula:
I know I'm being pretty pedantic here, for vce what you have said is perfectly fine, however I just wanted to clarify the use of the word "average" that you used, it is very important to distinguish between an average and an expected value, in general, they are not the same. In fact what you should say is the "variance is the
expected value of squared deviations from the mean." In general "average" is not the same as "expected value". If you used the word "average" then your sentence should be "the sample variance is the
average squared deviation from the sample mean". Although this doesn't mean much in VCE, but the implications are huge for beyond VCE purposes (which I think you have the mathematical maturity understand
). There is a huge difference between sample variance and actual variance (that is, the definition of variance from
^2))
). That is, whenever we talk about averages we are always talking about the sample variances, sample means, sample standard deviations, sample medians etc. You may ask, what is a "sample"? What is a "population"?
I will try show you the difference.
Let S denote the population set, that is S could include the income of every household on the planet earth, or the age of every dog in australia and so on.
A sample (or more formally, a random sample
http://en.wikipedia.org/wiki/Random_sample) is where you take a subset of S, ie, the income of every household in Australia, or the age of every dog in victoria etc.
Now the variance, denoted by
implies that we know the actual POPULATION variance, however in reality, we will NEVER know the population variance, it is impossible to survey every household in the world to ask for their income. Thus we often "estimate" the population variance with the sample variance (the correct term here to use is that the sample variance is an ESTIMATOR of the population variance). Thus the best approximation we can arrive at to approximate
is from
(where s^2 is the conventional notation for sample variance).
Now note the difference
^2))
BUT
^2)
where
is the SAMPLE mean (different from the POPULATION mean denoted by
) and n is the sample size
In fact
can be shown to be an "biased" estimator of the population variance, thus to correct for this bias we often divide by n-1. So
^2)
is the sample variance.
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