We are often asked "variance unbiased estimate of how to calculate? What and biased estimation difference is?", I thought "Oh, forgot." Benpian return to the essence of the problem with your understanding of the practical problems solved behind these terms.
First, the basic concept
The first step in problem-solving is understanding the problem, by following the example first figure out the concept.
If you want to research the university women's height, you stand in front of the toilet (girls usually fall in love with toilet ^ ~ ^), n randomly ask a girl (independent and identically distributed), and finally to reflect the height by which the value of it? Generally, we will use the mean.
But if in the research, you find some girls particularly high (guess is the school basketball team), the sample does not reflect real tall girls generally, which leads to abnormal data samples collected, then you can be measured by the variance differences in height.
Since the mean μ and variance of all tall girls school [sigma] 2 unknown sample obtained by the calculation here 
and S 2 , are only known distribution of an estimate of unknown parameters, that is the estimated amount. Sample mean and variance of the sample used in the estimation is used to describe data characteristics, it is called a statistic.
The following examples of the above-mentioned concept, strictly defined as follows:
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Expect
It refers to the sum of random variables and random events in which the probability of occurrence of the product reflects the average value of the random variable size, also known as "mean."
E (X-) = [Sigma I P (X I ) X I
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It is used to measure the degree of divergence between the random variables with mean and variance, the smaller the degree of deviation.
D(X) = E([X-E(X)]2)
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Statistics
Set of known samples, sample values calculated by a function, called a statistical amount, excluding unknown parameters. For example, the sample mean, sample variance, the sample standard deviation.
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Estimate
Provided overall sample distribution function is known, the unknown parameters. Known sample set, need to construct an appropriate statistic to estimate the approximate value of the unknown parameter, which is called the estimated amount.
Second, the question then comes
In the above example two indices calculated as follows:
Sample mean
Sample variance
Why denominator to calculate the variance is n-1, instead of n?
Statistics example is actually an estimate of the unknown parameters, the estimated amount of choice is the evaluation criteria, the following are three common evaluation, examine only the unbiased estimators of here.
Third, the estimated amount of evaluation criteria
1. Unbiasedness
If the estimated amount of the mathematical expectation is present, and is equal to the desired unknown parameters, called the estimated as parameter unbiased estimator.
Unbiased estimator means for some sample values, the estimate of the true value obtained compared with some too large, some small, but its average deviation 0. And the estimated amount of the desired true value is called a system error, unbiased estimate actually refer to non-systematic error.
2. Effectiveness
There are two unbiased estimators estimate, it is the true value, which estimates the amount of variance smaller than the large variance more effective.
The validity of the estimated amount of hope unbiased estimate of the value of the deviation from the true value of the smaller the better, so with a small variance estimators better.
3. Consistency
With an infinite number of samples increases, the estimated amount of convergence in probability to the true value, is called a consistent estimate of the amount.
More than two standards are based on the premise of a fixed number of samples, we hope that with the increase of the sample, the estimated value of the amount close to the true value of the parameter.
Fourth, the variance of unbiased
By definition of the above standards unbiased variance unbiased estimate of the mean variance is equal to an amount necessary to estimate the true value when the denominator is n, the following equation can be seen
1. Derivation formula
Therefore, only the sample mean equal to the true value of the mean, mean sample variance was equal to true variance. Due to the randomness of the sample, the sample does not necessarily mean value, the estimated amount of the denominator of the mean n <= variance of the true value, it is biased estimate.
The following formula is an unbiased estimate of variance
2. popular understanding (DOF)
Calculate the estimated amount of samples need to be independent and identically distributed, due to the distribution of the unknown parameters, when the sample variance is calculated using the sample mean, the sample mean is calculated from each sample. Assuming that the sample size is n, n-1 is known sample values, deduced by the sample mean value of the last sample, sample damage independence, so the degree of freedom of the sample set of n-1, so that the sample variance is calculated Sample number should subtract 1.
reference:
https://www.zhihu.com/question/20099757
"Probability and Statistics"