Sample Variance Value

A Sample Variance Value is a sample statistic that measures the variance of a dataset.

• Context:
  • It can be defined, for a sample [math]\displaystyle{ (x_1, \cdots, x_n) }[/math] of the random variable [math]\displaystyle{ x }[/math] as:

[math]\displaystyle{ \sigma_x=s^2_{x}=s(x,x)=\frac{1}{n-1} \sum_{i=1}^n [x_i − E(x)]^2 }[/math]

where [math]\displaystyle{ E(x) }[/math] is a central tendency measure (e.g. sample mean value) of and where [math]\displaystyle{ n }[/math] is the sample size.

• • …
• Counter-Example(s):
  • Population Variance.
  • Sample Covariance Value.
• See: Sample Mean, Covariance Matrix Estimate.

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References

2017a

• (Wikipedia, 2017) ⇒ https://www.wikiwand.com/en/Sample_mean_and_covariance
  • The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.

The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken.

The sample mean is a vector each of whose elements is the sample mean of one of the random variables that is, each of whose elements is the arithmetic average of the observed values of one of the variables. The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables. If only one variable has had values observed, then the sample mean is a single number (the arithmetic average of the observed values of that variable) and the sample covariance matrix is also simply a single value (a 1x1 matrix containing a single number, the sample variance of the observed values of that variable).

Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

2015

• (Siegrist et al.,2015) ⇒ Retrieved from http://www.math.uah.edu/stat/sample/Variance.html Copyright 1997-2015 Kyle Siegrist
  • Suppose that [math]\displaystyle{ x=(x_1,x_2,\cdots,x_n) }[/math] is a sample of size [math]\displaystyle{ n }[/math] from a real-valued variable [math]\displaystyle{ x }[/math]. Recall that the sample mean is [math]\displaystyle{ m=\frac{1}{n}\sum_{i=1}^nx_i }[/math] and is the most important measure of the center of the data set. The sample variance is defined to be

[math]\displaystyle{ s^2=\frac{1}{n−1}\sum_{i=1}^n(x_i−m)^2 }[/math]

If we need to indicate the dependence on the data vector [math]\displaystyle{ x }[/math], we write [math]\displaystyle{ s^2(x) }[/math]. The difference [math]\displaystyle{ x_i−m }[/math] is the deviation of [math]\displaystyle{ x_i }[/math] from the mean [math]\displaystyle{ m }[/math] of the data set. Thus, the variance is the mean square deviation and is a measure of the spread of the data set with respet to the mean. The reason for dividing by [math]\displaystyle{ n−1 }[/math] rather than [math]\displaystyle{ n }[/math] is best understood in terms of the inferential point of view that we discuss in the next section; this definition makes the sample variance an unbiased estimator of the distribution variance. However, the reason for the averaging can also be understood in terms of a related concept.

2013

• http://www.r-tutor.com/elementary-statistics/numerical-measures/variance
  • Variance - The variance is a numerical measure of how the data values is dispersed around the mean. In particular, the sample variance is defined as:

[math]\displaystyle{ s^2 =\frac{1}{n-1}\sum_{i=1}^n (x - \bar{x})^2 }[/math]

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