Sample mean and covariance

Sample mean

Let x i j be the i^th independently drawn observation (i=1,...,N) on the j^th random variable (j=1,...,K). These observations can be arranged into N column vectors, each with K entries, with the K ×1 column vector giving the i^th observations of all variables being denoted x i (i=1,...,N).

The sample mean vector x ¯ is a column vector whose j^th element x ¯ j is the average value of the N observations of the j^th variable:

Thus, the sample mean vector contains the average of the observations for each variable, and is written

Sample covariance

The sample covariance matrix is a K-by-K matrix Q = [ q j k ] with entries

where q j k is an estimate of the covariance between the j^th variable and the k^th variable of the population underlying the data. In terms of the observation vectors, the sample covariance is

Alternatively, arranging the observation vectors as the columns of a matrix, so that

which is a matrix of K rows and N columns. Here, the sample covariance matrix can be computed as

where 1 N is an N by 1 vector of ones. If the observations are arranged as rows instead of columns, so x ¯ is now a 1×K row vector and M = F T is an N×K matrix whose column j is the vector of N observations on variable j, then applying transposes in the appropriate places yields

Like covariance matrices for random vector, sample covariance matrices are positive semi-definite. To prove it, one may notice its similarity with the covariance matrix associated with the random choice among the N observation vectors. Furthermore, a covariance matrix is positive definite if and only if the rank of the x i . − x ¯ vectors is K.

Discussion

The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector X , a row vector whose j^th element (j = 1, ..., K) is one of the random variables. The sample covariance matrix has N − 1 in the denominator rather than N due to a variant of Bessel's correction: In short, the sample covariance relies on the difference between each observation and the sample mean, but the sample mean is slightly correlated with each observation since it is defined in terms of all observations. If the population mean E ⁡ ( X ) is known, the analogous unbiased estimate

using the population mean, has N in the denominator. This is an example of why in probability and statistics it is essential to distinguish between random variables (upper case letters) and realizations of the random variables (lower case letters).

The maximum likelihood estimate of the covariance

for the Gaussian distribution case has N in the denominator as well. The ratio of 1/N to 1/(N − 1) approaches 1 for large N, so the maximum likelihood estimate approximately equals the unbiased estimate when the sample is large.

Variance of the sample mean

For each random variable, the sample mean is a good estimator of the population mean, where a "good" estimator is defined as being efficient and unbiased. Of course the estimator will likely not be the true value of the population mean since different samples drawn from the same distribution will give different sample means and hence different estimates of the true mean. Thus the sample mean is a random variable, not a constant, and consequently has its own distribution. For a random sample of N observations on the j^th random variable, the sample mean's distribution itself has mean equal to the population mean E ( X j ) and variance equal to σ j 2 N , where σ j 2 is the variance of the random variable X_j.

Weighted samples

In a weighted sample, each vector x i (each set of single observations on each of the K random variables) is assigned a weight w i ≥ 0 . Without loss of generality, assume that the weights are normalized:

(If they are not, divide the weights by their sum). Then the weighted mean vector x ¯ is given by

and the elements q j k of the weighted covariance matrix Q are

If all weights are the same, w i = 1 / N , the weighted mean and covariance reduce to the sample mean and covariance above.

Criticism

The sample mean and sample covariance are not robust statistics, meaning that they are sensitive to outliers. As robustness is often a desired trait, particularly in real-world applications, robust alternatives may prove desirable, notably quantile-based statistics such as the sample median for location, and interquartile range (IQR) for dispersion. Other alternatives include trimming and Winsorising, as in the trimmed mean and the Winsorized mean.