# Mean,-Variance,-and-Standard-Deviation

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Statistics Examples
Mean, Variance, and Standard Deviation Let X 1 , X 2 , , X n be n observations of a random variable X. We wish to measure the average of X 1 , X 2 , , X n in some sense. One of the most commonly used statistics is the mean,  X , defined by the formula 
1 n  Xi n i 1

X  X 

Next, we wish to obtain some measure of the variability of the data. The statistics most often
2 2 used are the variance  X and the standard deviation  X   X . We have

X 

1 n 2 1 n    X i    X i  n  i 1 n  i 1  

2

    

It is easy to show that the variance is simply the mean squared deviation from the mean. Covariance and Correlation Next, let be n pairs of values of two random variables X and Y. We wish to measure the degree to which X and Y vary together, as opposed to being independent. The first statistic we will calculate is the covariance  XY given by

 XY 

1 n 1 n  n   X iYi    X i   Yi    n  i 1 n  i 1  i 1  

Actually, a much better measure of correlation can be obtained from the formula
1 n  n  X iYi    X i   Yi   n  i 1  i 1  i 1 2 2  n 2 1 n   n 2 1  n       X i    X i    Yi    Yi   n  i 1    i 1 n  i 1    i 1   
n

 XY 

The quantity  XY is known as the coefficient of correlation of X and Y.

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The Covariance Matrix Covariances and variances are sometimes arranged in a matrix known as a covariance matrix. In our case, the covariance matrix will be a 2  2 matrix:

 2 C X  XY

 XY  2  Y 

The eigenvalues of the covariance matrix are sometimes of interest. These are obtained in the usual way by solving the characteristic equation:

p     det  I  C  

2   X  XY

 XY 2  Y

Statistics Examples

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