Given a sample of observations on say a variable X. We usually calculate how each of the observations is deviated from the mean of the sample. So you first calculate the mean. Then subtract each observation from the mean. If you sum those deviation from the mean the result will be zero, and you will lose the information in the data. To keep the information, you square the deviations and then sum them. The variance, Var(X), is the sum of squared differences divided by the number of observations (say T) minus one (the the one is substracted because you used up one degrees of freedom in calculation the mean).

Co-variance need two variables such as X and Y. To get the Covariance of X and Y, Cov(X,Y), proceed thus. First, deviate each observations from their respective means. Then, multiply the deviated observations by observation. Sum the cross-products. Then, divide by (T-1)

When you have many variables, you can use the matrix function of Excel (see Matrix Multiply, and Inverse function of Excel). Form a spreadsheet with one variable in each column, and call the whole spreadsheet  X. [You need some basics about transpose, and dimension when you add, multiply, and take inverse of matrix].

1.     First pre- multiply X by its transpose.

2.     Take the matrix inverse of the result of step 1.

3.     The result will be a square matrix, with information for the variances of linear coefficients on the main diagonal for each coefficient successively, and co-variances on the off-diagonal elements.

There are sometimes problem with interpretation variances and covariances. You may want to look at the correlation coefficient as well, defined as the cov(X,Y) divided by the product of the two standard devitions.

Here is a non-matrix example calculated in Excel:

the difference is that the variance is the average of the residuals between the data and average of single variable s^2=1/n S_i=1,^n(x_i-X)^2.  Covariance is the degree of variation between two variables cov(x,y)=1/n S_i=1,^2 (x_i-X)(y_i-Y) 

best regards

variance is  average of residuals of the same variable, but the co-variance is the  degree  of variation (+,-) between two variables say (x,y).

you can compute by Excel or matlab function  var_x=var(x), cov_XY=cov(x,y) and also  and var>0 but  cov(+,-)

Given a sample of observations on say a variable X. We usually calculate how each of the observations is deviated from the mean of the sample. So you first calculate the mean. Then subtract each observation from the mean. If you sum those deviation from the mean the result will be zero, and you will lose the information in the data. To keep the information, you square the deviations and then sum them. The variance, Var(X), is the sum of squared differences divided by the number of observations (say T) minus one (the the one is substracted because you used up one degrees of freedom in calculation the mean).

Co-variance need two variables such as X and Y. To get the Covariance of X and Y, Cov(X,Y), proceed thus. First, deviate each observations from their respective means. Then, multiply the deviated observations by observation. Sum the cross-products. Then, divide by (T-1)

When you have many variables, you can use the matrix function of Excel (see Matrix Multiply, and Inverse function of Excel). Form a spreadsheet with one variable in each column, and call the whole spreadsheet  X. [You need some basics about transpose, and dimension when you add, multiply, and take inverse of matrix].

1.     First pre- multiply X by its transpose.

2.     Take the matrix inverse of the result of step 1.

3.     The result will be a square matrix, with information for the variances of linear coefficients on the main diagonal for each coefficient successively, and co-variances on the off-diagonal elements.

There are sometimes problem with interpretation variances and covariances. You may want to look at the correlation coefficient as well, defined as the cov(X,Y) divided by the product of the two standard devitions.

Here is a non-matrix example calculated in Excel:

Covariance is a measure of dispersion between two variables. The variance of a variable is the covariance of the variable with itself.

Unfortunately, the mean doesn’t tell us a lot about the data except for a sort of middle point. For example, these two data sets have exactly the same mean (10), but are obviously quite different: [0, 8, 12, 20] and [ 8, 9, 11, 12]

So what is different about these two sets? It is the spread of the data that is different. The Standard Deviation (SD) of a data set is a measure of how spread out the data is.

Variance is another measure of the spread of data in a data set. In fact it is almost identical to the standard deviation. You will notice that this is simply the standard deviation squared.

The last two measures we have looked at are purely 1-dimensional. Data sets like this could be: heights of all the people in the room, marks for the last exam etc.

However many data sets have more than one dimension, and the aim of the statistical analysis of these data sets is usually to see if there is any relationship between the dimensions. For example, we might have as our data set both the height of all the students in a class, and the mark they received for that paper. We could then perform statistical analysis to see if the height of a student has any effect on their mark.

Standard deviation and variance only operate on 1 dimension, so that you could only calculate the standard deviation for each dimension of the data set independently of the other dimensions. However, it is useful to have a similar measure to find out how much the dimensions vary from the mean with respect to each other.

Covariance is such a measure. Covariance is always measured between 2 dimensions. If you calculate the covariance between one dimension and itself, you get the variance.

Variance measures the variability of ONE random variable. Covariance measures the Co-variability between TWO random variables.

variance (var)  is the expectation of the squared deviation of a random variable from its mean. covariance (cov) is a measure of the joint variability of two random variables. The variance of a random variable X  is the expected value of the squared deviation from the mean of X, here just a random variable. However, the covariance between two jointly distributed real-valued random variables X and Y with finite second moments is defined as the expected product of their deviations from their individual expected values, here two variables of X and Y...

This may confuse but I sometimes find it helpful

the variance of a variable is the co-variance  of a variable and itself . Consequently when this co-variance is standardized ( by dividing by the square root of the product of the variance(s)) it will give a correlation of 1

Variance and covariance are mathematical terms frequently used in statistics, and despite the similar sounding names they actually have quite different meanings. A covariance refers to the measure of how two random variables will change together and is used to calculate the correlation between variables. The variance refers to the spread of the data set — how far apart the numbers are in relation to the mean, for instance. Variance is particularly useful when calculating the probability of future events or performance.

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