I think this is a very common debate.
In the context of statistical models, variance refers to the randomness within each of the values in a data set. This is the same concept as what I had in mind when I said that you can’t be completely random. However, variance is very different from covariance, which represents that spread between pairs of values. For example, you might have a data set with a mean and a standard deviation of 100, and a sample size of 10.
Covariance is when you have a total of 20 variables, but each variable is only used to estimate a single parameter (say the mean). Variance is the spread in the individual values, and is much more descriptive of how the data is collected. Covariance is what you get if you have more than 20 variables, but you have a lot of of them.
If you have a lot of variables, then variance is the easiest to calculate, but variance is the least descriptive of the three. Covariance is probably the best because we can see the spread of the individual variables, and if we just don’t know what they are, then we don’t know what the spread is.
The spread of a variable is the range of values it contains. This is a crucial distinction that is often lost in the equation, because what we’re really measuring is the mean. The mean is basically a mathematical concept that describes the mean of a distribution. A mean is the average of all the data points, and is therefore the number that is equivalent to dividing the number of data points by the number of observations.
If we take a look at the spread of a variable, we can see that the spread varies depending on the number of observations. For example, a variable with a normal distribution will have a spread that is dependent on the number of values it contains. If a variable is normally distributed, then the spread of values for the number of observations should be the standard deviation, which is the square root of the mean.
This is known as the variance. A variance will be dependent on the number of observations, which is why we can say that the spread of a variable is the standard deviation times the number of observations. You will also see that the standard deviation is the same thing as the standard error of the mean, which is equal to the square root of the mean squared.
The standard error of the mean is the standard deviation times the square root of the mean squared. The variance is the spread of the observations divided by the mean squared. So the spread of a variable is the square root of the variance times the number of observations.
Variance is important because it is what determines the range of values a variable can take. For example, if you have two values of x and you want them to be anywhere from 1 to 5, you cannot use the variance to define where to put the two values. Instead, you have to use the standard deviation, which is the average of the two values and is equal to the square root of the variance times the number of observations squared.