Question

What is the meaning of variance and standard deviation? (I understand how to calculate them, but understanding the concept behind them is confusing.)

What happens if the variance calculated is the same as them mean? What is the meaning behind these numbers? For example, you roll a 6 sided dice and each possibility is equally likely. Then both the mean and variance is 3.5.

Answer 1

Variance is a measure of variability/spread of data. For example, let's say we have two data sets:

1,4,8,10,7 and 6,6,6,6,6

I have deliberately chosen these two examples. As you may observe, both these have 5 observations with mean = 6. But the first data varies from 1 to 10 while the second one remains constant at 6. The first data varies considerably around its mean 6 while the second one doesn't vary at all. Thus we can say that the first data has got some variance while the second has none.

The standard deviation is the square root of the variance. Since the variance contains the mean of squared errors, we can't use it directly to compare with our raw data. For ex. assume that the above data is the marks of 5 students out of 10. The variance will have a unit of (marks^2). When we take its square root, then only we can compare (see how far a certain score lies from the mean, normalised according to the standard deviation).

Generally, variance being equal to the mean doesn't have any mathematical significance. Both should be interpreted individually.