What Can we obtain from correlation coefficient and Sample variance and Standard deviation?

Question

what do they show us? how do we describe the number that we get from it?

Answer 1

Sample variance and standard deviation tell us how variable the data is. Usually we think of this in terms of the standard deviation, since that has the same units as the data. A useful rule of thumb is that for many data sets around 95% of the data is within two standard deviations for the mean (at least 75% of the data must be within two standard deviations from the mean, for ANY data set).

The correlation coefficient describes how closely two paired sets of data track each other. It always has a value between -1 and 1. If it is close to -1 or 1 it means the two have a very nearly linear relationship (-1 means that as one increases, the other decreases; +1 means that they increase or decrease together). If it is close to 0 then the data sets are not correlated linearly; they may be not correlated at all or be correlated in some non-linear way. The significance of any particular value depends on the number of data points used.

Answer 2

To calculate the correlation coefficient of two variables (X and Y), you need the sample variance or standard deviation of both (the standard deviation is the square root of the variance, so knowing one tells you the other) AND the sample covariance of the two variables. Then, the correlation is: r = Cov(X, Y)/[SD(X)*SD(Y)] If r is positive, then an increase in X is ASSOCIATED with an increase in Y (one doesn't necessarily CAUSE the other). If r is negative, then an increase in X is ASSOCIATED with a decrease in Y (one doesn't necessarily CAUSE the other). If r is close to 1 or -1, the relationship between the two is strong.