What is correlation coefficient?

Answer 1

'Correlation coefficient' means a statistic representing how closely two variables co-vary; it can vary from -1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation)

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A key piece of information that is left out of the answer by True Knowledge (which casts very serious doubts about its name!) is that the statistic only is a measure of linearrelationship. A symmetric non-linear relationship (a parabola, for example) will show zero correlation but show anyone a graph of a parabola and then try convincing them that there is no relationship between the two variables!
A correlation for two variables is a measure of the strength of a linear relationship between them. It is a measure that ranges from -1 (the variables move perfectly together but in opposite directions) to 1 (the variables move perfectly together and in the same direction). A correlation coefficient of 0 indicates no linear relationship between the variables.

Two important points to note:

1. Correlation measures linear relationship: not any other relationships. Thus a perfect relationship that is symmetric (y = x^2, for example) will have a correlation coefficient of 0.
2. Correlation coefficient is a measure of association, not of causality. In the UK, ice cream sales and swimming accidents are correlated. This is not because eating ice cream causes swimming accidents not because people recover from swimming accidents by eating ice cream. In reality, both events are more likely on warm days - such as they are!