Question

If the sample correlation coefficient is equal to 1 , is it necessarily true that \(\rho=1 ?\) If \(\rho=1\), is it necessarily true that \(r=1 ?\)

Short answer

No, if r is equal to 1 it does not necessarily mean that \(\rho\) is equal to 1 because the sample data may not perfectly represent the population. Yes, if \(\rho\) is equal to 1, it means that r should be 1 as the sample must represent the population correlation.

Step-by-step solution

Understanding Correlations

In correlation, '1' often represents a perfect positive correlation. This means that for every positive increase in one variable, there is a positive increase of a fixed proportion in the other. In terms of 'r' and '\(\rho\)', it's possible that both can hit the extreme value of '1' but each under different circumstances.

Correlation Coefficient r=1

The sample correlation coefficient 'r' equals 1 when there is a perfect positive relationship in the sample data i.e., an increase in one variable corresponds to a proportional increase in the other. This, however, does not necessarily mean '\(\rho=1\)' because the sample may not perfectly represent the population.

Population Correlation Coefficient \(\rho=1\)

The population correlation coefficient '\(\rho\)' equals 1 when there is a perfect positive correlation between two variables across the entire population. If this is the case, any sample taken from the population should also show a perfect positive correlation, thus, 'r' should also be equal to 1.