To identify the true statements about the correlation coefficient $r$, let’s examine each statement carefully:

1. The value of $r$ ranges from negative one to positive one.

   • This is true. The correlation coefficient $r$ can only take values between -1 and 1, inclusive.

2. The sign of $r$ describes the direction of the association between two variables.

   • This is true. A positive $r$ indicates a positive association (as one variable increases, the other also increases), while a negative $r$ indicates a negative association (as one variable increases, the other decreases).

3. The correlation coefficient is not affected by outliers.

   • This is false. Outliers can have a significant impact on the value of the correlation coefficient, often distorting the perceived strength and direction of the association.

4. When the data points in a scatter plot fall closely around a straight line that is either increasing or decreasing, the correlation between the two variables is strong.

   • This is true. When data points cluster tightly around a straight line, the correlation is close to -1 or 1, indicating a strong relationship.

5. If two variables are positively correlated, when one variable increases, the other variable decreases.

   • This is false. Positive correlation means that as one variable increases, the other also increases.

6. A correlation coefficient of zero means that no relationship exists between the two variables.

   • This is partially true but requires clarification. A correlation coefficient of zero indicates no linear relationship, but there might still be a non-linear relationship between the variables.

Summary of True Statements:

• Statement 1: True
• Statement 2: True
• Statement 4: True
• Statement 6: Partially true but requires caution

Would you like more details on any of these points or have questions on correlation? Here are some related questions you might find helpful:

1. What are the effects of outliers on correlation coefficients?
2. Can a correlation of zero indicate a different type of relationship?
3. How does the correlation coefficient differ from causation?
4. What is the difference between positive and negative correlation?
5. How does the correlation coefficient apply in different fields like economics or psychology?

Tip: Remember, correlation does not imply causation; it only describes the strength and direction of the linear relationship between two variables.