Correlation coefficient is a number between:
(1) +1 and +2
(2) 0 and +1
(3) -1 and 0
(4) -1 and +1

📘 Problem Statement

The correlation coefficient is a measure used in statistics to determine the relationship between two variables. It is essential for understanding the direction and strength of the correlation.

Which of the following is the correct range for the correlation coefficient?

Options:

1. +1 and +2

2. 0 and +1

3. -1 and 0

4. -1 and +1

🔬 What is the Correlation Coefficient?

The correlation coefficient is a statistical measure that quantifies the degree to which two variables are related. It tells you how strongly the variables move together and in which direction. This value is represented by r and ranges from -1 to +1. The sign and the absolute value of r give us important information:

• Positive correlation (+1): Both variables increase or decrease together.

• Negative correlation (-1): As one variable increases, the other decreases.

• No correlation (0): There is no relationship between the two variables.

📏 Range of the Correlation Coefficient

The correlation coefficient (r) has the following possible range:

• +1 indicates a perfect positive correlation. As one variable increases, the other increases in perfect proportion.

• -1 indicates a perfect negative correlation. As one variable increases, the other decreases in perfect proportion.

• 0 indicates no correlation. The variables do not move together in any discernible pattern.

Thus, the correlation coefficient is always between -1 and +1.

🧠 Breakdown of the Answer Choices:

1. +1 and +2:
   This is incorrect. The correlation coefficient cannot exceed +1 or go beyond +2. The correct range is between -1 and +1.

2. 0 and +1:
   While this describes a positive correlation where the variables are positively related, it does not include the negative correlation range of -1. Hence, this is not the full range.

3. -1 and 0:
   This option describes a negative correlation, but it does not reflect the full possible range, which also includes positive correlations. Hence, it’s incomplete.

4. -1 and +1:
   This is the correct range of the correlation coefficient. It encompasses both positive and negative correlations, as well as no correlation (0).

✅ Correct Answer: (4) -1 and +1

The correlation coefficient always lies between -1 and +1. It indicates the strength and direction of the relationship between two variables.

💡 Why This Concept Matters

Understanding the correlation coefficient is crucial for:

• Data Analysis: Identifying and quantifying relationships between variables.

• Predictive Modeling: Establishing whether one variable can be used to predict another.

• Research and Experimentation: Determining how variables interact in various fields like economics, medicine, psychology, etc.

✅ Key Takeaways

• The correlation coefficient ranges from -1 to +1.

• +1 means a perfect positive relationship, -1 means a perfect negative relationship, and 0 means no relationship.

• This measure is fundamental in understanding the strength and direction of relationships in statistics