To find the linear correlation coefficient (also known as Pearson's correlation coefficient, denoted as r), follow these steps:

Steps to Calculate Linear Correlation Coefficient:

1. Gather Data:

   • You need two sets of data, let's call them $X = \{x_1, x_2, ..., x_n\}$ and $Y = \{y_1, y_2, ..., y_n\}$, where $n$ is the number of data points.

2. Calculate the Mean of $X$ and $Y$:

   • $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$
   • $\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i$

3. Calculate the Numerator:

   • Compute the covariance between $X$ and $Y$: $\text{Cov}(X, Y) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$

4. Calculate the Denominator:

   • Compute the standard deviation of $X$ and $Y$: $\sigma_x = \sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2}$ $\sigma_y = \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}$

5. Compute the Correlation Coefficient:

   • The Pearson correlation coefficient is given by: $r = \frac{\text{Cov}(X, Y)}{\sigma_x \sigma_y}$ This value will range between $-1$ and $1$:
   • $r = 1$: Perfect positive correlation.
   • $r = -1$: Perfect negative correlation.
   • $r = 0$: No linear correlation.

Example:

Suppose you have two sets of dataTo find the linear correlation coefficient (also known as Pearson's correlation coefficient, denoted as r), follow these steps:

Steps to Calculate Linear Correlation Coefficient:

1. Gather Data:

   • You need two sets of data, let's call them $X = \{x_1, x_2, ..., x_n\}$ and $Y = \{y_1, y_2, ..., y_n\}$, where $n$ is the number of data points.

2. Calculate the Mean of $X$ and $Y$:

   • $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$
   • $\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i$

3. Calculate the Numerator:

   • Compute the covariance between $X$ and $Y$: $\text{Cov}(X, Y) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$

4. Calculate the Denominator:

   • Compute the standard deviation of $X$ and $Y$: $\sigma_x = \sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2}$ $\sigma_y = \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}$

5. Compute the Correlation Coefficient:

   • The Pearson correlation coefficient is given by: $r = \frac{\text{Cov}(X, Y)}{\sigma_x \sigma_y}$ This value will range between $-1$ and $1$:
   • $r = 1$: Perfect positive correlation.
   • $r = -1$: Perfect negative correlation.
   • $r = 0$: No linear correlation.

Example:

Suppose you have two sets of

• $X = \{1, 2, 3, 4, 5\}$
• $Y = \{2, 4, 6, 8, 10\}$

Follow the steps to calculate $r$ and it will give you a value of $r = 1$, indicating a perfect positive correlation.

Would you like a worked-out example or any clarifications?

5 Related Questions:

1. How does the correlation coefficient differ from covariance?
2. What does a correlation coefficient of 0.5 indicate?
3. How do outliers affect the value of the correlation coefficient?
4. Can you calculate the correlation coefficient for non-linear relationships?
5. How is the correlation coefficient used in regression analysis?

Tip: A correlation coefficient close to 0 does not always mean no relationship; it could indicate a non-linear relationship.