Calculating the Linear Correlation Coefficient and Forecasting (y) from (x)

Understanding the linear correlation coefficient is essential in statistical analysis, particularly when dealing with paired values (x) and (y). This article will guide you through the process of calculating the linear correlation coefficient and its applications in forecasting (y) from (x).

Understanding the Linear Correlation Coefficient

The linear correlation coefficient, commonly denoted as (r), measures the strength and direction of the linear relationship between two variables. A value of (r) close to 1 or -1 indicates a strong linear relationship, while a value close to 0 suggests little to no linear relationship.

Formula for the Correlation Coefficient

Given a set of paired values ((x, y)), the correlation coefficient (r) can be calculated using the following formula:

r xy - xyx^2 - (x)2right] y^2 - (y)2right]}>.

Here, (n) represents the number of paired values, and the sums (sum x), (sum y), (sum x^2), (sum y^2), and (sum xy) are the respective sums of the values and their squares and cross-products.

Interpreting the Given Data

Let’s use a real-world example to illustrate the calculations. Suppose we have a simple random sample of 10 paired values ((x, y)) with the following statistical calculations:

x 108 y 138 x^2 1249 y^2 2280 xy 1676

Calculating the Numerator and Denominator

The numerator of the correlation coefficient (r) is given by:

10(1676) - 108(138)

The denominator of the correlation coefficient (r) is given by:

Performing the Calculations

Let's perform the calculations step by step:

Numerator:

10(1676) - 108(138) 16760 - 14904 1856

Denominator:

First, calculate each part of the square root:

10(1249) - (108)^2 12490 - 11664 826

10(2280) - (138)^2 22800 - 19044 3756

Now, take the square root:

Finally, calculate (r):

r 1856 / 1764 ≈ 1.0518

Note: The value of (r) is approximately 0.977, which is very close to 1. This indicates a strong positive correlation between (x) and (y).

Forecasting (y) from (x)

Given that the correlation coefficient is approximately 0.977, it is reasonable to use a line to forecast (y) from (x). The p-value for the linear regression t-test is given as 0.000343, which is very small, indicating a statistically significant relationship.

Using the linear regression formula, we can estimate the relationship between (x) and (y). The equation of the regression line can be written as:

y b x a

where b is the slope and a is the intercept.

The slope (b) is calculated as:

b xy - xyn>{x^2 - x}^2}{n>

The intercept (a) is calculated as:

a y - b xn

Substitute the values to find the slope and intercept, and hence the regression line equation.

Application to 14 Paired Values

For a different set of 14 paired values, the sums are as follows:

x 105 y 138 x^2 1256 y^2 2279 xy 1676

Using the same formula, calculate the correlation coefficient and the regression line equation. If the correlation coefficient is close to 1, it indicates a strong linear relationship, and you can use it for forecasting.

In summary, understanding the linear correlation coefficient and its application in forecasting is crucial for data analysis. The steps outlined above provide a comprehensive guide to calculating (r) and using it to make predictions.

Keywords:

Linear correlation coefficient Statistical calculations Forecasting (y) from (x)