To compute the Pearson correlation coefficient $r$ between two sets of numbers, we use the following formula:

[ r = \frac{n \sum{xy} - \sum{x} \sum{y}}{\sqrt{[n \sum{x^2} - (\sum{x})^2][n \sum{y^2} - (\sum{y})^2]}} ]

Where:

• $n$ is the number of paired scores
• $x$ and $y$ are the individual values in the two datasets
• $\sum$ denotes the summation over the values

Let me calculate this step by step:

Given Data:

x values: $24.7, 15.2, 12.3, 10.9, 10.8, 9.9, 8.3, 7.2, 6.6, 5.8, 5.7, 5.6, 4.2, 3.9, 3.1$

y values: $46.1, 23.6, 23.7, 7, 12.3, 14.2, 7.4, 3.0, 7.2, 10.6, 3.7, 3.4, 4.3, 3.6, 5.4$

Now, let me calculate the Pearson correlation coefficient $r$ and then determine the significance. numpy as np from scipy.stats import pearsonr

Given x and y data

x = np.array([24.7, 15.2, 12.3, 10.9, 10.8, 9.9, 8.3, 7.2, 6.6, 5.8, 5.7, 5.6, 4.2, 3.9, 3.1]) y = np.array([46.1, 23.6, 23.7, 7, 12.3, 14.2, 7.4, 3.0, 7.2, 10.6, 3.7, 3.4, 4.3, 3.6, 5.4])

Calculate Pearson correlation coefficient and p-value for significance

r, p_value = pearsonr(x, y) r, p_valueThe Pearson correlation coefficient ($r$) between the two datasets is approximately 0.939. This value indicates a strong positive linear correlation between the x and y values, meaning that as one variable increases, the other tends to increase as well.

The p-value associated with this correlation is approximately 2.2 × 10⁻⁷. Since this p-value is extremely small (much lower than the typical significance level of 0.05), we can conclude that the correlation is statistically significant. In other words, the observed relationship between x and y is highly unlikely to have occurred by chance.

Interpretation:

• The positive value of $r$ suggests a strong, direct relationship between the variables.
• Given the p-value is very small, we can confidently say that this correlation is statistically significant and not due to random variation.

Let me know if you would like further details or have any additional questions.

Related Questions:

1. How does the Pearson correlation coefficient relate to the strength and direction of a linear relationship?
2. What does a p-value indicate about the significance of a statistical result?
3. What are other methods for measuring the relationship between two variables?
4. How can the Pearson correlation be interpreted in different fields, like economics or biology?
5. How does sample size affect the significance of the Pearson correlation coefficient?

Tip: Always ensure that the data being analyzed meets the assumptions of Pearson’s correlation (e.g., linearity, normality, homoscedasticity) for the results to be reliable.