16.6. Worked Example

Given two datasets:

• 𝑋 = {2,4,6,8}
• 𝑌 = {1,3,5,7} Find the Covariance and Correlation.

This section introduces a practical example involving two datasets, X and Y. The goal is to calculate the covariance and correlation between these two sets of numbers. Knowing the covariance and correlation helps us understand the relationship between the two datasets — whether they move together, in opposite directions, or not at all.

Imagine two friends who tend to go out together on weekends. Their joint activity shows how one friend's decisions might affect the other. Similarly, by calculating covariance and correlation, we can see how one dataset (like one friend's activities) influences the other.

𝑋‾ = (2 + 4 + 6 + 8) / 4 = 5 𝑌‾ = (1 + 3 + 5 + 7) / 4 = 4

In this step, we calculate the means (averages) of both datasets. The mean of X is calculated as the sum of its values divided by the number of values (4 in this case). The same calculation is applied to Y. Finding the mean helps establish a central point around which the other values can be compared.

Think of the mean like the average score in a class. If students have varying scores, the average gives us a sense of how well the class is performing overall.

Cov(𝑋,𝑌) = [ (2−5)(1−4) + (4−5)(3−4) + (6−5)(5−4) + (8−5)(7−4) ] / 4 = (1/4) [(−3)(−3) + (−1)(−1) + (1)(1) + (3)(3)] = (1/4) [9 + 1 + 1 + 9] = 5

Here, covariance is calculated using the means obtained in the previous step. Each term in the covariance formula considers how much each pair of values (one from X and one from Y) deviates from their respective means. A positive covariance indicates that when one dataset increases, the other tends to increase as well.

Consider two plants growing together: if one gets more sunlight (increasing), the other might also flourish (indicating positive covariance), but if one wilts while the other thrives, that's a negative relationship reflected in negative covariance.

Standard Deviation for X: 𝜎𝑋 = √[ (2−5)² + (4−5)² + (6−5)² + (8−5)² ] / 4 = √(9 + 1 + 1 + 9) / 4 = √5 Standard Deviation for Y: 𝜎𝑌 = √[ (1−4)² + (3−4)² + (5−4)² + (7−4)² ] / 4 = √(9 + 1 + 1 + 9) / 4 = √5

In this step, we find the standard deviations of both datasets X and Y. The process involves calculating the squared deviations of each value from the mean, summing those, averaging them, and then taking the square root. The result provides insight into how spread out the values are in each dataset.

Think of standard deviation like measuring how varied the heights of students are in a classroom. If everyone is roughly the same height, the standard deviation is low. If there's a wide variation, the standard deviation is high.

Corr(𝑋,𝑌) = Cov(𝑋,𝑌) / (𝜎𝑋 * 𝜎𝑌) = 5 / (√5 * √5) = 1

The final step involves calculating the correlation using the previously found covariance and standard deviations. The correlation gives us a scaled measure of the relationship. Since the result is 1, it indicates a perfect positive linear relationship between datasets X and Y.

Picture two friends who always hold hands when walking; they are perfectly in sync (correlation of 1). If one pulls ahead, the other does as well, showcasing the strong connection between their behaviors.

🔎 Interpretation: Perfect positive linear relationship between 𝑋 and 𝑌.

This interpretation summarizes the findings: since the correlation is 1, we understand that whenever X increases, Y does too, in a perfectly linear manner. It reinforces the consistency in how both datasets relate to each other.

Imagine two cars traveling on a straight highway at constant speed. If the first car speeds up and maintains its speed, the second car unerringly follows suit, depicting a perfect correlation. Their relationship is predictable and reliable.

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