To calculate the correlation coefficient ($r$) between $x$ and $y$, we use the formula:

$r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \cdot \sum{(y_i - \bar{y})^2}}}$

Step-by-step process:

1. Compute the means of $x$ and $y$: $\bar{x} = \frac{\sum{x_i}}{n}, \quad \bar{y} = \frac{\sum{y_i}}{n}$

2. Calculate deviations from the mean: For each $x_i$ and $y_i$, calculate: $(x_i - \bar{x}) \quad \text{and} \quad (y_i - \bar{y})$

3. Compute the cross-product of deviations and square of deviations: Calculate: $(x_i - \bar{x})(y_i - \bar{y}), \quad (x_i - \bar{x})^2, \quad \text{and} \quad (y_i - \bar{y})^2$

4. Substitute into the correlation formula:

   • Numerator: Sum of $(x_i - \bar{x})(y_i - \bar{y})$
   • Denominator: Square root of the product of $\sum{(x_i - \bar{x})^2}$ and $\sum{(y_i - \bar{y})^2}$

Let's calculate this step-by-step! numpy as np

Data points

x = np.array([5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]) y = np.array([46.95, 43.7, 40.75, 36.2, 34.35, 32.6, 30.65, 27.1, 25.65, 20.3, 18.65, 17.2])

Calculate means

mean_x = np.mean(x) mean_y = np.mean(y)

Calculate deviations

x_deviation = x - mean_x y_deviation = y - mean_y

Calculate terms

numerator = np.sum(x_deviation * y_deviation) denominator = np.sqrt(np.sum(x_deviation2) * np.sum(y_deviation2))

Correlation coefficient

r = numerator / denominator r_rounded = round(r, 3) r_roundedThe correlation coefficient $r$ between $x$ and $y$ is approximately $-0.995$. This indicates a very strong negative linear relationship between the two variables.

Would you like further explanation or details about the calculation? Here are some related questions for exploration:

1. How does a correlation coefficient of $-0.995$ affect predictions between $x$ and $y$?
2. Can we derive the line of best fit for these data points?
3. What would happen to $r$ if the data included significant outliers?
4. How does this result compare to a perfect correlation of $-1$?
5. What are the limitations of correlation analysis?

Tip: Always visualize the data with a scatter plot to confirm the linearity of the relationship when interpreting $r$.