Coefficient of Correlation Calculation:

To determine the coefficient of correlation between the x and y series, we need to calculate the necessary values and apply the formula for correlation. Let's break down the calculation step by step:

1. Calculate the mean for x and y series:
- Mean of x series (mean_x) = sum of x series / number of items in x series
= 136 / 15
= 9.07
- Mean of y series (mean_y) = sum of y series / number of items in y series
= 138 / 15
= 9.2

2. Calculate the deviation of each value in the x and y series from their respective means:
- Deviation of x series (d_x) = x - mean_x
- Deviation of y series (d_y) = y - mean_y

3. Calculate the sum of the square of deviations of x and y series from their means:
- Sum of the square of deviation of x series (sum_sq_dev_x) = Σ(d_x^2)
= (d_x1^2 + d_x2^2 + ... + d_x15^2)
- Sum of the square of deviation of y series (sum_sq_dev_y) = Σ(d_y^2)
= (d_y1^2 + d_y2^2 + ... + d_y15^2)

4. Calculate the sum of the product of deviations of x and y series from their means:
- Sum of the product of deviation of x and y series (sum_prod_dev_xy) = Σ(d_x * d_y)
= (d_x1 * d_y1 + d_x2 * d_y2 + ... + d_x15 * d_y15)
= 122

5. Calculate the coefficient of correlation (r):
- Coefficient of correlation (r) = sum_prod_dev_xy / sqrt(sum_sq_dev_x * sum_sq_dev_y)

Explanation:

The coefficient of correlation measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 represents a perfect negative correlation, 0 represents no correlation, and +1 represents a perfect positive correlation.

In this case, we have calculated the mean values for both x and y series. Then, we calculated the deviation of each value in the x and y series from their respective means. By squaring these deviations and summing them up, we obtained the sum of the square of deviations for both x and y series.

Additionally, we calculated the sum of the product of deviations of x and y series, which represents the covariance between the two variables.

Finally, we applied the formula for correlation, dividing the sum of the product of deviations by the square root of the product of the sum of square deviations for x and y series. This gave us the coefficient of correlation (r).

By substituting the given values into the formula, we can calculate the coefficient of correlation between the x and y series.