Result Details & Calculation

X Values
∑ = 804.3
Mean = 53.62
∑(X - Mx)2 = SSx = 15194.724

Y Values
∑ = 16907.5
Mean = 1127.167
∑(Y - My)2 = SSy = 199520070.413

X and Y Combined
N = 15
∑(X - Mx)(Y - My) = 1614964.21

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = 1614964.21 / √((15194.724)(199520070.413)) = 0.9275

Meta Numerics (cross-check)
r = 0.9275
What is the correlation coefficient without the outlier?
rwo =

The correlation coefficient \( r \) you calculated with the given data is \( r = 0.9275 \). To calculate the correlation coefficient without the outlier, we would need to identify the outlier and then remove its effect on the sums and products used in the correlation calculation.

Let's assume that you have identified and removed the outlier. To calculate the new correlation coefficient \( r_{wo} \) (correlation without the outlier), the following steps would be needed:

### Steps to Recalculate the Correlation Coefficient Without the Outlier:
1. **Adjust the sums** \( \sum X \) and \( \sum Y \) by removing the values of the outlier.
2. **Recalculate the means** \( M_x \) and \( M_y \) without the outlier:
   \[
   M_x = \frac{\sum X}{N-1}, \quad M_y = \frac{\sum Y}{N-1}
   \]
3. **Recalculate the sum of squares** \( SS_x \) and \( SS_y \) without the outlier:
   \[
   SS_x = \sum (X - M_x)^2, \quad SS_y = \sum (Y - M_y)^2
   \]
4. **Recalculate the combined sum** \( \sum (X - M_x)(Y - M_y) \) without the outlier.
5. **Calculate the new correlation coefficient** \( r_{wo} \) using the formula:
   \[
   r_{wo} = \frac{\sum (X - M_x)(Y - M_y)}{\sqrt{SS_x \times SS_y}}
   \]

Since you haven't provided the specific data points or which one is the outlier, I cannot perform these steps exactly. However, if you can identify the outlier or provide the data, I can help you with the detailed calculations.

Would you like more details on identifying and removing the outlier, or do you have any specific data points to work with?

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### Follow-up Questions:
1. How would you identify an outlier in a dataset?
2. What happens to the mean of a dataset when you remove an outlier?
3. How does removing an outlier generally affect the correlation coefficient?
4. What are some methods to detect outliers in a dataset?
5. How would the sum of squares \( SS_x \) change if you removed the highest value?
6. What is the effect of outliers on the standard deviation?
7. Can removing an outlier ever decrease the correlation coefficient?
8. What is the difference between Pearson's correlation and Spearman's correlation?

**Tip:** When dealing with outliers, consider plotting the data to visually inspect their effect on the correlation.

Learn how to calculate the correlation coefficient with and without outliers using Pearson's formula. Understand the impact of outliers on statistical analysis with a step-by-step explanation. Suitable for advanced high school or college-level students.

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