Consider the following data and find Co variance and Coefficient of Correlation.
X 15 29 87 11 43
Y 34 10 30 22 50

To calculate **Covariance** and the **Coefficient of Correlation**, let's proceed step by step:

### Given data:
| X   | 15 | 29 | 87 | 11 | 43 |
|-----|----|----|----|----|----|
| Y   | 34 | 10 | 30 | 22 | 50 |

---

### 1. Formulas:

#### Covariance:
\[
\text{Cov}(X, Y) = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{n}
\]
where:
- \( X_i, Y_i \) are individual data points.
- \( \bar{X}, \bar{Y} \) are the means of \( X \) and \( Y \), respectively.
- \( n \) is the number of data points.

#### Coefficient of Correlation (\( r \)):
\[
r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}
\]
where:
- \( \sigma_X, \sigma_Y \) are the standard deviations of \( X \) and \( Y \).

---

### 2. Calculate Means (\( \bar{X} \) and \( \bar{Y} \)):

\[
\bar{X} = \frac{\sum{X}}{n} = \frac{15 + 29 + 87 + 11 + 43}{5} = \frac{185}{5} = 37
\]
\[
\bar{Y} = \frac{\sum{Y}}{n} = \frac{34 + 10 + 30 + 22 + 50}{5} = \frac{146}{5} = 29.2
\]

---

### 3. Calculate Deviations and Cross Products:

| \( X_i \) | \( Y_i \) | \( X_i - \bar{X} \) | \( Y_i - \bar{Y} \) | \( (X_i - \bar{X})(Y_i - \bar{Y}) \) |
|-----------|-----------|---------------------|---------------------|-------------------------------------|
| 15        | 34        | -22                | 4.8                 | -105.6                              |
| 29        | 10        | -8                 | -19.2               | 153.6                               |
| 87        | 30        | 50                 | 0.8                 | 40                                  |
| 11        | 22        | -26                | -7.2                | 187.2                               |
| 43        | 50        | 6                  | 20.8                | 124.8                               |

\[
\sum (X_i - \bar{X})(Y_i - \bar{Y}) = -105.6 + 153.6 + 40 + 187.2 + 124.8 = 400
\]

---

### 4. Covariance:
\[
\text{Cov}(X, Y) = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{n} = \frac{400}{5} = 80
\]

---

### 5. Standard Deviations (\( \sigma_X \) and \( \sigma_Y \)):

\[
\sigma_X = \sqrt{\frac{\sum{(X_i - \bar{X})^2}}{n}} \quad \text{and} \quad \sigma_Y = \sqrt{\frac{\sum{(Y_i - \bar{Y})^2}}{n}}
\]

**For \( X \):**
\[
\sum{(X_i - \bar{X})^2} = (-22)^2 + (-8)^2 + 50^2 + (-26)^2 + 6^2 = 484 + 64 + 2500 + 676 + 36 = 3760
\]
\[
\sigma_X = \sqrt{\frac{3760}{5}} = \sqrt{752} \approx 27.42
\]

**For \( Y \):**
\[
\sum{(Y_i - \bar{Y})^2} = (4.8)^2 + (-19.2)^2 + (0.8)^2 + (-7.2)^2 + (20.8)^2
\]
\[
= 23.04 + 368.64 + 0.64 + 51.84 + 432.64 = 876.8
\]
\[
\sigma_Y = \sqrt{\frac{876.8}{5}} = \sqrt{175.36} \approx 13.24
\]

---

### 6. Coefficient of Correlation:
\[
r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} = \frac{80}{27.42 \cdot 13.24} \approx \frac{80}{363.12} \approx 0.22
\]

---

### Final Results:
1. **Covariance**: \( \text{Cov}(X, Y) = 80 \)
2. **Coefficient of Correlation**: \( r \approx 0.22 \)

---

Do you have any questions or want a deeper explanation?  
Here are 5 follow-up questions:  

1. How do you interpret the positive covariance value?  
2. Why is the correlation coefficient only 0.22 despite the positive covariance?  
3. Can covariance be compared across different datasets?  
4. How would the values change if all \( X \) and \( Y \) values were doubled?  
5. What does the sign of \( r \) indicate about the relationship between \( X \) and \( Y \)?  

**Tip**: Covariance measures the directional relationship, while the correlation coefficient standardizes it for comparison.

Consider the following data and find Co variance and Coefficient of Correlation. X 15 29 87 11 43 Y 34 10 30 22 50

Learn how to calculate the covariance and correlation coefficient between two datasets X = [15, 29, 87, 11, 43] and Y = [34, 10, 30, 22, 50]. This step-by-step guide covers key statistical concepts like mean, covariance, and correlation, suitable for students in Grades 11-12.

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