To calculate the coefficient of correlation (denoted as $r$) between two variables $X$ and $Y$, we use the formula:

[ r = \frac{n \sum{XY} - \sum{X} \sum{Y}}{\sqrt{[n \sum{X^2} - (\sum{X})^2][n \sum{Y^2} - (\sum{Y})^2]}} ]

Where:

• $n$ is the number of data points
• $\sum{X}$ is the sum of all values in $X$
• $\sum{Y}$ is the sum of all values in $Y$
• $\sum{XY}$ is the sum of the product of corresponding values of $X$ and $Y$
• $\sum{X^2}$ is the sum of the squares of the values in $X$
• $\sum{Y^2}$ is the sum of the squares of the values in $Y$

Let's first calculate the necessary sums for the given data:

Data:

• $X = [1, 2, 3, 4, 5]$
• $Y = [2, 4, 6, 8, 10]$

Step 1: Calculate the sums

• $\sum{X} = 1 + 2 + 3 + 4 + 5 = 15$
• $\sum{Y} = 2 + 4 + 6 + 8 + 10 = 30$
• $\sum{X^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$
• $\sum{Y^2} = 2^2 + 4^2 + 6^2 + 8^2 + 10^2 = 4 + 16 + 36 + 64 + 100 = 220$
• $\sum{XY} = 1 \cdot 2 + 2 \cdot 4 + 3 \cdot 6 + 4 \cdot 8 + 5 \cdot 10 = 2 + 8 + 18 + 32 + 50 = 110$

Step 2: Substitute these values into the correlation formula

Using $n = 5$ (since there are 5 data points):

[ r = \frac{5 \cdot 110 - 15 \cdot 30}{\sqrt{[5 \cdot 55 - 15^2][5 \cdot 220 - 30^2]}} ]

Now, let's break down the calculations step-by-step:

1. Numerator: $5 \cdot 110 - 15 \cdot 30 = 550 - 450 = 100$

2. Denominator: $\text{First part of denominator: } 5 \cdot 55 - 15^2 = 275 - 225 = 50$ $\text{Second part of denominator: } 5 \cdot 220 - 30^2 = 1100 - 900 = 200$ Now multiply these two results: $\sqrt{50 \cdot 200} = \sqrt{10000} = 100$

Step 3: Final calculation

$r = \frac{100}{100} = 1$

Conclusion:

The coefficient of correlation $r$ between the given data sets $X$ and $Y$ is 1, indicating a perfect positive linear relationship between the two variables.

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Would you like more details or have any questions on this?

Here are some related questions:

1. How would the correlation coefficient change if the relationship between $X$ and $Y$ were negative?
2. How do outliers affect the coefficient of correlation?
3. What is the interpretation of a correlation coefficient of 0?
4. How can you calculate the correlation coefficient using software like Excel or Python?
5. What are the limitations of using the coefficient of correlation in statistical analysis?

Tip: A correlation coefficient of $1$ (or $-1$) signifies a perfect linear relationship, but real-world data often shows some degree of noise, leading to values less than 1.