The correlation coefficient ($r$) measures the strength and direction of a linear relationship between two variables. When the chart provides the sums of $x^2$, $y^2$, $xy$, and other values (like $\Sigma x$ and $\Sigma y$), you can calculate $r$ using the formula:

$r = \frac{n\Sigma xy - \Sigma x \Sigma y}{\sqrt{\left(n\Sigma x^2 - (\Sigma x)^2\right)\left(n\Sigma y^2 - (\Sigma y)^2\right)}}$

Steps:

1. Identify values: Gather the following from the chart:

   • $n$: Number of data points
   • $\Sigma x$: Sum of all $x$ values
   • $\Sigma y$: Sum of all $y$ values
   • $\Sigma xy$: Sum of the product of each $x$ and $y$
   • $\Sigma x^2$: Sum of the squares of $x$
   • $\Sigma y^2$: Sum of the squares of $y$

2. Substitute values into the formula:

   • Calculate the numerator: $n\Sigma xy - \Sigma x \Sigma y$
   • Calculate the denominator: $\sqrt{\left(n\Sigma x^2 - (\Sigma x)^2\right)\left(n\Sigma y^2 - (\Sigma y)^2\right)}$

3. Compute $r$: Divide the numerator by the denominator to find the correlation coefficient.

Example:

Suppose:

• $n = 5$,
• $\Sigma x = 15$,
• $\Sigma y = 20$,
• $\Sigma xy = 80$,
• $\Sigma x^2 = 55$,
• $\Sigma y^2 = 100$.

Step-by-step:

1. Numerator: $n\Sigma xy - \Sigma x \Sigma y = 5(80) - (15)(20) = 400 - 300 = 100$

2. Denominator: $\sqrt{\left(n\Sigma x^2 - (\Sigma x)^2\right)\left(n\Sigma y^2 - (\Sigma y)^2\right)}$ Calculate each term:

   • $n\Sigma x^2 - (\Sigma x)^2 = 5(55) - (15)^2 = 275 - 225 = 50$
   • $n\Sigma y^2 - (\Sigma y)^2 = 5(100) - (20)^2 = 500 - 400 = 100$ So, the denominator is: $\sqrt{50 \cdot 100} = \sqrt{5000} \approx 70.71$

3. Correlation coefficient: $r = \frac{100}{70.71} \approx 1.41$

If you receive a number like $r > 1$ or $r < -1$, double-check your calculations, as $r$ should always lie between $-1$ and $1$.

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Would you like me to demonstrate on your own data or explain more? 😊

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5 Related Questions:

1. What does the correlation coefficient tell us about a dataset?
2. How is the coefficient of determination ($r^2$) related to $r$?
3. How does changing one data point affect $r$?
4. Can you calculate $r$ without a chart? If so, how?
5. What assumptions are made when using the Pearson correlation coefficient?

Tip: Always plot your data first to check for linear relationships before computing $r$.