Calculating Pearson's r, also known as the Pearson correlation coefficient, helps us understand how two sets of data are related. Let's break this down into easy steps.

Step 1: Gather Your Data

Start by collecting two sets of information. For example:

• X (like Hours Studied): [1, 2, 3, 4, 5]
• Y (like Exam Scores): [50, 60, 65, 70, 80]

Step 2: Calculate the Averages

Next, find the average (mean) for both sets of data:

• Mean of X: $\text{Mean of X} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3$

• Mean of Y: $\text{Mean of Y} = \frac{50 + 60 + 65 + 70 + 80}{5} = 65$

Step 3: Find Deviations

Now, for each number, subtract the mean from the value to see how far each point is from the average:

• For X:

  • 1: $1 - 3 = -2$
  • 2: $2 - 3 = -1$
  • 3: $3 - 3 = 0$
  • 4: $4 - 3 = 1$
  • 5: $5 - 3 = 2$

• For Y:

  • 50: $50 - 65 = -15$
  • 60: $60 - 65 = -5$
  • 65: $65 - 65 = 0$
  • 70: $70 - 65 = 5$
  • 80: $80 - 65 = 15$

Step 4: Multiply Deviations

Next, multiply each pair of deviations:

• $(-2)(-15) = 30$
• $(-1)(-5) = 5$
• $(0)(0) = 0$
• $(1)(5) = 5$
• $(2)(15) = 30$

Step 5: Add Them Up

Now, add all these products together: $30 + 5 + 0 + 5 + 30 = 70$

Step 6: Calculate Squared Deviations

Let’s square the deviations for each set:

• For X: $\text{Squared Deviations} = 4 + 1 + 0 + 1 + 4 = 10$

• For Y: $\text{Squared Deviations} = 225 + 25 + 0 + 25 + 225 = 500$

Step 7: Calculate Pearson's r

Finally, use the formula for Pearson's r: $r = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sqrt{\sum{(X_i - \bar{X})^2} \sum{(Y_i - \bar{Y})^2}}}$

Plugging in what we found: $r = \frac{70}{\sqrt{10 \cdot 500}} = \frac{70}{\sqrt{5000}} \approx 0.99$

This means there is a strong positive relationship between hours studied and exam scores! While calculating Pearson's r by hand can take some time, it’s a great way to learn about statistics.