# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/statistics/product-moment-correlation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/statistics/product-moment-correlation/en.mdx

Output docs content for large language models.

---

import { ScatterDiagram } from "@repo/design-system/components/contents/scatter-diagram";

export const metadata = {
  title: "Product Moment Correlation",
  description: "Calculate Pearson's correlation coefficient (r) to measure linear relationships. Learn formulas, scatter plot analysis, and interpret r values from -1 to +1.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/30/2025",
  subject: "Statistics",
};

## What Is Product Moment Correlation?

Product Moment Correlation, often called **Pearson Correlation** or simply denoted by **<InlineMath math="r" />**, is the most commonly used statistical measure to determine how strong and in what direction the **linear relationship** (straight-line pattern) is between two quantitative variables (numbers).

The value of <InlineMath math="r" /> tells us whether the two variables tend to move in the same direction (positive), opposite directions (negative), or if there is no linear relationship at all.

## Correlation from Scatter Diagrams

The most intuitive way to understand the value of <InlineMath math="r" /> is by looking at how the data points are scattered on a diagram:

### Strong Positive Correlation

<InlineMath math="r" /> approaching +1 means both variables tend to move in the same
direction.

<ScatterDiagram
  title="Example of Strong Positive Correlation"
  description="Data points cluster very closely forming an upward-sloping straight line pattern."
  xAxisLabel="Variable X"
  yAxisLabel="Variable Y"
  datasets={[
    {
      name: "Data",
      color: "var(--chart-1)",
      points: [
        { x: 1, y: 2 },
        { x: 2, y: 3.1 },
        { x: 3, y: 3.9 },
        { x: 4, y: 5.2 },
        { x: 5, y: 6.1 },
        { x: 6, y: 6.8 },
        { x: 7, y: 8.1 },
        { x: 8, y: 9.0 },
      ],
    },
  ]}
  calculateRegressionLine
  showResiduals
  regressionLineStyle={{ color: "var(--chart-4)" }}
/>

If your data points look like this (rising from bottom left to top right and tightly clustered), the <InlineMath math="r" /> value will be close to +1.

### Weak Positive Correlation

<InlineMath math="r" /> being positive but close to 0 means both variables tend to
move in the same direction, but not very strongly.

<ScatterDiagram
  title="Example of Weak Positive Correlation"
  description="Points tend to rise, but are more spread out from the straight line."
  xAxisLabel="Variable X"
  yAxisLabel="Variable Y"
  datasets={[
    {
      name: "Data",
      color: "var(--chart-2)",
      points: [
        { x: 1, y: 1 },
        { x: 2, y: 4 },
        { x: 3, y: 2 },
        { x: 4, y: 6 },
        { x: 5, y: 5 },
        { x: 6, y: 8 },
        { x: 7, y: 6 },
        { x: 8, y: 9 },
      ],
    },
  ]}
  calculateRegressionLine
  showResiduals
  regressionLineStyle={{ color: "var(--chart-4)" }}
/>

If the points still show an upward trend but are more scattered like this, the <InlineMath math="r" /> value is positive but smaller (closer to 0).

### Strong Negative Correlation

<InlineMath math="r" /> approaching -1 means both variables tend to move in opposite
directions.

<ScatterDiagram
  title="Example of Strong Negative Correlation"
  description="Data points cluster very closely forming a downward-sloping straight line pattern."
  xAxisLabel="Variable X"
  yAxisLabel="Variable Y"
  datasets={[
    {
      name: "Data",
      color: "var(--chart-3)",
      points: [
        { x: 1, y: 10 },
        { x: 2, y: 8.9 },
        { x: 3, y: 8.1 },
        { x: 4, y: 6.8 },
        { x: 5, y: 6.0 },
        { x: 6, y: 5.1 },
        { x: 7, y: 4.0 },
        { x: 8, y: 3.1 },
      ],
    },
  ]}
  calculateRegressionLine
  showResiduals
  regressionLineStyle={{ color: "var(--chart-4)" }}
/>

If the points fall from top left to bottom right and are very tightly clustered, the <InlineMath math="r" /> value will be close to -1.

### No Linear Correlation

<InlineMath math="r" /> approaching 0 means the two variables have no linear relationship.

<ScatterDiagram
  title="Example of No Linear Correlation"
  description="Data points are scattered randomly without forming a straight line pattern."
  xAxisLabel="Variable X"
  yAxisLabel="Variable Y"
  datasets={[
    {
      name: "Data",
      color: "var(--chart-5)",
      points: [
        { x: 1, y: 5 },
        { x: 2, y: 2 },
        { x: 3, y: 7 },
        { x: 4, y: 4 },
        { x: 5, y: 9 },
        { x: 6, y: 3 },
        { x: 7, y: 8 },
        { x: 8, y: 1 },
      ],
    },
  ]}
  calculateRegressionLine
  showResiduals
  regressionLineStyle={{ color: "var(--chart-4)" }}
/>

When the points are scattered randomly without a clear linear pattern, the <InlineMath math="r" /> value will be close to 0.

## How is r Calculated?

The Pearson correlation coefficient (<InlineMath math="r" />) essentially measures how **synchronously** two variables (X and Y) move relative to their own variations.

**Imagine this:**

1.  **Individual Variation:**

    Each variable (X and Y) has its own variability. Some values fluctuate a lot (large variation), while others are stable (small variation). This is measured by <InlineMath math="SS_{xx}" /> for X and <InlineMath math="SS_{yy}" /> for Y (formulas below).

2.  **Joint Variation (Covariance):**

    We also need to know how X and Y vary _together_. When X increases, does Y also tend to increase? Or decrease? This measure of joint variation is called **covariance**, calculated using <InlineMath math="SS_{xy}" />.

    - If <InlineMath math="SS_{xy}" /> is large and positive: X and Y often move in the same direction.
    - If <InlineMath math="SS_{xy}" /> is large and negative: X and Y often move in opposite directions.
    - If <InlineMath math="SS_{xy}" /> is close to zero: No clear pattern of joint movement.

3.  **Standardizing the Measure:**

    The problem is that the value of <InlineMath math="SS_{xy}" /> (covariance) is heavily influenced by the units of the data. For example, the covariance between height (cm) and weight (kg) will have a different value if we measure height in meters and weight in grams, even if the relationship is the same.

    To overcome this, we need to **standardize** the covariance measure. This is done by **dividing the covariance (<InlineMath math="SS_{xy}" />) by a measure of the individual variations** (adjusted using square roots: <InlineMath math="\sqrt{SS_{xx} SS_{yy}}" />).

    <BlockMath math="r = \frac{\text{How much X and Y vary together}}{\text{Standardized measure of individual X and Y variations}} = \frac{SS_{xy}}{\sqrt{SS_{xx} SS_{yy}}}" />

The result of this division is **<InlineMath math="r" />**, the Pearson Correlation Coefficient. Because it's standardized, its value will always be between -1 and +1, regardless of the original data units. This allows us to compare the strength of linear relationships between different pairs of variables.

So, the value of <InlineMath math="r" /> is determined by comparing how strongly X and Y move together relative to how much they move individually.

## Product Moment Correlation Formula

To calculate the value of <InlineMath math="r" /> precisely, we use formulas involving the **Sum of Squares**:

<BlockMath math="r = \frac{SS_{xy}}{\sqrt{SS_{xx} SS_{yy}}}" />

**What are <InlineMath math="SS_{xy}" />, <InlineMath math="SS_{xx}" />, and <InlineMath math="SS_{yy}" />?**

These measure how varied our data is:

1.  **<InlineMath math="SS_{xx}" /> (Sum of Squares for x):** Measures how spread out the x data is from its mean.

    <BlockMath math="SS_{xx} = \sum (x - \bar{x})^2 = \sum x^2 - \frac{(\sum x)^2}{n}" />

2.  **<InlineMath math="SS_{yy}" /> (Sum of Squares for y):** Measures how spread out the y data is from its mean.

    <BlockMath math="SS_{yy} = \sum (y - \bar{y})^2 = \sum y^2 - \frac{(\sum y)^2}{n}" />

3.  **<InlineMath math="SS_{xy}" /> (Sum of Products of deviations for x and y):** Measures how x and y vary _together_.
    <BlockMath math="SS_{xy} = \sum (x - \bar{x})(y - \bar{y}) = \sum xy - \frac{(\sum x)(\sum y)}{n}" />

Key:

- <InlineMath math="n" />: Number of data pairs (x, y).
- <InlineMath math="\sum x" />, <InlineMath math="\sum y" />: Sum of all x and y
  values.
- <InlineMath math="\sum x^2" />, <InlineMath math="\sum y^2" />: Sum of the squares
  of each x and y value.
- <InlineMath math="\sum xy" />: Sum of the product of each x and y pair.
- <InlineMath math="\bar{x}" />, <InlineMath math="\bar{y}" />: Mean of x and y values.

By calculating these three SS values and plugging them into the formula for <InlineMath math="r" />, we get the Product Moment Correlation Coefficient.

## Interpreting the Value of r

Once we have the value of <InlineMath math="r" />, we can interpret its strength and direction using the following general guidelines:

| Value of <InlineMath math="r" />      | Correlation Strength  | Description                                        |
| :------------------------------------ | :-------------------- | :------------------------------------------------- |
| <InlineMath math="1" />               | Perfect Positive      | All points lie exactly on an upward sloping line.  |
| <InlineMath math="0.7 \le r < 1" />   | Strong Positive       | Clear and strong positive linear relationship.     |
| <InlineMath math="0.3 < r < 0.7" />   | Moderate Positive     | Moderately visible positive linear relationship.   |
| <InlineMath math="0 < r \le 0.3" />   | Weak Positive         | Very low positive linear relationship.             |
| <InlineMath math="0" />               | No Linear Correlation | No linear relationship at all.                     |
| <InlineMath math="-0.3 \le r < 0" />  | Weak Negative         | Very low negative linear relationship.             |
| <InlineMath math="-0.7 < r < -0.3" /> | Moderate Negative     | Moderately visible negative linear relationship.   |
| <InlineMath math="-1 < r \le -0.7" /> | Strong Negative       | Clear and strong negative linear relationship.     |
| <InlineMath math="-1" />              | Perfect Negative      | All points lie exactly on a downward sloping line. |