The **least squares method** is a mathematical approach used in regression analysis to find the best-fitting line (or curve) to a set of data points. The goal of the least squares method is to minimize the sum of the squares of the vertical distances (residuals) between the observed values (data points) and the values predicted by the line.

### Steps to Calculate the Least Squares Regression Line

Let's assume you have a set of \( n \) data points \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\), and you want to find the line of best fit, which has the equation:
\[
y = mx + c
\]
where:
- \( y \) is the dependent variable (response variable).
- \( x \) is the independent variable (predictor variable).
- \( m \) is the slope of the line.
- \( c \) is the y-intercept of the line.

#### 1. **Calculate the Slope \( m \):**

The slope \( m \) is calculated using the formula:
\[
m = \frac{n \sum{xy} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}
\]
where:
- \( n \) is the number of data points.
- \( \sum{xy} \) is the sum of the product of each \( x \) and \( y \) value.
- \( \sum{x} \) is the sum of all \( x \) values.
- \( \sum{y} \) is the sum of all \( y \) values.
- \( \sum{x^2} \) is the sum of the squares of the \( x \) values.

#### 2. **Calculate the Y-intercept \( c \):**

The y-intercept \( c \) is calculated using the formula:
\[
c = \frac{\sum{y} - m \sum{x}}{n}
\]
where:
- The symbols \( \sum{x} \) and \( \sum{y} \) have the same meanings as above.

#### 3. **Construct the Regression Line:**

With the values of \( m \) and \( c \) calculated, you can write the equation of the line of best fit as:
\[
y = mx + c
\]

### Example

Suppose you have the following data points:

| \( x \) | \( y \) |
|---------|---------|
| 1       | 2       |
| 2       | 3       |
| 3       | 5       |
| 4       | 4       |
| 5       | 6       |

**Step 1: Calculate Sums**
\[
\sum{x} = 1 + 2 + 3 + 4 + 5 = 15
\]
\[
\sum{y} = 2 + 3 + 5 + 4 + 6 = 20
\]
\[
\sum{xy} = (1 \times 2) + (2 \times 3) + (3 \times 5) + (4 \times 4) + (5 \times 6) = 2 + 6 + 15 + 16 + 30 = 69
\]
\[
\sum{x^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55
\]

**Step 2: Calculate the Slope \( m \)**
\[
m = \frac{5 \times 69 - 15 \times 20}{5 \times 55 - 15^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9
\]

**Step 3: Calculate

Explore the least squares method used in regression analysis to find the best-fitting line to a set of data points. Learn how to calculate the slope and y-intercept step-by-step with an example. Suitable for advanced high school students and beyond.

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