1. What is the Least Square Estimator?

The least square estimator (β̂) calculates the slope of the best-fit line for a set of data points (x,y) by minimizing the sum of squared residuals. It's a fundamental tool in linear regression analysis.

2. How Does the Calculator Work?

The calculator uses the least squares formula:

Where:

• \( \hat\beta \) — Estimated slope coefficient
• \( x_i \) — Individual x values
• \( y_i \) — Individual y values
• \( \bar{x} \) — Mean of x values
• \( \bar{y} \) — Mean of y values

Explanation: The formula measures how much y changes on average when x changes by one unit, accounting for the covariance between x and y relative to the variance of x.

3. Importance of Least Squares Estimation

Details: Least squares estimation is the most common method for fitting linear regression models. It provides unbiased, minimum-variance estimates under standard regression assumptions.

4. Using the Calculator

Tips: Enter comma-separated x and y values of equal length. The calculator will compute the mean-centered covariance and variance to estimate the slope.

5. Frequently Asked Questions (FAQ)

Q1: What assumptions does least squares make?
A: Linearity, independence, homoscedasticity, and normally distributed errors are key assumptions.

Q2: How is this different from correlation?
A: Correlation measures strength of linear relationship, while regression quantifies the slope of that relationship.

Q3: What if my denominator is zero?
A: This occurs when x has no variance (all values identical), making slope estimation impossible.

Q4: Can I use this for nonlinear relationships?
A: Not directly - consider transformations or nonlinear regression for non-linear patterns.

Q5: How do I interpret the slope value?
A: The slope represents the expected change in y for a one-unit increase in x.