Standard Error Regression Calculator

Standard Error of Regression Formula:

\[ SE = \frac{\sqrt{\frac{\sum (y - \hat{y})^2}{n - 2}}}{\sqrt{\sum (x - \bar{x})^2}} \]

Standard Error (SE):

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1. What is Standard Error of Regression?

The Standard Error of Regression measures the precision of the regression coefficients in a linear regression model. It indicates how much the estimated coefficients are expected to vary from the true population values.

2. How Does the Calculator Work?

The calculator uses the standard error formula:

\[ SE = \frac{\sqrt{\frac{\sum (y - \hat{y})^2}{n - 2}}}{\sqrt{\sum (x - \bar{x})^2}} \]

Where:

\( y \) — Observed values
\( \hat{y} \) — Predicted values from regression
\( n \) — Number of observations
\( x \) — Independent variable values
\( \bar{x} \) — Mean of x values

Explanation: The numerator represents the standard error of the estimate, while the denominator accounts for the variability in the independent variable.

3. Importance of Standard Error Calculation

Details: Standard error is crucial for constructing confidence intervals and conducting hypothesis tests for regression coefficients. A smaller standard error indicates more precise coefficient estimates.

4. Using the Calculator

Tips: Enter comma-separated values for observed, predicted, and x values. All arrays must have the same length and contain at least 3 values for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What does a high standard error indicate?
A: A high standard error suggests that the coefficient estimate is less precise and may vary significantly from the true population value.

Q2: How is standard error different from R-squared?
A: While R-squared measures the proportion of variance explained, standard error measures the precision of coefficient estimates.

Q3: When is standard error most useful?
A: Standard error is particularly useful when comparing different regression models or when assessing the reliability of coefficient estimates.

Q4: What affects the standard error value?
A: Sample size, variability in the independent variable, and the goodness of fit all influence the standard error value.

Q5: How is standard error used in hypothesis testing?
A: Standard error is used to calculate t-statistics for testing whether regression coefficients are statistically significant.