Standard Error Of Slope Calculator Graph

Standard Error of Slope Formula:

\[ SE_b = \sqrt{ \frac{ \frac{\sum (y_i - \hat{y}_i)^2}{n - 2} }{ \sum (x_i - \bar{x})^2 } } \]

Observed Y Values (y_i):

comma separated

Predicted Y Values (ŷ_i):

comma separated

X Values (x_i):

comma separated

Standard Error of Slope (SE_b):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Standard Error of Slope?

The Standard Error of Slope (SE_b) measures the precision of the estimated slope coefficient in linear regression. It quantifies how much the slope estimate would vary across different samples from the same population.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ SE_b = \sqrt{ \frac{ \frac{\sum (y_i - \hat{y}_i)^2}{n - 2} }{ \sum (x_i - \bar{x})^2 } } \]

Where:

\( y_i \) — Observed y values
\( \hat{y}_i \) — Predicted y values from regression
\( n \) — Number of observations
\( x_i \) — X values
\( \bar{x} \) — Mean of x values

Explanation: The numerator represents the mean squared error of the regression, while the denominator represents the variability in the independent variable.

3. Importance of Standard Error of Slope

Details: SE_b is crucial for constructing confidence intervals around the slope estimate and for hypothesis testing (e.g., testing if the slope is significantly different from zero). A smaller SE_b indicates a more precise estimate of the slope.

4. Using the Calculator

Tips: Enter comma-separated values for observed y, predicted y, and x. All arrays must have the same number of values (n > 2). Ensure values are numeric and properly formatted.

5. Frequently Asked Questions (FAQ)

Q1: What does a high standard error of slope indicate?
A: A high SE_b suggests that the slope estimate is imprecise and may vary considerably across different samples from the same population.

Q2: How is SE_b used in hypothesis testing?
A: SE_b is used to calculate the t-statistic (t = b/SE_b) for testing whether the slope is statistically significantly different from zero.

Q3: What affects the magnitude of SE_b?
A: SE_b decreases with larger sample size, greater variability in x, and better model fit (smaller residuals).

Q4: Can SE_b be negative?
A: No, standard error is always a non-negative value as it represents a measure of variability.

Q5: How does SE_b relate to confidence intervals?
A: The 95% confidence interval for the slope is calculated as: b ± t_α/2 × SE_b, where t_α/2 is the critical value from the t-distribution.