Updated Regression & Correlation Tool

Regression Line Calculator

Perform simple linear regression from raw X and Y data. This calculator finds the best-fit line \(\hat{y} = b_0 + b_1 x\), reports the slope and intercept, correlation, \(R^2\), standard error, t-test for the slope, and allows prediction at a chosen value of \(x_0\) with confidence and prediction intervals.

Simple Linear Regression Best-Fit Line Equation Correlation & \(R^2\) Prediction & Intervals

Compute the Regression Line, Correlation and Predictions

Enter paired X and Y data to fit a simple linear regression model. The Regression Line Calculator computes the least-squares slope \(b_1\), intercept \(b_0\), the correlation \(r\), coefficient of determination \(R^2\), standard error of estimate and a t-test for the slope. You can also specify a value \(x_0\) to obtain the predicted value \(\hat{y}(x_0)\), a confidence interval for the mean response and a prediction interval for an individual future observation.

Enter X and Y data as comma, space or line-break separated lists. Both lists must have the same length and at least 3 paired observations to allow regression with standard error and t-tests.

Used to compute \(\hat{y}(x_0)\), a confidence interval for the mean response and a prediction interval for an individual value at that X.

Regression Line Calculator – Complete Guide to Simple Linear Regression

The Regression Line Calculator on MyTimeCalculator performs simple linear regression on raw paired X and Y data. It finds the least-squares regression line \[ \hat{y} = b_0 + b_1 x, \] and reports the slope, intercept, correlation coefficient \(r\), coefficient of determination \(R^2\), standard error of estimate, and a t-test for the slope. You can also obtain a predicted value \(\hat{y}(x_0)\) at a chosen \(x_0\), together with a confidence interval for the mean response and a prediction interval for an individual future observation.

1. Core Formulas for Simple Linear Regression

Suppose you have \(n\) paired observations \((x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\). Let the sample means be \(\bar{x}\) and \(\bar{y}\). Define the sums of squares and cross-products:

\[ S_{xx} = \sum_{i=1}^{n} (x_i - \bar{x})^2, \quad S_{yy} = \sum_{i=1}^{n} (y_i - \bar{y})^2, \quad S_{xy} = \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}). \]

The least-squares estimates for the slope \(b_1\) and intercept \(b_0\) of the regression line are:

\[ b_1 = \frac{S_{xy}}{S_{xx}}, \qquad b_0 = \bar{y} - b_1 \bar{x}. \]

Once the regression line is fitted, the predicted value at any \(x\) is

\[ \hat{y}(x) = b_0 + b_1 x. \]

2. Correlation, \(R^2\) and Standard Error

The sample correlation coefficient \(r\) between X and Y is

\[ r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}. \]

The coefficient of determination \(R^2\) measures the proportion of variance in Y explained by the linear relationship with X:

\[ R^2 = \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\text{SSE}}{\text{SST}}, \]

where \(\text{SST} = S_{yy}\) is the total sum of squares, \(\text{SSR}\) is the regression sum of squares and \(\text{SSE}\) is the residual sum of squares. For simple linear regression, you can show that \(\text{SSR} = b_1^2 S_{xx}\) and \(\text{SSE} = \text{SST} - \text{SSR}\).

The standard error of estimate (also called the residual standard deviation) is

\[ s = \sqrt{\frac{\text{SSE}}{n - 2}}, \]

where \(n - 2\) is the residual degrees of freedom for a simple linear regression model with two fitted parameters \(b_0\) and \(b_1\).

3. t-Test for the Slope

To test whether there is a linear relationship between X and Y, we often test the null and alternative hypotheses:

\[ H_0: \beta_1 = 0 \quad\text{vs.}\quad H_1: \beta_1 \ne 0. \]

The standard error of the slope estimate \(b_1\) is

\[ \text{SE}(b_1) = \frac{s}{\sqrt{S_{xx}}}, \]

and the t-statistic for testing \(H_0\) is

\[ t = \frac{b_1 - 0}{\text{SE}(b_1)} = \frac{b_1}{\text{SE}(b_1)}, \]

with \(df = n - 2\) degrees of freedom. A large positive or negative value of \(t\) provides evidence against \(H_0\). The calculator reports an approximate two-sided p-value and compares it with the chosen significance level \(\alpha\).

4. Prediction, Confidence Interval and Prediction Interval

Once you have fitted the regression line, you can predict the mean response at a specific value \(x_0\). The point prediction is

\[ \hat{y}(x_0) = b_0 + b_1 x_0. \]

The standard error of the estimated mean response at \(x_0\) is

\[ s_{\hat{y}}(x_0) = s \sqrt{\frac{1}{n} + \frac{(x_0 - \bar{x})^2}{S_{xx}}}. \]

A \(100(1 - \alpha)\%\) confidence interval for the mean response at \(x_0\) is

\[ \hat{y}(x_0) \pm t_{\alpha/2,\,n-2} \, s_{\hat{y}}(x_0), \]

where \(t_{\alpha/2,\,n-2}\) is the critical value from the t-distribution with \(n - 2\) degrees of freedom for a two-sided interval.

If you are interested in predicting an individual future observation at \(x_0\), the variability is larger. The standard error for a prediction interval is

\[ s_{\text{pred}}(x_0) = s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{S_{xx}}}, \]

and a \(100(1 - \alpha)\%\) prediction interval for a single new observation is

\[ \hat{y}(x_0) \pm t_{\alpha/2,\,n-2} \, s_{\text{pred}}(x_0). \]

5. How to Use the Regression Line Calculator

  1. Enter X and Y data: paste or type your X values and Y values into the input areas. They can be separated by commas, spaces or line breaks, but both lists must have the same length.
  2. Choose a significance level \(\alpha\): set \(\alpha\) (for example 0.05). This is used for the t-test on the slope and for constructing confidence and prediction intervals.
  3. Specify a prediction point \(x_0\): enter the X value at which you want to predict the corresponding Y value and intervals.
  4. Run the calculation: click the button to compute the regression line, correlation, \(R^2\), standard error and slope t-test. The calculator also returns \(\hat{y}(x_0)\), a confidence interval for the mean response and a prediction interval for an individual value at \(x_0\).
  5. Interpret results: use the sign and magnitude of the slope and correlation to understand the direction and strength of the linear relationship, check the p-value to see whether the slope is statistically different from zero, and use the intervals to quantify uncertainty around predictions.

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Regression Line Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about simple linear regression, regression lines, correlation and prediction intervals.

The regression line is the straight line that best fits the relationship between X and Y in the least-squares sense. For any given value of X, the line produces a predicted value \(\hat{y}\). The slope \(b_1\) describes how much Y is expected to change, on average, for a one-unit increase in X, while the intercept \(b_0\) is the predicted value of Y when X is zero (if that is meaningful in context).

In simple linear regression with a single X and a single Y, the coefficient of determination satisfies \(R^2 = r^2\). The value \(r\) measures the strength and direction of the linear relationship between X and Y (positive or negative), while \(R^2\) measures the proportion of variation in Y explained by that linear relationship, ignoring the sign. For example, \(r = -0.8\) corresponds to \(R^2 = 0.64\), meaning 64% of the variance in Y is explained by X through the fitted line.

Simple linear regression fits two parameters: the intercept \(b_0\) and the slope \(b_1\). The residual degrees of freedom are \(n - 2\), where \(n\) is the number of observations. To estimate the residual variance and construct t-tests or confidence intervals, you need at least 1 degree of freedom, which requires \(n \ge 3\). With only two points, the line is defined exactly and there is no remaining information to estimate variability.

A confidence interval for the mean response at \(x_0\) describes the uncertainty about the average value of Y for all units with X = \(x_0\). A prediction interval is wider and describes the uncertainty for a single new observation at \(x_0\), which includes both the uncertainty in the regression line and the natural scatter of individual points around that line. The calculator reports both intervals so you can choose the one that matches your question.

Yes. You can enter the same X and Y data from your exercise and compare the slope, intercept, correlation, \(R^2\), standard error and p-value with your own calculations. This is especially helpful for verifying numerical work in regression problems. For written solutions, you should still show your formulas, algebra and graphs, since the calculator focuses on numerical outputs rather than detailed derivations.

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