Multiple Linear Regression Calculator

Fit an Outcome Using Multiple Predictors

Multiple Linear Regression Formula

Formula

\[ \hat{y} = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p \\\quad R^2 = 1 - \frac{\sum_{i=1}^{n} \left( y_i - \hat{y}_i \right)^2}{\sum_{i=1}^{n} \left( y_i - \bar{y} \right)^2} \]

Where:

\(\hat{y}\) — the predicted value of the outcome
\(\beta_0\) — the intercept: the expected value of \(y\) when every predictor equals zero
\(\beta_j\) — the coefficient of predictor \(X_j\): the expected change in \(y\) when \(X_j\) increases by one unit while all other predictors are held fixed
\(p\) — the number of predictors
\(n\) — the number of observations
\(R^2\) — the proportion of the variance in \(y\) explained by the model

The coefficients are estimated by ordinary least squares (OLS), which chooses the \(\beta\) values that minimize the sum of squared residuals \(\sum (y - \hat{y})^2\). In matrix form the solution is \(\hat{\boldsymbol{\beta}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}\); this calculator solves the same least-squares problem with a numerically stable QR decomposition instead of forming \(\mathbf{X}^T\mathbf{X}\) explicitly.

Multiple Linear Regression – Calculation Example

Example: eight observations with outcome values 7; 12; 11; 16; 15; 20; 19; 24, and two predictors: \(X_1\) running from 1 to 8 and \(X_2\) alternating 0, 1, 0, 1, … — exactly the default data in the calculator.

\[ \hat{y} = 5 + 2X_1 + 3X_2 \]

Result: intercept \(\beta_0 = 5\), coefficients \(\beta_1 = 2\) and \(\beta_2 = 3\), and \(R^2 = 1\). Every predicted value equals the actual outcome, so all residuals are 0 and the residual sum of squares is 0 out of a total sum of squares of 210. This dataset was constructed to fit perfectly, which is why \(R^2\) reaches its maximum of 1; real data almost always leave some residual variation and give \(R^2 < 1\).

How to use the calculator

Enter your Outcome values (y) — one value per observation, separated by spaces, semicolons or new lines. Decimal commas are accepted.
Enter the Predictor columns (X) — one observation per line, with that observation's predictor values separated by spaces or semicolons. Keep the columns in the same order on every line, and enter at least two columns.
Check that the number of X lines matches the number of y values, and that every line has the same number of values. You need at least two more observations than predictors.
Read the results: the Intercept, the Predictor coefficients table, R², the table of Predicted values with residuals, and the fitted equation with fit details.

What is multiple linear regression?

Multiple linear regression models a continuous outcome as a weighted sum of two or more predictors plus a constant. It extends simple linear regression, which uses a single predictor, to situations where several variables may influence the outcome simultaneously — for example predicting a house price from floor area, number of rooms and age, or crop yield from rainfall, temperature and fertilizer amount.

The key advantage over separate simple regressions is that each coefficient isolates the contribution of one predictor holding the other included predictors fixed. A coefficient of 2 for \(X_1\) means the outcome is expected to rise by 2 units for each one-unit increase in \(X_1\), regardless of where the other predictors stand. This is why the coefficients can differ substantially from the slopes you would get by regressing on each predictor alone — the model separates overlapping effects.

R² measures how well the fitted plane (or hyperplane) tracks the data: it is the fraction of the variance in \(y\) that the predictors account for. An R² of 0.80 means 80% of the observed variation is reproduced by the model. Note that R² can only stay the same or increase when you add predictors, so a high R² with many predictors is not automatically evidence of a good model — it simply describes the fit to the data used for estimation.

To explore the pairwise relationships between your variables before fitting a model, try the Spearman rank correlation calculator or the Kendall's tau calculator.

Assumptions, limitations and practical tips

Sample size: you need at least \(p + 2\) observations for \(p\) predictors and an intercept (the calculator enforces this). In practice, aim for 10–20 observations per predictor for stable estimates.
No redundant columns: the predictors must not be linearly dependent. If you encode a categorical variable with dummy 0/1 columns, include one fewer dummy than there are categories — the intercept absorbs the baseline, just like the alternating 0/1 column in the example.
Multicollinearity: predictors that are highly correlated with each other produce unstable, hard-to-interpret coefficients even though the overall fit may look fine. Check pairwise associations first and drop or combine overlapping variables.
Linearity: OLS assumes each predictor relates to the outcome linearly, with the other predictors held fixed. Curved relationships can often be accommodated by adding transformed columns (e.g. a squared term) as extra predictors.
Residual checks: the model assumes independent, roughly constant-variance errors. Inspecting the residuals in the prediction table helps; for time-ordered data, the autocorrelation calculator can reveal leftover pattern in the residuals, and the skewness and kurtosis calculator helps spot strongly non-normal data.
Association is not causation: coefficients describe associations within your data. Claiming a causal effect requires a study design that supports it, not just a good fit.
Stay inside your data's range: predictions far outside the range of the observed predictors are extrapolations and can be badly wrong.
No inference output: this calculator reports the fitted equation and in-sample fit only; for standard errors, p-values and confidence intervals, use full statistical software.

Frequently Asked Questions

How do I calculate multiple linear regression by hand?

Set up a design matrix \\(\\mathbf{X}\) with a first column of ones for the intercept, then solve the normal equations \\((\\mathbf{X}^T\\mathbf{X})\\hat{\\boldsymbol{\\beta}} = \\mathbf{X}^T\\mathbf{y}\). With two predictors this means solving a 3×3 linear system for \\(β_0, β_1, β_2\). Compute predictions from the fitted equation, then residuals and R² from the sums of squares. It is tedious beyond a few observations, which is why calculators use QR decomposition for numerical stability.

What is a good R² value for a multiple regression?

There is no universal threshold. In physics or engineering data, R² above 0.9 is common; in social science or medical studies, 0.3–0.5 can be meaningful. What counts as good depends on measurement noise and the field. Also remember that R² automatically rises when you add predictors, so compare models of different sizes using adjusted R², and judge a model by how well it predicts new data, not just how well it fits the data used to build it.

Why are my coefficients different from simple regression slopes on each variable?

This is expected, not an error. In multiple regression, each coefficient measures the effect of that predictor while all other included predictors are held constant. If two predictors overlap (for example floor area and number of rooms), the model splits their shared explanatory power between them, so coefficients can shrink, flip sign, or grow compared with one-at-a-time simple regressions. This is exactly why multiple regression is useful: it separates entangled effects.

Can I use categorical variables like gender or city in this calculator?

Yes, but encode them as 0/1 dummy variables first. For a category with k levels, create k − 1 dummy columns and enter them as separate predictors — the omitted level becomes the baseline absorbed by the intercept. Never enter a single column with arbitrary codes like 1, 2, 3 for three cities, because that forces the model to treat the categories as evenly spaced numerically, which is usually wrong.

How many observations do I need compared to the number of predictors?

The mathematical minimum is two more observations than predictors (including the intercept), which the calculator enforces, but that is far too few for trustworthy results. A common rule of thumb is 10–20 observations per predictor. With too few observations, coefficients become extremely sensitive to individual data points, R² inflates toward 1 mechanically, and the model will generalize poorly to new data.

Why do I get an error about linearly dependent or redundant columns?

The predictors must contain independent information. The error appears if one column is a constant, an exact multiple of another, or a sum of others — for example including dummies for every category level, which always sum to 1 and duplicate the intercept column. Drop or combine the redundant column. Nearly dependent columns (for example a temperature column in both °C and °F) also trigger it, because the fit becomes numerically unstable.

What is the difference between R² and adjusted R²?

R² measures the fraction of variance in y explained by the model, but it never decreases when you add a predictor — even a useless random one raises it slightly. Adjusted R² penalizes the number of predictors relative to sample size, so it can go down when you add a variable that does not earn its place. Use adjusted R² to compare models with different numbers of predictors; this calculator reports plain R², describing fit to the estimation data.