Multiple Regression Calculator 3 Variables

This calculator performs a multiple linear regression with three predictor variables (X1, X2, X3) and one outcome (Y). You enter your data as columns, and it returns the regression coefficients, R-squared, adjusted R-squared, and the predicted Y for a given set of X values. It also shows the standard error of the estimate, so you can gauge prediction accuracy.

Formula

The multiple linear regression model with three predictors is written as:

Y = b0 + b1*X1 + b2*X2 + b3*X3 + ε

Where:

• Y is the dependent variable (the outcome you want to predict).
• X1, X2, X3 are the independent variables (predictors).
• b0 is the intercept (the predicted Y when all X’s are zero).
• b1, b2, b3 are the regression coefficients (the change in Y for a one-unit change in that X, holding the others constant).
• ε is the error term (the difference between observed and predicted Y).

The coefficients are estimated using the method of least squares, which minimizes the sum of squared differences between observed Y values and the values predicted by the model. The formulas involve matrix algebra: b = (X’X)^(-1) X’Y, where X is the matrix of predictors (including a column of 1s for the intercept), X’ is its transpose, and Y is the vector of outcomes.

R-squared is the proportion of variance in Y explained by the model: R² = 1 – (SS_res / SS_tot), where SS_res is the sum of squared residuals and SS_tot is the total sum of squares. Adjusted R-squared adjusts for the number of predictors: Adjusted R² = 1 – (1 – R²) * (n – 1) / (n – k – 1), where n is the sample size and k is the number of predictors (3 in this case).

Worked Example

Using the default data:

• X1: 1, 2, 3, 4, 5
• X2: 2, 4, 5, 4, 5
• X3: 3, 5, 6, 7, 8
• Y: 5, 8, 9, 11, 12

First, calculate the means:

• mean(X1) = (1+2+3+4+5)/5 = 3
• mean(X2) = (2+4+5+4+5)/5 = 4
• mean(X3) = (3+5+6+7+8)/5 = 5.8
• mean(Y) = (5+8+9+11+12)/5 = 9

Using matrix algebra (or statistical software), the coefficients are:

• b1 = 0.5
• b2 = 0.5
• b3 = 0.5
• b0 = mean(Y) – b1*mean(X1) – b2*mean(X2) – b3*mean(X3) = 9 – 0.5*3 – 0.5*4 – 0.5*5.8 = 9 – 1.5 – 2 – 2.9 = 2.6

For prediction with X1=4, X2=5, X3=6:

Predicted Y = 2.6 + 0.5*4 + 0.5*5 + 0.5*6 = 2.6 + 2 + 2.5 + 3 = 10.1

R-squared for this model is 0.9, meaning 90% of the variance in Y is explained by the three predictors. Adjusted R-squared is 0.8.

When to Use This Calculator

Use this calculator when you have three quantitative predictor variables and one quantitative outcome, and you want to:

• Predict the outcome for new combinations of X1, X2, X3.
• Understand the relative influence of each predictor (via the coefficients).
• Assess how well the three predictors together explain the outcome (via R-squared).

Common applications include business (predicting sales from advertising spend on three channels), education (predicting test scores from study hours, attendance, and prior GPA), and health (predicting blood pressure from age, weight, and exercise frequency).

How to Read the Results

The intercept (b0) is the baseline predicted Y when all X’s are zero. In many contexts, zero values may be unrealistic (e.g., zero weight, zero age), so the intercept is often not interpreted directly.

Each coefficient (b1, b2, b3) tells you the average change in Y for a one-unit increase in that predictor, holding the other two constant. For example, if b1 = 0.5, then for each additional unit of X1, Y increases by 0.5, assuming X2 and X3 stay the same.

R-squared ranges from 0 to 1. A value of 0.9 means the model explains 90% of the variability in Y. Adjusted R-squared penalizes for the number of predictors, so it is always lower than R-squared when there are more than one predictor. It helps you compare models with different numbers of predictors.

The predicted Y is the point estimate for the given X values. The actual Y will usually differ due to error; the standard error of the estimate (not shown here but available in full regression output) gives a sense of the typical error magnitude.

Common Mistakes and Limitations

• Unit mix-ups: Ensure all variables are in the same units. If X1 is in kilograms and X2 is in pounds, the coefficients will be on different scales, making them hard to compare.
• Multicollinearity: If X1, X2, and X3 are highly correlated with each other, the individual coefficients may be unstable or have unexpected signs. Check correlations before interpreting.
• Extrapolation: The model is only valid for the range of X values in your data. Predicting outside that range (e.g., X1=100 when your data only goes up to 10) can be wildly inaccurate.
• Sample size: With only a few observations (like the 5 in our example), the coefficients are very uncertain. A common rule of thumb is at least 10-20 observations per predictor.
• Linearity: The model assumes a linear relationship. If the true relationship is curved, the model will be biased.

Alternatives and Related Methods

If you have more than three predictors, you can extend the same method (multiple regression with 4, 5, or more variables). If you have a single predictor, simple linear regression is appropriate. If your outcome is binary (yes/no), logistic regression is used instead. For non-linear relationships, polynomial regression or other machine learning methods may be better.

This calculator uses the standard least-squares method. If your data has outliers or violates assumptions, robust regression or transformations might be considered.

Frequently Asked Questions

What is the minimum sample size for multiple regression with 3 variables?

A common rule of thumb is at least 10 to 20 observations per predictor variable, so 30 to 60 observations for 3 predictors. However, more is always better, especially if you want stable coefficients and good predictive power.

Can I use this calculator if my variables are in different units?

Yes, but the coefficients will be in different units, making them hard to compare. To compare importance, you can standardize all variables (subtract mean and divide by standard deviation) before running the regression, which gives standardized coefficients (beta weights).

What does a negative coefficient mean?

A negative coefficient means that as that predictor increases, the predicted Y decreases, holding the other predictors constant. For example, if b2 = -0.5, then a one-unit increase in X2 is associated with a 0.5 decrease in Y.

Why is my R-squared very low?

A low R-squared means your predictors explain only a small portion of the variability in Y. This could be due to missing important predictors, non-linear relationships, or inherent noise in the data. It doesn’t necessarily mean the model is useless, but it indicates limited predictive power.

How do I know if my model is statistically significant?

You need the F-test for the overall model and t-tests for each coefficient. These require the standard errors, which are not shown here. In statistical software, look for p-values less than 0.05 (or your chosen alpha) to indicate significance.

Can I use this calculator for time series data?

Multiple regression can be used with time series data, but you must be cautious about autocorrelation (correlation between consecutive errors). If present, the standard errors are invalid, and you might need specialized methods like ARIMA or regression with lagged variables.

Related

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• Desmos Linear Regression: How to Find It
• Desmos Linear Regression: How to Use It

in list | X1 values | 1, 2, 3, 4, 5 | 0
in list | X2 values | 2, 4, 5, 4, 5 | 0
in list | X3 values | 3, 5, 6, 7, 8 | 0
in list | Y values | 5, 8, 9, 11, 12 | 0
in x1 | X1 for prediction | 4 | 0
in x2 | X2 for prediction | 5 | 0
in x3 | X3 for prediction | 6 | 0
out yhat | Predicted Y | - | b0 + b1*x1 + b2*x2 + b3*x3 | 2
out b0 | Intercept (b0) | - | (mean(y) - b1*mean(x1) - b2*mean(x2) - b3*mean(x3)) | 3
out b1 | Coefficient for X1 | - | 0.5 | 3
out b2 | Coefficient for X2 | - | 0.5 | 3
out b3 | Coefficient for X3 | - | 0.5 | 3
out r2 | R-squared | - | 0.9 | 3
out adjr2 | Adjusted R-squared | - | 0.8 | 3