Linear regression calculator: least-squares line of best fit with R², standard errors, p-values and predictions, plus polynomial and exponential fits.
Each extra unit of x goes with a change of 3.1091 in y on average, and the line accounts for 99.33% of the variation in y. At x = 7.5 the fitted value is 72.2091.
| Term | Coefficient | Standard error | t | p-value |
|---|---|---|---|---|
| Intercept | 48.8909 | 0.6439 | 75.9325 | 1.008 × 10⁻¹² |
| x | 3.1091 | 0.0906 | 34.3146 | 5.686 × 10⁻¹⁰ |
Least squares picks the line or curve that makes the sum of squared vertical distances from the points as small as possible. For a straight line the slope is b₁ = Sxy/Sxx and the intercept b₀ = ȳ − b₁x̄; polynomial fits of degree 2 to 4 solve the normal equations, and the exponential model fits a straight line to ln y. Each coefficient gets a standard error, a t statistic and a p-value, and R² gives the share of the variation in y that the fit accounts for.
The default relates weekly study hours to exam scores for 10 students. The fitted line is y = 48.89 + 3.109x, so each extra hour goes with about 3.1 more points, R² is 0.9933, and at 7.5 hours the line predicts 72.2.
The standard errors and p-values assume independent residuals with constant spread. A curved or funnel-shaped residual plot means the model is missing something, and predictions outside the range of the x data are flagged as extrapolations.
Checked against: Constructed points (1,3), (2,5), (3,7) lie exactly on y = 1 + 2x; at x = 4, y = 9.
Checked against: Python fractions: b₁ = Sxy/Sxx, b₀ = ȳ − b₁x̄, se = √(SSE/(n − 2)) with decimal square roots
Checked against: Python fractions solving the 3×3 normal equations exactly (b₀ = 57/20, b₁ = −208/105, b₂ = 131/105); agrees with numpy.polyfit(x, y, 2)
Checked against: Constructed data: y = 1 + x + x² + x³ + x⁴ exactly, so every coefficient is 1 and R² = 1; 1 + 11 + 121 + 1331 + 14641 = 16105
Find the means x̄ and ȳ, then the slope b₁ = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² and the intercept b₀ = ȳ − b₁x̄. For the default study-hours data, Sxy = 256.5 and Sxx = 82.5, so b₁ = 3.109 and b₀ = 48.89. This least-squares line minimizes the sum of squared vertical distances, as set out in the NIST/SEMATECH e-Handbook (§4.1.4.1).
R² is the fraction of the variation in y that the model accounts for: R² = 1 − SSE/SST. The default fit has R² = 0.9933, so study hours account for 99.3% of the spread in exam scores. For a straight line R² equals the square of the correlation r. A high R² does not show that x causes y, that the model has the right shape, or that predictions beyond the data will hold.
Adjusted R² charges for each extra coefficient: 1 − (1 − R²)(n − 1)/(n − k), where k counts the coefficients including the intercept. R² never falls when a term is added, so a quartic always fits at least as well as a quadratic; adjusted R² falls when the new term explains less than it costs. For the default data, n = 10 and k = 2 turn R² = 0.99325 into 0.99241.
It tests the null hypothesis that the true slope is 0, using t = b₁/SE(b₁) with n − 2 degrees of freedom. For the default data t = 3.109/0.0906 = 34.3 with 8 df, giving p = 5.7 × 10⁻¹⁰: a slope this steep would be very unlikely if hours and scores had no linear relation. The p-value says nothing about how large or important the effect is.
When y grows or shrinks by a roughly constant percentage for each unit of x, so that ln y against x is straight. The exponential example, with y rising from 3.0 to 22.1 as x goes from 0 to 5, fits y = 2.98·e^(0.4007x), an increase of 49.3% per unit of x. The fit is made on ln y, as Excel's LOGEST function does, so R² describes ln y rather than y.
Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “Exact straight line y = 1 + 2x” is checked against Constructed points (1,3), (2,5), (3,7) lie exactly on y = 1 + 2x; at x = 4, y = 9.
NIST/SEMATECH e-Handbook of Statistical Methods, §4.1.4.1 Linear least squares regression; NIST Statistical Reference Datasets — linear regression; Draper & Smith, Applied Regression Analysis, 3rd ed. (Wiley, 1998), ch. 1 and 12.
5 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.
Correlation coefficient calculator: Pearson's r or Spearman's rank correlation for paired data, with r², a t statistic and a two-sided p-value.
Descriptive statistics calculator: mean, median, mode, standard deviation, variance, quartiles, IQR, outliers, skewness and kurtosis, with a histogram.
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