Breusch-Pagan Test
Computes BP heteroscedasticity statistic from n and auxiliary R squared.
resultado
—
The Breusch-Pagan LM statistic by hand
You regressed the squared residuals on the regressors, wrote down the auxiliary R squared, and now you need the test statistic to decide whether the errors are heteroscedastic. Statistical software hands this over automatically, but when the numbers come from a spreadsheet, a problem set or a paper that only reported n and R squared, the arithmetic is yours. That is the gap this page fills.
The statistic is LM = n × R², printed to three decimals. This is the Lagrange multiplier form of the test, in Koenker's studentized version, which drops the normality assumption the 1979 original leaned on. Under the null of constant variance, LM follows a chi-square distribution with k degrees of freedom, where k is the number of regressors in the auxiliary regression, intercept excluded. With n = 100 and R² = 0.15 you get LM = 15.000.
The page stops at the statistic: it asks for no degrees of freedom and returns no p-value, so the comparison is on you. In the example above, with two regressors in the auxiliary equation the 5% chi-square critical value is 5.99, so 15.000 rejects homoscedasticity comfortably. Small samples make the test unreliable, and rejection never names the guilty variable, so plot residuals against fitted values before reaching for robust standard errors. Runs in the browser.
Frequently asked questions
Which R squared goes in?
How many degrees of freedom?
How is this different from White's test?
Related Tools
Prime Number Checker: Trial Division Test
Tells you whether a whole number is prime by dividing up to its square root, naming the first divisor when it is not. Practical up to about 10^12.
Exact Binomial Test
Computes the exact binomial test, which checks whether the observed proportion of successes is compatible with a hypothesized probability, without resorting to the normal approximation. For small samples, where the z-test fails, this is the correct test: it sums the exact probabilities, given by the binomial distribution, of all outcomes as likely or less likely than the observed one. For example: are 8 heads in 10 tosses compatible with a fair coin? Enter the successes, the total and the hypothesized probability.
Brown-Forsythe Test (Variances)
Computes the Brown-Forsythe test, a robust version of Levene's test for checking whether several groups share the same variance. The difference lies in one decisive detail: instead of measuring how far each point sits from the group mean, it uses the distance from the median. Because the median resists extreme values and skewed distributions, the test becomes much more reliable when the data aren't normal. Enter the groups: values by comma, groups by semicolon.
One-Sample Variance Test (Chi-Square)
Computes the chi-square test for the variance of a single sample, which checks whether the population variance equals a reference value. It's the dispersion counterpart of the one-sample t-test for the mean. It shows up often in quality control: is a process's variability within the specified limit, or has it increased? The statistic compares the sample variance with the hypothesized one and follows a chi-square distribution. Enter the sample and the hypothesized variance (σ₀²).
Bartlett's Test (Equality of Variances)
Computes Bartlett's test, the classic test for checking whether several groups share the same variance. It compares the pooled variance of all groups against the individual variances through a statistic that approximately follows a chi-square distribution, with a bias correction that improves the approximation in small samples. It's more powerful than Levene's test when the data are genuinely normal, but more sensitive to departures from normality. Enter the groups: values by comma, groups by semicolon.
D'Agostino-Pearson Normality Test (K²)
Computes the D'Agostino-Pearson omnibus normality test, which combines two clues of non-normality into a single verdict: skewness (do the data lean to one side?) and kurtosis (are the tails too heavy or too light?). Each becomes a standardized z score, and the sum of their squares, the K² statistic, follows a chi-square with 2 degrees of freedom. It's more informative than looking at skewness and kurtosis separately. Enter the sample (at least 8 values).
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.