What is the Tukey 1.5 IQR rule for outliers?

Any observation more than 1.5 times the interquartile range below the first quartile or above the third quartile is flagged as an outlier. It is the rule behind the whiskers on a boxplot. Use 3 * IQR for "extreme" outliers. This rule is non-parametric and works on any continuous distribution.

When should I use Grubbs' test versus Tukey's IQR rule?

Grubbs' test assumes a normal distribution and tests one outlier at a time with a formal p-value, use it when you need a hypothesis test result. Tukey's rule is non-parametric and flags multiple outliers at once but does not give a p-value. For exploratory work, start with Tukey; for reporting, use Grubbs.

Should I always remove outliers before fitting a model?

No. Remove only data-entry errors and impossible values. Genuine extreme observations may carry the most information. Use robust regression (rlm), bootstrapping, or trimming if outliers distort your fit. Always report what you removed, why, and how the analysis changes if you keep them.

Outlier Detection

Outliers can quietly distort means, standard deviations, and regression slopes. Grubbs, ESD, Hampel (MAD), and Tukey IQR each flag suspicious values with different rules. Paste your data, pick a method, and see exactly which points get flagged, by what statistic, and at what alpha level.

Flag values that drift far from the bulk of your data using Grubbs, Generalized ESD, the Hampel (MAD) filter, or the Tukey IQR rule. Reproducible R code, runs in your browser.

Try a real-world example to load.

Detection

Outliers detected

No values flagged at the current threshold.

Per-test summary

Flagged values

none

R Reproducible code

# Outlier detection in R
library(outliers)

x <- c(2, 3, 3, 4, 4, 5, 5, 6, 7, 30)

# Grubbs (single outlier, two-sided)
grubbs.test(x, two.sided = TRUE)

# Generalized ESD (manual loop, k = max outliers)
esd <- function(x, k = 5, alpha = 0.05) {
  n <- length(x); xx <- x; idx <- seq_along(x)
  out <- integer(0)
  for (i in 1:k) {
    m <- mean(xx); s <- sd(xx); ni <- length(xx)
    R <- max(abs(xx - m)) / s
    p <- 1 - alpha / (2 * ni)
    t <- qt(p, ni - 2)
    lam <- (ni - 1) * t / sqrt(ni * (ni - 2 + t^2))
    if (R > lam) {
      j <- which.max(abs(xx - m))
      out <- c(out, idx[j]); xx <- xx[-j]; idx <- idx[-j]
    }
  }
  out
}
esd(x, k = 5)

# Hampel filter (median absolute deviation, k = 3)
mads <- abs(x - median(x)) / mad(x)
which(mads > 3)

# Tukey IQR rule (1.5 * IQR fences)
boxplot.stats(x)$out

Plot Dot plot with thresholds

Each dot is one observation. Red dots are flagged; dashed lines are method thresholds.

Flagged: 0 · n=10 · method: Grubbs

αalpha 0.05

kscale 3.0

Inference