Should I use a z-score threshold of 2 or 3?

A threshold of 3 is more conservative (only extreme outliers), capturing 99.7% of normal data. A threshold of 2 captures 95.4% and flags more points. Use 2 for exploratory analysis, 3 for production filtering. Some domains use 2.5 as a compromise.

Can z-score miss outliers in small datasets?

Yes. In small samples, the mean and standard deviation are heavily influenced by outliers, which "mask" themselves by pulling the statistics toward them. This is why modified z-score (MAD) or Grubbs test is preferred for small datasets.

Why is 0.6745 used in the modified z-score formula?

The constant 0.6745 is the 75th percentile of the standard normal distribution (the value where P(Z <= 0.6745) = 0.75). It makes the MAD a consistent estimator of the standard deviation for normal data: for a normal distribution, MAD * 1/0.6745 = MAD * 1.4826 equals the standard deviation.

What if MAD equals zero?

MAD = 0 occurs when more than 50% of the data has the same value. In this case, the modified z-score is undefined. A common fix is to use the mean absolute deviation instead, or to add a small epsilon (e.g., MAD + 1e-6) to avoid division by zero.

What are the assumptions of Grubbs' test?

Grubbs' test assumes the data comes from a normally distributed population. It tests the null hypothesis that there are no outliers vs. the alternative that there is exactly one outlier. For non-normal data, consider the IQR method or generalized ESD test.

Can Grubbs' test detect multiple outliers?

The basic Grubbs test identifies only one outlier at a time. For multiple outliers, apply it iteratively: remove the detected outlier, then re-run. However, this can suffer from masking. The generalized ESD (Extreme Studentized Deviate) test is better for multiple outliers.

All Tools

Outlier Detection Calculator

Identify outliers using IQR, z-score, modified z-score (MAD), and Grubbs' test with step-by-step solutions and Python code.

Enter Data

Dataset (comma or space separated)

Sample Data:

Enter your data and click Detect Outliers.

Available Methods:

IQR: Q1 - 1.5*IQR to Q3 + 1.5*IQR
Z-Score: |z| > threshold (2 or 3)
Modified Z: MAD-based, robust
Grubbs': Hypothesis test for single outlier

Frequently Asked Questions

Why use 1.5 as the IQR multiplier?

The 1.5 * IQR rule was proposed by John Tukey. For normally distributed data, this captures about 99.3% of values. A multiplier of 3.0 identifies "extreme outliers." The 1.5 value balances sensitivity and specificity for most practical datasets.

How does the IQR method differ from z-score for outlier detection?

The IQR method is non-parametric and uses quartiles, making it robust to skewed data and resistant to the influence of outliers themselves. Z-score assumes normality and uses mean/std dev, which are heavily affected by outliers. Use IQR for skewed data, z-score for approximately normal data.

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