The Toolkit

Twelve values around 10, plus 25 and 300. The z-score rule flags only 300: that one value inflates the standard deviation so much that 25 hides behind it (“masking”). The median-based modified z-score flags both 25 and 300. Outlier rules should not be built from statistics the outliers themselves distort.

• An outlier is defined relative to a model; a value extreme in one feature can be normal given the others.
• Deleting outliers automatically can delete the most important cases (fraud, failures, rare events).
• Thresholds such as 3 or 3.5 are conventions; the expected number of false flags grows with the data.

x = np.array([10, 11, 9, 10, 12, 11, 10, 9, 11, 10, 25, 300.0])   # two outliers: 25 and 300
z = (x - x.mean()) / x.std()
mad = np.median(np.abs(x - np.median(x)))
robust = 0.6745 * (x - np.median(x)) / mad          # modified z-score
flag_z = x[np.abs(z) > 3].tolist()
flag_robust = x[np.abs(robust) > 3.5].tolist()

Assumes import numpy as np and import pandas as pd (and SciPy where imported). Each snippet simulates or makes up its own data, as the comments say.

# Python — outlier detection
from sklearn.ensemble import IsolationForest
import numpy as np

# IQR method
Q1, Q3 = np.percentile(data, [25, 75])
IQR = Q3 - Q1
mask = (data < Q1 - 1.5*IQR) | (data > Q3 + 1.5*IQR)

# Isolation Forest — multi-dimensional
iso = IsolationForest(contamination=0.05)
labels = iso.fit_predict(X)  # -1 = outlier