The Toolkit

10,000 incomes with a true mean of 43,741, where higher earners skip the question more often (17% missing overall). The mean of the answers is 36,509 — and imputing the mean for the gaps keeps exactly that bias, while also shrinking the spread. When missingness depends on the value itself, no method on the observed data alone can fully recover it.

• Missing not at random (MNAR) cannot be detected from the data; it needs domain knowledge or a sensitivity analysis.
• Multiple imputation handles missing-at-random well; it does not fix MNAR.
• A missingness indicator can itself be predictive, and dropping it throws information away.

rng = np.random.default_rng(9)
income = rng.lognormal(10.5, 0.6, 10_000)
p_missing = np.clip((income - 30_000) / 100_000, 0, 0.8)     # higher earners skip the question more
observed = np.where(rng.random(10_000) < p_missing, np.nan, income)
true_mean = income.mean()
complete_case = np.nanmean(observed)                          # what mean imputation also centres on
share_missing = np.isnan(observed).mean()

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 — imputation strategies
from sklearn.impute import SimpleImputer, KNNImputer
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer

# Simple: median (robust to outliers)
imp = SimpleImputer(strategy='median')

# KNN: uses similar rows
imp = KNNImputer(n_neighbors=5)

# MICE: iterative multivariate
imp = IterativeImputer(max_iter=10)
X_filled = imp.fit_transform(X)