The Toolkit

Bootstrapping the maximum of 50 values: in 63.8% of resamples the “new” maximum is just the original maximum, matching the theoretical 1 − (1 − 1/50)⁵⁰ = 63.6%. The bootstrap distribution piles up on one value and cannot see beyond the sample. The bootstrap works for smooth statistics like means; it fails for extremes.

• Other failures: very small samples, heavy tails with infinite variance, and dependent data resampled as if independent.
• Dependent data need block or stationary bootstraps that keep neighbours together.
• The bootstrap estimates sampling variability, not bias from a flawed design.

rng = np.random.default_rng(15)
x = rng.uniform(0, 1, 50)
boot_max = np.array([rng.choice(x, 50).max() for _ in range(10_000)])
share_equal = (boot_max == x.max()).mean()            # resamples whose max is just the sample max
theory = 1 - (1 - 1 / 50) ** 50

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 — bootstrap for any statistic
import numpy as np

def bootstrap(data, stat_fn, B=10000):
    n = len(data)
    estimates = [stat_fn(np.random.choice(data, n, replace=True))
                 for _ in range(B)]
    return np.array(estimates)

# Bootstrap the median (no formula needed!)
boots = bootstrap(data, np.median)
ci = np.percentile(boots, [2.5, 97.5])
se = boots.std()