How to compute a bootstrap confidence interval with SciPy

A point estimate reduces a sample to one number, but it does not show how that estimate could vary across repeated samples. scipy.stats.bootstrap() resamples observed values with replacement and returns confidence bounds for a statistic without requiring a normal-distribution formula.

For a one-sample mean, bootstrap() still expects the data as a one-item tuple such as (latency_ms,). Giving the statistic an axis parameter lets SciPy process batches of resamples, while a fixed numpy.random.Generator makes the simulated distribution reproducible.

The bias-corrected and accelerated method, abbreviated BCa, adjusts the interval limits for bias and skewness. A degenerate statistic can make BCa bounds nan, so the program checks for finite ordered bounds, a finite standard error, and the requested resample count before reporting success.

1. Create the initial bootstrap_ci.py file with the sample and repeatable random-number generator.

   bootstrap_ci.py

   import numpy as np
   from scipy.stats import bootstrap
    
   latency_ms = np.array([
       117, 121, 113, 125,
       119, 128, 115, 122,
       120, 118, 124, 116,
   ])
   rng = np.random.default_rng(20260625)

2. Append the vectorized mean statistic to bootstrap_ci.py.

    
   def mean_latency(sample, axis):
       return np.mean(sample, axis=axis)

3. Append the BCa resampling call below mean_latency().

    
   result = bootstrap(
       (latency_ms,),
       mean_latency,
       confidence_level=0.95,
       n_resamples=9999,
       method="BCa",
       rng=rng,
   )

4. Complete bootstrap_ci.py with the final result-handling section.

   bootstrap_ci.py

   import numpy as np
   from scipy.stats import bootstrap
    
   latency_ms = np.array([
       117, 121, 113, 125,
       119, 128, 115, 122,
       120, 118, 124, 116,
   ])
   rng = np.random.default_rng(20260625)
    
    
   def mean_latency(sample, axis):
       return np.mean(sample, axis=axis)
    
    
   result = bootstrap(
       (latency_ms,),
       mean_latency,
       confidence_level=0.95,
       n_resamples=9999,
       method="BCa",
       rng=rng,
   )
    
   interval = result.confidence_interval
   sample_mean = np.mean(latency_ms)
   checks_passed = (
       np.isfinite([interval.low, interval.high]).all()
       and np.isfinite(result.standard_error)
       and interval.low < interval.high
       and result.bootstrap_distribution.shape[-1] == 9999
   )
    
   if not checks_passed:
       raise RuntimeError("bootstrap result failed validation")
    
   print(f"sample mean: {sample_mean:.2f} ms")
   print(f"95% BCa interval: {interval.low:.2f} to {interval.high:.2f} ms")
   print(f"bootstrap standard error: {result.standard_error:.2f} ms")
   print(f"resamples: {result.bootstrap_distribution.shape[-1]}")
   print("validation checks: passed")

   BCa can return nan bounds when identical observations or a statistic that does not vary produce a degenerate bootstrap distribution.

5. Confirm the confidence interval by running bootstrap_ci.py.

   $ python3 bootstrap_ci.py
   sample mean: 119.83 ms
   95% BCa interval: 117.58 to 122.25 ms
   bootstrap standard error: 1.20 ms
   resamples: 9999
   validation checks: passed