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Bootstrap

Also known as: bootstrap resampling, resampling.

Estimating how much a statistic would vary from sample to sample by resampling the observed data, with replacement, many times.

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Synthetic dataExamples on this page use synthetic data created for illustration.

Definition

The bootstrap estimates how much a statistic would vary from sample to sample by resampling the observed data with replacement many times and recomputing the statistic each time. The spread of the results approximates its uncertainty without needing a formula.

In plain terms

You have one sample and want to know how reliable a number computed from it is. The bootstrap treats the sample as a stand-in for the population: draw a new sample of the same size from your own data, putting each value back after it is drawn, and compute the statistic. Do this thousands of times. The middle 95% of the results is a rough 95% interval.

It works for statistics with no tidy formula, such as a median or a ratio. It does not rescue a tiny or biased sample, and it needs adapting when observations are dependent, as in a time series.

Example

Eight prices: 12, 15, 14, 13, 16, 15, 14 and 40. The mean is 17.4 and the median is 14.5. A bootstrap with 10,000 resamples gives:

StatisticEstimateBootstrap 95% interval
Mean17.4about 13.5 to 24.1 (wide and lopsided: the 40 is sometimes drawn, sometimes not)
Median14.5about 13 to 16 (narrow: the 40 barely matters)

The intervals show that the median is the steadier summary of these data.

Common misunderstandings

  • “The bootstrap creates new data.” It reuses the data you have; it reveals variability but adds no information.
  • “It works on any sample, however small.” With a handful of observations the resamples have little to vary.
  • “Resampling with and without replacement are the same.” Replacement is what makes each resample a plausible new sample.
  • Confidence Interval

    A range calculated from a sample that expresses how precisely a quantity has been estimated. A 95% interval comes from a procedure that captures the true value in 95% of repetitions.

  • Population and Sample

    The population is everything you want to know about; the sample is the part you observed. Statistics asks what the second says about the first.

  • Median and Mean

    The mean is the sum divided by the count; the median is the middle value. Extreme values pull the mean but not the median.

References

  1. Bradley Efron. Bootstrap Methods: Another Look at the Jackknife . The Annals of Statistics, 1979. DOI: 10.1214/aos/1176344552
  2. Bradley Efron, Robert J. Tibshirani. An Introduction to the Bootstrap . Chapman and Hall/CRC, 1993. ISBN 978-0412042317.