Bootstrap resampling is a practical way to estimate uncertainty when you only have one sample from a population. Instead of assuming a perfect formula for the sampling distribution, the bootstrap uses the data you already collected as a stand-in for the population. It is especially useful for estimating standard errors and confidence intervals for statistics such as means, medians, proportions, and differences between groups.
The main idea is to resample from one sample many times to see how much the statistic changes.
Key Facts
- A bootstrap sample is made by sampling n observations with replacement from the original sample of size n.
- Resampling with replacement means the same data point can appear more than once in one bootstrap sample.
- For each bootstrap sample, compute the statistic of interest, such as x̄, median, p̂, or r.
- The bootstrap sampling distribution is the distribution of the statistic values from many bootstrap samples.
- Bootstrap standard error can be estimated by SE_boot = standard deviation of bootstrap statistics.
- A 95% percentile bootstrap confidence interval is approximately the 2.5th percentile to the 97.5th percentile of the bootstrap statistics.
Vocabulary
- Bootstrap resampling
- A method that repeatedly samples from the original data with replacement to estimate the variability of a statistic.
- Resampling with replacement
- A sampling process where each selected observation is put back before the next draw, so it can be chosen again.
- Bootstrap sample
- One new sample of the same size as the original sample, created by drawing observations with replacement from the original data.
- Sampling distribution
- The distribution of a statistic over many repeated samples or simulated resamples.
- Confidence interval
- A range of plausible values for a population parameter based on the observed data and its estimated uncertainty.
Common Mistakes to Avoid
- Sampling without replacement, because this only reshuffles the original data and does not create realistic variation in the statistic.
- Changing the bootstrap sample size, because each bootstrap sample should usually have the same size n as the original sample.
- Bootstrapping raw data that are not representative, because the method cannot fix bias from a poor sampling design.
- Treating a 95% confidence interval as a 95% probability that the true parameter is inside this one interval, because the parameter is fixed and the interval is random.
Practice Questions
- 1 An original sample has 8 values. You create 1000 bootstrap samples. How many observations should be drawn in each bootstrap sample, and are repeated values allowed?
- 2 The 2.5th percentile of 2000 bootstrap means is 12.4 and the 97.5th percentile is 18.9. What is the 95% percentile bootstrap confidence interval?
- 3 Explain why resampling with replacement from the original sample can help estimate uncertainty even when you do not know the population distribution.