Question

Why is the standard deviation of the bootstrapped distribution smaller than that of the sample?

Answer 1

StDev (bootstrap sample)

The standard deviation is the most common measure of dispersion /or how spread out the data are about the mean. The σ (sigma) sign is often used to represent the standard deviation of a population, while s is used to represent the standard deviation of a sample. Variation that is random or natural to a process is often referred to as noise. Because the standard deviation is in the same units as the data, it is usually easier to interpret than the variance.

The standard deviation of the bootstrap samples (also known as bootstrap standard error) is an estimate of the standard deviation of the sampling distribution of the mean.
Because the bootstrap standard error is the variation of sample means, whereas the standard deviation of the observed samples is the variation of individual observations, the bootstrap standard error is smaller.

Interpretation

Use the SD(standard deviation) to determine how spread out the means from the bootstrap sample are from the overall mean. A higher SDvalue indicates greater spread in the means. A good rule of thumb for the normal distribution is that approximately 68% of the values fall within one standard deviation of the overall mean, 95% of the values fall within two standard deviations, and 99.7% of the values fall within three standard deviations.

Use the standard deviation of the bootstrap samples to determine how precisely the bootstrap means estimate the population mean. A smaller value indicates a more precise estimate of the population mean. Usually the larger standard deviation results in a larger bootstrap standard error and a less precise estimate of the population mean. A larger sample size results in a smaller bootstrap standard error and a more precise estimate of the population mean