Bootstrap Simulation

Simulations

Bootstrap Resampling Simulation

Parameters

Sample Dataset

Sample size:20

Sample mean:79.7500

Sample median:80.0000

Sample std dev:9.4025

Bootstrap Statistic

Number of Resamples

Confidence Level (%)

Bootstrap Distribution of the Mean

Run the simulation to see the bootstrap distribution

Learn More

What is the Bootstrap Method?

The bootstrap is a powerful resampling method introduced by Bradley Efron in 1979. It estimates the sampling distribution of a statistic by repeatedly sampling with replacement from the original data. This lets you quantify uncertainty (standard errors, confidence intervals) without making assumptions about the population distribution.

"The bootstrap treats the sample as if it were the population, and resampling from it simulates the process of drawing new samples from the population."

How Bootstrap Resampling Works

Start with your original sample of $n$ observations
Draw a bootstrap sample of size $n$ by sampling with replacement from the original data
Compute the statistic of interest (mean, median, etc.) on the bootstrap sample
Repeat steps 2-3 a large number of times (e.g., 1,000 or 10,000)
The collection of bootstrap statistics approximates the sampling distribution

Mathematical Foundation

Given a sample $x_1, x_2, ..., x_n$ , each bootstrap sample $x_1^*, x_2^*, ..., x_n^*$ is drawn with replacement. For each bootstrap sample $b = 1, 2, ..., B$ , compute the statistic:

\hat{\theta}^*_b = s(x_1^*, x_2^*, ..., x_n^*)

The bootstrap standard error is the standard deviation of the bootstrap statistics:

SE_{boot} = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\theta}^*_b - \bar{\hat{\theta}}^*)^2}

The percentile confidence interval uses the quantiles of the bootstrap distribution directly:

CI_{1-\alpha} = [\hat{\theta}^*_{\alpha/2}, \hat{\theta}^*_{1-\alpha/2}]

When to Use Bootstrap

When the theoretical sampling distribution is unknown or complex
When you cannot assume normality (small samples, skewed data)
For statistics without simple closed-form standard errors (e.g., median, ratios, correlation)
When you want a non-parametric confidence interval that doesn't depend on distributional assumptions
In machine learning for estimating model performance variability

Related Calculators

Confidence Interval for Mean

Sample Size & Power Analysis Calculator

Confidence Interval Simulation

View All Statistics Calculators

Bootstrap Simulation

Simulations

Bootstrap Resampling Simulation

Parameters

Sample Dataset

Sample size:20

Sample mean:79.7500

Sample median:80.0000

Sample std dev:9.4025

Bootstrap Statistic

Number of Resamples

Confidence Level (%)

Bootstrap Distribution of the Mean

Run the simulation to see the bootstrap distribution

Learn More

What is the Bootstrap Method?

"The bootstrap treats the sample as if it were the population, and resampling from it simulates the process of drawing new samples from the population."

How Bootstrap Resampling Works

Start with your original sample of $n$ observations
Draw a bootstrap sample of size $n$ by sampling with replacement from the original data
Compute the statistic of interest (mean, median, etc.) on the bootstrap sample
Repeat steps 2-3 a large number of times (e.g., 1,000 or 10,000)
The collection of bootstrap statistics approximates the sampling distribution

Mathematical Foundation

Given a sample $x_1, x_2, ..., x_n$ , each bootstrap sample $x_1^*, x_2^*, ..., x_n^*$ is drawn with replacement. For each bootstrap sample $b = 1, 2, ..., B$ , compute the statistic:

\hat{\theta}^*_b = s(x_1^*, x_2^*, ..., x_n^*)

The bootstrap standard error is the standard deviation of the bootstrap statistics:

SE_{boot} = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\theta}^*_b - \bar{\hat{\theta}}^*)^2}

The percentile confidence interval uses the quantiles of the bootstrap distribution directly:

CI_{1-\alpha} = [\hat{\theta}^*_{\alpha/2}, \hat{\theta}^*_{1-\alpha/2}]

When to Use Bootstrap

When the theoretical sampling distribution is unknown or complex
When you cannot assume normality (small samples, skewed data)
For statistics without simple closed-form standard errors (e.g., median, ratios, correlation)
When you want a non-parametric confidence interval that doesn't depend on distributional assumptions
In machine learning for estimating model performance variability

Related Calculators

Confidence Interval for Mean

Sample Size & Power Analysis Calculator

Confidence Interval Simulation

View All Statistics Calculators

Bootstrap Simulation

Simulations

Bootstrap Resampling Simulation

Parameters

Sample Dataset

Sample size:20

Sample mean:79.7500

Sample median:80.0000

Sample std dev:9.4025

Bootstrap Statistic

Number of Resamples

Confidence Level (%)

Bootstrap Distribution of the Mean

Run the simulation to see the bootstrap distribution

Learn More

What is the Bootstrap Method?

"The bootstrap treats the sample as if it were the population, and resampling from it simulates the process of drawing new samples from the population."

How Bootstrap Resampling Works

Start with your original sample of $n$ observations
Draw a bootstrap sample of size $n$ by sampling with replacement from the original data
Compute the statistic of interest (mean, median, etc.) on the bootstrap sample
Repeat steps 2-3 a large number of times (e.g., 1,000 or 10,000)
The collection of bootstrap statistics approximates the sampling distribution

Mathematical Foundation

Given a sample $x_1, x_2, ..., x_n$ , each bootstrap sample $x_1^*, x_2^*, ..., x_n^*$ is drawn with replacement. For each bootstrap sample $b = 1, 2, ..., B$ , compute the statistic:

\hat{\theta}^*_b = s(x_1^*, x_2^*, ..., x_n^*)

The bootstrap standard error is the standard deviation of the bootstrap statistics:

SE_{boot} = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\theta}^*_b - \bar{\hat{\theta}}^*)^2}

The percentile confidence interval uses the quantiles of the bootstrap distribution directly:

CI_{1-\alpha} = [\hat{\theta}^*_{\alpha/2}, \hat{\theta}^*_{1-\alpha/2}]

When to Use Bootstrap

When the theoretical sampling distribution is unknown or complex
When you cannot assume normality (small samples, skewed data)
For statistics without simple closed-form standard errors (e.g., median, ratios, correlation)
When you want a non-parametric confidence interval that doesn't depend on distributional assumptions
In machine learning for estimating model performance variability

Bootstrap Simulation

Simulations

Bootstrap Resampling Simulation

Parameters

Sample Dataset

Sample size:20

Sample mean:79.7500

Sample median:80.0000

Sample std dev:9.4025

Bootstrap Statistic

Number of Resamples

Confidence Level (%)

Bootstrap Distribution of the Mean

Run the simulation to see the bootstrap distribution

Learn More

What is the Bootstrap Method?

"The bootstrap treats the sample as if it were the population, and resampling from it simulates the process of drawing new samples from the population."

How Bootstrap Resampling Works

Start with your original sample of $n$ observations
Draw a bootstrap sample of size $n$ by sampling with replacement from the original data
Compute the statistic of interest (mean, median, etc.) on the bootstrap sample
Repeat steps 2-3 a large number of times (e.g., 1,000 or 10,000)
The collection of bootstrap statistics approximates the sampling distribution

Mathematical Foundation

Given a sample $x_1, x_2, ..., x_n$ , each bootstrap sample $x_1^*, x_2^*, ..., x_n^*$ is drawn with replacement. For each bootstrap sample $b = 1, 2, ..., B$ , compute the statistic:

\hat{\theta}^*_b = s(x_1^*, x_2^*, ..., x_n^*)

The bootstrap standard error is the standard deviation of the bootstrap statistics:

SE_{boot} = \sqrt{\frac{1}{B-1} \sum_{b=1}^{B} (\hat{\theta}^*_b - \bar{\hat{\theta}}^*)^2}

The percentile confidence interval uses the quantiles of the bootstrap distribution directly:

CI_{1-\alpha} = [\hat{\theta}^*_{\alpha/2}, \hat{\theta}^*_{1-\alpha/2}]

When to Use Bootstrap

When the theoretical sampling distribution is unknown or complex
When you cannot assume normality (small samples, skewed data)
For statistics without simple closed-form standard errors (e.g., median, ratios, correlation)
When you want a non-parametric confidence interval that doesn't depend on distributional assumptions
In machine learning for estimating model performance variability