Bootstrap sample statistics and graphs for Bootstrapping for 1-Sample Mean

Find definitions and interpretation guidance for every bootstrap sample statistic and graph that is provided with bootstrapping for 1-sample mean.

Histogram

A histogram divides sample values into many intervals and represents the frequency of data values in each interval with a bar.

Interpretation

Use the histogram to examine the shape of your bootstrap distribution. The bootstrap distribution is the distribution of means from each resample. The bootstrap distribution should appear to be normal. If the bootstrap distribution is non-normal, you cannot trust the results.
50 resamples
1000 resamples

The distribution is usually easier to determine with more resamples. For example, in these data, the distribution is ambiguous for 50 resamples. With 1000 resamples, the shape looks approximately normal.

In this histogram, the bootstrap distribution appears to be normal.

Individual value plot

An individual value plot displays the individual values in the sample. Each circle represents one observation. An individual value plot is especially useful when you have relatively few observations and when you also need to assess the effect of each observation.

Note

Minitab displays an individual value plot only when you take only one resample. Minitab displays both the original data and the resample data.

Interpretation

With a large sample size, the bootstrap sample will usually have a similar center and spread as the original sample. However, a small sample size may result in a bootstrap sample that is not similar to the original sample. If your bootstrap sample does not look like your original sample, you should consider increasing your sample size.
Sample size of 8
Sample size of 50

Number of Resamples

The number of resamples is the number of times Minitab takes a random sample with replacement from your original data set. Usually, a large number of resamples works best. The sample size for each resample is equal to the sample size of the original data set. The number of resamples equals the number of observations on the histogram.

Average

The average is the sum of all the means in the bootstrapping sample divided by the number of resamples.

Interpretation

Minitab displays two different mean values, the mean of the observed sample and the mean of the bootstrap distribution (Average). Both these values are an estimate of the population mean and will usually be similar. If there is a large difference between these two values, you should increase the sample size of your original sample.

Because the mean is based on sample data and not on the entire population, it is unlikely that the sample mean equals the population mean. To better estimate the population mean, use the confidence interval.

StDev (bootstrap sample)

The standard deviation is the most common measure of dispersion, or how spread out the data are about the mean. The symbol σ (sigma) 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 the 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 standard deviation to determine how spread out the means from the bootstrap sample are from the overall mean. A higher standard deviation value indicates greater spread in the means. A good rule of thumb for a 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, a 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.

The standard deviation can also be used to establish a benchmark for estimating the overall variation of a process.
Hospital 1
Hospital 2
Hospital discharge times

Administrators track the discharge time for patients who are treated in the emergency departments of two hospitals. Although the average discharge times are about the same (35 minutes), the standard deviations are significantly different. The standard deviation for hospital 1 is about 6. On average, a patient's discharge time deviates from the mean (dashed line) by about 6 minutes. The standard deviation for hospital 2 is about 20. On average, a patient's discharge time deviates from the mean (dashed line) by about 20 minutes.

Confidence interval (CI) and bounds

Confidence intervals are based on the sampling distribution of a statistic. If a statistic has no bias as an estimator of a parameter, its sampling distribution is centered at the true value of the parameter. A bootstrapping distribution approximates the sampling distribution of the statistic. Therefore, the middle 95% of values from the bootstrapping distribution provide a 95% confidence interval for the parameter. The confidence interval helps you assess the practical significance of your estimate for the population parameter. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation.

Note

Minitab does not calculate the confidence interval when the number of resamples is too small to obtain an accurate confidence interval.

Bootstrap Samples for Mean
Number of Resamples
Mean
StDev

In these results, the estimate for the population mean is approximately 11.3. You can be 95% confident that the population mean is between approximately 9.9 and 12.9.

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