Updated for the 2026 CFA® Level I curriculum.
Resampling methods help analysts learn more from an observed data sample by reusing its observations in a structured way. The bootstrap draws repeated samples with replacement, while the jackknife removes one observation at a time and recalculates the statistic.
For CFA Level I, you should understand how each process works, what the resulting estimates can tell you, and why both methods remain dependent on the quality of the original sample.
Quick Answer
Bootstrap resampling repeatedly draws samples with replacement from the observed data, usually keeping the original sample size. Jackknife resampling creates leave-one-out samples by removing each observation in turn. The resulting statistics help analysts estimate standard error, bias, or the shape of a sampling distribution when a direct analytical approach is difficult.
Key Takeaways About Resampling Methods
Resampling methods reuse observations from an existing sample.
Bootstrap samples are drawn with replacement.
A bootstrap sample usually contains the same number of observations as the original sample.
Jackknife resampling removes one observation at a time.
A sample containing observations produces jackknife samples, each containing observations.
Bootstrap resampling is generally more flexible for approximating sampling distributions.
Jackknife resampling is commonly used to estimate standard error and bias.
Neither method creates genuinely new information or corrects an unrepresentative original sample.
What You Need to Know for CFA Level I
For CFA Level I, focus on:
Explaining why analysts use resampling.
Distinguishing random bootstrap samples from systematic jackknife samples.
Recognizing that bootstrap observations may appear more than once.
Recognizing that some observations may be absent from a bootstrap sample.
Determining the number and size of jackknife samples.
Explaining how resampled statistics provide information about estimation uncertainty.
Understanding that resampling quality depends on the original data.
The process and interpretation matter more than memorizing every formula. The equations below show how the resampled estimates are summarized.
What Does Resampling Mean?
Resampling means repeatedly reusing observations from an available sample to study how a statistic might behave across different samples.
Suppose an analyst observes the following sample:
The analyst uses these observations to calculate a statistic:
The statistic could represent a mean, median, correlation, regression coefficient, or portfolio risk measure.
Resampling creates modified versions of the original sample. The analyst recalculates for each version and studies how the resulting estimates vary. A wider spread suggests greater uncertainty around the original estimate.
How Does Bootstrap Resampling Work?
Bootstrap resampling draws observations randomly with replacement from the original sample.
A standard bootstrap process follows these steps:
Begin with an observed sample containing values.
Draw observations from that sample with replacement.
Calculate the chosen statistic for the bootstrap sample.
Repeat the process times.
Use the calculated statistics to approximate the statistic’s sampling distribution.
Sampling with replacement allows the same observation to appear more than once. Some original observations may also be left out of a particular bootstrap sample.
For example, suppose the original sample is:
One possible bootstrap sample is:
The 2% observation appears twice, while 4%does not appear. The resample still contains four observations, matching the size of the original sample.
If the bootstrap statistics are:
their average is:
The estimated bootstrap standard error is:
Where:
= total number of bootstrap resamples
= one particular bootstrap resample
= statistic calculated from bootstrap sample $b$
= average of all bootstrap statistics
= estimated bootstrap standard error
The standard error measures the spread of the bootstrap statistics around their average. A larger spread means the estimate varies more across the resamples.
How Does Jackknife Resampling Work?
Jackknife resampling creates a series of leave-one-out samples.
For an original sample containing observations:
Remove the first observation and calculate the statistic.
Restore the first observation.
Remove the second observation and recalculate the statistic.
Continue until every observation has been omitted once.
Compare the resulting $n$ estimates.
The estimate calculated after observation $i$ has been removed is:
The average of the jackknife estimates is:
The jackknife standard error is:
The estimated jackknife bias is:
Where:
= number of observations in the original sample
= observation omitted from a jackknife sample
= statistic calculated from the full sample
= statistic calculated after observation $i$ is removed
= average of all jackknife estimates
= estimated jackknife standard error
= estimated jackknife bias
The jackknife follows a fixed process rather than random sampling. An original sample with observations always produces exactly leave-one-out samples, each containing observations.
Bootstrap vs Jackknife Resampling: Main Difference
Area | Bootstrap Resampling | Jackknife Resampling |
|---|---|---|
Resampling rule | Draw observations with replacement | Remove one observation at a time |
Resample size | Usually | |
Number of resamples | Chosen by the analyst, often a large number | Exactly |
Selection process | Random | Systematic |
Repeated observations | Allowed | No |
Omitted observations | Possible in each resample | Exactly one per resample |
Main use | Estimate a sampling distribution, standard error, confidence interval, or bias | Estimate standard error and bias |
Flexibility | High | More limited |
Main limitation | May require many calculations and depends on the original sample | May perform poorly for non-smooth or unstable statistics |
A useful way to remember the distinction is to focus on the action:
Bootstrap: draw repeatedly with replacement.
Jackknife: leave one observation out at a time.
Worked Example: Jackknife Standard Error
Assume an analyst observes four annual returns:
The full-sample mean is:
The analyst removes each observation once and recalculates the mean.
Omitted Return | Remaining Returns | Jackknife Mean |
|---|---|---|
2% | 4%, 6%, 8% | 6.00% |
4% | 2%, 6%, 8% | 5.33% |
6% | 2%, 4%, 8% | 4.67% |
8% | 2%, 4%, 6% | 4.00% |
Step 1: Calculate the Average Jackknife Estimate
The average jackknife estimate equals the full-sample mean.
Step 2: Estimate the Jackknife Bias
The estimated bias is zero for this sample mean.
Step 3: Calculate the Jackknife Standard Error
The estimated standard error is approximately 1.29 percentage points.
The leave-one-out means show how much the sample mean changes when each return is removed. An observation that causes a large change when omitted may have a strong influence on the estimate.
Bootstrap Resampling Example
Using the same four returns, one possible bootstrap sample is:
Its mean is:
A second bootstrap sample might be:
Its mean is:
Two bootstrap samples are not enough to estimate a reliable sampling distribution. In practice, the analyst repeats the process many times.
The resulting collection of bootstrap means shows:
Where the estimates tend to be centered.
How widely they vary.
Whether the estimated distribution appears symmetric or skewed.
How much uncertainty surrounds the original sample statistic.
When Are Resampling Methods Useful in Investment Analysis?
Resampling methods are useful when analysts want to study estimation uncertainty without repeatedly collecting new samples from the full population.
Possible investment applications include:
Estimating the standard error of a portfolio statistic.
Examining the stability of a correlation estimate.
Evaluating how sensitive a regression coefficient is to the available observations.
Approximating the sampling distribution of a risk measure.
Estimating confidence intervals when analytical formulas are difficult.
Identifying whether particular observations strongly influence an estimate.
Conducting simulations based on patterns found in observed data.
Bootstrap resampling is especially useful for statistics with complicated sampling distributions. Jackknife resampling provides a more direct way to examine how each observation affects an estimate.
Why Does the Original Sample Still Matter?
Every resample comes from the observations in the original dataset. The resampling method can only reproduce values and relationships that are already present.
A representative original sample gives the analysis a stronger foundation. A sample affected by selection bias, missing data, measurement errors, or an unusual market period carries those weaknesses into the resampled results.
Increasing the number of bootstrap resamples can reduce simulation noise within the bootstrap procedure. It cannot add missing information or correct systematic problems in the original sample.
Bootstrap Resampling vs Monte Carlo Simulation
Bootstrap resampling and Monte Carlo simulation both generate repeated outcomes, but they use different sources.
Bootstrap resampling draws from the observed data. It is an empirical method because the resampled values come directly from the original sample.
Monte Carlo simulation draws outcomes from an assumed probability model. The analyst selects a distribution and its parameters, then generates simulated observations from that model.
Area | Bootstrap Resampling | Monte Carlo Simulation |
|---|---|---|
Source of outcomes | Observed sample | Assumed probability model |
Main assumption | Original sample represents the population reasonably well | Selected model represents the process reasonably well |
Possible simulated values | Usually limited to observed values | May include values not present in the historical sample |
Main use | Empirical sampling-distribution estimation | Model-based scenario and outcome simulation |
The exam may test whether you can identify which method relies on observed data and which relies on a specified model.
Common Exam Traps
Common mistakes include:
Saying bootstrap samples are drawn without replacement.
Assuming each observation appears exactly once in every bootstrap sample.
Forgetting that bootstrap samples may repeat some observations and omit others.
Saying a jackknife sample contains observations.
Forgetting that an original sample of size creates jackknife samples.
Treating resampled observations as newly collected independent data.
Confusing bootstrap resampling with Monte Carlo simulation.
Assuming that more resamples correct selection bias.
Assuming jackknife and bootstrap procedures construct samples in the same way.
Memorizing formulas without understanding how the resamples are formed.
Practice Question
An analyst has a sample containing 20 observations and uses jackknife resampling to estimate the standard error of a statistic. Which statement is most accurate?
The analyst creates 19 samples containing 20 observations each.
The analyst creates 20 samples containing 19 observations each.
The analyst creates a large number of random samples containing 20 observations each.
Correct Answer: B
Jackknife resampling removes each observation once. A sample containing 20 observations therefore produces 20 leave-one-out samples, each containing 19 observations.
Option A. reverses the number and size of the jackknife samples.
Option C. describes bootstrap resampling more closely. Bootstrap procedures usually create many random samples with replacement, each with the same size as the original sample.
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FAQs About Resampling Methods
What Is Resampling in Simple Terms?
Resampling reuses observations from an existing sample to create alternative samples. Analysts calculate the same statistic from each one and compare the results to learn about sampling variation, standard error, or bias.
What Is the Main Difference Between Bootstrap and Jackknife Resampling?
Bootstrap resampling randomly draws observations with replacement. Jackknife resampling systematically removes one observation at a time.
A bootstrap procedure usually produces many random samples of size . A jackknife procedure produces exactly $n$ samples of size .
Does Bootstrap Resampling Use Replacement?
Yes. Bootstrap samples are drawn with replacement. An observation may appear more than once in a bootstrap sample, while another observation may not appear at all.
How Many Samples Does the Jackknife Method Produce?
An original sample containing observations produces exactly jackknife samples. Each jackknife sample contains observations because one observation is removed at a time.
Can Resampling Fix a Biased Original Sample?
Resampling cannot correct systematic bias in the original sample. It reuses the available observations, including any problems created by poor sample selection, missing data, or measurement errors