CFA Level I Exam · Estimation and Hypothesis Testing

Bootstrap vs Jackknife Resampling for CFA Level I

Updated 7 October 2026 · Fact-checked

Resampling estimates the sampling distribution of a statistic using only your own sample. Bootstrap draws many random samples with replacement, each the same size as the original. Jackknife leaves out one observation at a time, giving n samples of size n − 1. Both estimate standard error without assuming a formula.

Understand Resampling: Bootstrap and Jackknife

A statistic, such as a sample mean or median, changes from sample to sample. Its spread is the standard error. For the sample mean, you have a formula: s ÷ √n. For many other statistics, such as the median or a correlation, no simple formula exists or the data are not normal. Resampling solves this.

The idea is simple. Treat your sample as a stand-in for the population. Draw new samples from it, compute the statistic each time, and look at how the results vary. That variation estimates the sampling distribution.

Bootstrap draws samples with replacement from the original sample. Each new sample has the same size n as the original. Because of replacement, one observation can appear twice or more in a resample, and others not at all. You repeat this many times, often thousands. The standard deviation of the resulting statistics is the bootstrap estimate of the standard error. It is simulation-based, so the answer differs slightly each run.

Jackknife is systematic, not random. From a sample of n observations, you create n new samples by leaving out one observation each time. Each has n − 1 observations. You compute the statistic for each one and use the variation to estimate the standard error (and bias). Because it is fixed, the jackknife gives the same result every time.

The jackknife standard error is not a plain standard deviation of the leave-one-out statistics. The sum of squared deviations is scaled by (n − 1) ÷ n, as shown in the formulas below. The leave-one-out statistics are very similar to each other, so this scaling is what turns their small spread into a proper standard error estimate.

Both are nonparametric: they make no assumption about the shape of the population. Both use only the data you already have. They differ in the draw rule, the number of resamples and the sample size of each resample. Bootstrap can also be used for confidence intervals, and it is more flexible. Neither fixes a sample that is biased or too small to represent the population.

Key formulas to remember

Standard error of the sample mean (analytical): s_X̄ = s ÷ √n; The benchmark formula. Resampling is used when no such formula is practical.
Bootstrap standard error: SE(bootstrap) = standard deviation of the B resampled statistics; Each resample is drawn with replacement and has size n. The usual convention is the sample standard deviation of the B statistics.
Jackknife resample count and size: n resamples, each of size n − 1; Each resample omits exactly one observation. Result is deterministic.
Jackknife mean of the statistic: θ̄ = (1 ÷ n) × Σ θ_i; θ_i is the statistic computed with observation i left out.
Jackknife standard error: SE(jackknife) = √[ ((n − 1) ÷ n) × Σ(θ_i − θ̄)² ]; Sum runs over i = 1 to n. It is not a plain standard deviation of the θ_i: the sum of squares is scaled by (n − 1) ÷ n.

How to solve Resampling: Bootstrap and Jackknife questions

Use this method for any question that asks you to identify, compare or apply bootstrap or jackknife.

1Read what is being drawn: with replacement and random, or leave-one-out and systematic.
2If random with replacement, same size n as the original sample, it is bootstrap.
3If one observation is dropped each time, giving n−1 observations per sample, it is jackknife.
4For counting questions, jackknife gives exactly n resamples. Bootstrap count is chosen by the analyst.
5For calculation questions, compute the statistic for each resample, then find the mean or standard deviation as asked. For a jackknife standard error, use the (n − 1) ÷ n scaled formula, not a plain standard deviation.
6Check the purpose: estimating standard error, bias or a confidence interval without a distribution assumption.
7Eliminate options that claim resampling fixes sampling bias or creates new information beyond the sample.

Quickest way: Replacement and Leave-One-Out Test

When to use it: Use for any definition or comparison question where you must name the method.

Ask: is any observation removed? If yes, think jackknife.
Ask: can an observation appear twice? If yes, think bootstrap.
Ask: is the result the same each run? Jackknife yes, bootstrap no.
Pick the option matching these three tests and discard the rest.

Common mistakes in Resampling: Bootstrap and Jackknife

Saying the jackknife samples with replacement.
Both are called resampling, so the rules blur together.
Fix: Jackknife leaves one out and has no repeats. Replacement belongs to bootstrap.
Thinking jackknife resamples have size n.
Bootstrap resamples have size n, so students carry that over.
Fix: Each jackknife sample has n − 1 observations, and there are n of them.
Believing the jackknife result changes between runs.
Students link all resampling with randomness.
Fix: Jackknife is deterministic. Only bootstrap involves random draws.
Thinking resampling corrects a biased or unrepresentative sample.
It feels like creating more data.
Fix: Resampling only reuses the existing data. A poor sample gives poor estimates.
Assuming bootstrap needs a normal population.
Confusing it with parametric tests such as the z-test.
Fix: Bootstrap is nonparametric and makes no distribution assumption about the population.
Computing the jackknife standard error as a plain standard deviation of the leave-one-out statistics.
It is natural to copy the bootstrap rule, which does use the standard deviation of the resampled statistics.
Fix: Use √[((n − 1) ÷ n) × Σ(θ_i − θ̄)²]. The sum of squares is scaled by (n − 1) ÷ n.

Worked examples

Example 1

An analyst has 25 monthly returns on a fund and uses the jackknife to estimate the standard error of the median return. How many resamples are created, and how many observations does each contain?

Show the solution

Jackknife leaves out one observation at a time.
With n = 25, there are 25 resamples.
Each resample omits one observation, so it has 25 − 1 = 24 observations.

Answer: 25 resamples, each with 24 observations.

Example 2

A bootstrap with 4 resamples of a sample gives sample means of 2, 4, 6 and 8 percent. The question specifies the population standard deviation of these four values as the standard error. What is the bootstrap estimate of the standard error?

Show the solution

Mean = (2 + 4 + 6 + 8) ÷ 4 = 5.
Deviations: −3, −1, 1, 3.
Squared deviations: 9, 1, 1, 9, sum = 20.
Population variance = 20 ÷ 4 = 5.
Population standard deviation = √5 ≈ 2.24.

Answer: About 2.24 percent, because the question specifies the population standard deviation. The usual sample standard deviation of the resampled statistics would divide by 3: √(20 ÷ 3) = √6.667 ≈ 2.58 percent.

Exam tips

Memorise the contrast: bootstrap is random, with replacement, size n; jackknife is leave-one-out, size n − 1, n samples.
Questions are usually definitional. Look for the option that says 'with replacement' or 'one observation removed'.
Remember both are nonparametric and estimate standard error without a formula.
Reject any option claiming resampling removes sampling bias or adds independent data.
Do not treat the jackknife standard error as a plain standard deviation of the leave-one-out statistics. Remember the (n − 1) ÷ n scaling of the sum of squares.
Link this topic to Monte Carlo simulation: Monte Carlo draws from an assumed distribution, bootstrap draws from your own data.

Practice questions from Estimation and Hypothesis Testing

Resampling: Bootstrap and Jackknife: frequently asked questions

What is bootstrap resampling in simple words?

You take your sample and repeatedly draw new samples from it with replacement, each the same size as the original. You compute the statistic for each one. The spread of those values estimates the standard error.

What is the main difference between jackknife and bootstrap?

Jackknife removes one observation at a time, giving n samples of size n − 1, and is deterministic. Bootstrap draws random samples with replacement of size n, and the number of draws is your choice. Results from bootstrap vary slightly between runs.

Do bootstrap and jackknife need a normal distribution?

No. Both are nonparametric and rely only on the observed sample. This is why they are useful for statistics without a simple standard error formula.

How is bootstrap different from Monte Carlo simulation?

Monte Carlo generates random values from a distribution you specify with chosen parameters. Bootstrap generates samples from your observed data, so it does not require assuming a distribution.

How do I calculate the jackknife standard error?

Compute the statistic with each observation left out, then find the mean of those n values. The standard error is the square root of ((n − 1) ÷ n) times the sum of squared deviations from that mean. It is not a plain standard deviation of the leave-one-out statistics.