You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

Learning Module 7 of Quantitative Methods 6–9% of the exam

Estimation and Inference

3 lessons. About 20 minutes in cram mode, 90 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
Sampling methods and sampling error
compare 6 min cram, 30 min full
2
The central limit theorem and the standard error of the mean
explain 8 min cram, 30 min full
3
Bootstrap and jackknife: resampling to estimate a standard error
describe 6 min cram, 30 min full

What the exam expects you to do

Compare and contrast simple random, stratified random, cluster, convenience, and judgmental sampling and their implications for sampling error in an investment problem.Sampling methods and sampling error
Explain the central limit theorem and its importance for the distribution and standard error of the sample mean.The central limit theorem and the standard error of the mean
Describe the use of resampling (bootstrap, jackknife) to estimate the sampling distribution of a statistic.Bootstrap and jackknife: resampling to estimate a standard error

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

Hide what I know

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Sampling methods and sampling error

Known

MemorizeThe six sampling methods

Method	Type	How	Error
Simple random	probability	every member equally likely	measurable
Systematic	probability	every kth member	measurable
Stratified random	probability	random draw in every group, proportional	smallest
Cluster	probability	random whole groups	larger
Convenience	non-probability	easiest to reach	not measurable
Judgmental	non-probability	analyst hand-picks	not measurable

Memory hook Strata are Sampled in every layer; Clusters are Chosen whole. S for sample inside, C for chosen as a block.

Know it cold Sampling error = sample statistic − population parameter ( $\bar{X} - \mu$ ). Only probability methods let you measure it.

Key terms

parameter Use it
sample statistic Use it
sampling error Memorize
sampling plan Know the idea
probability sampling Use it
non-probability sampling Use it
simple random sampling Memorize
systematic sampling Use it
stratified random sampling Memorize
cluster sampling Memorize
convenience sampling Memorize
judgmental sampling Memorize
indexing Use it

Sampling methods

Sampling methodshow the sample is picked

Probability sampling Memorize known chance of selection, error can be measured
- Simple random Memorize every member equally likely
- Systematic Use it every kth member
- Stratified random Memorize sample within every group, in proportion
- Cluster Memorize pick whole groups at random
  - One-stagesurvey every member of the chosen groups
  - Two-stagerandom subset inside each chosen group
Non-probability sampling Memorize depends on access or judgment, bias risk
- Convenience Memorize whoever is easy to reach
- Judgmental Memorize hand-picked by the analyst

Stratified versus cluster: which groups are sampled?

Population split into groups

Stratified Memorize every group is sampled
- Inside each group a random draw, in proportion to its size
- Result every group represented, smallest error
Cluster Memorize only some groups are sampled
- Chosen groups taken whole, or sampled again inside
- Result cheap and quick, larger error

Formulas

Sampling error Memorize

\text{Sampling error} = \bar{X} - \mu

Quick example

A sample of 40 funds has a mean return of 9.1%. The population mean is 8.4%. Sampling error = 9.1 − 8.4 = 0.7 percentage points.

On your calculator

9.1−8.4= 0.7

9.1ENTER8.4− 0.7

ENTER enter (on the HP, puts the number on the stack)

Items drawn from one stratum Use it

n_i = n \times \dfrac{N_i}{N}

Quick example

A population of 5,000 stocks has 1,250 in one cell. A proportional stratified sample of 200 takes 200 × 1,250 ÷ 5,000 = 50 from that cell.

On your calculator

200×1250÷5000= 50

200ENTER1250×5000÷ 50

ENTER enter (on the HP, puts the number on the stack)

Systematic sampling interval Use it

k = \dfrac{N}{n}

Quick example

To pull 80 trades from a log of 2,000, take every 2,000 ÷ 80 = 25th trade.

On your calculator

2000÷80= 25

2000ENTER80÷ 25

ENTER enter (on the HP, puts the number on the stack)

The central limit theorem and the standard error of the mean

Known

MemorizeCentral limit theorem and standard error

Item	Value
Shape of X̄	approximately normal, n ≥ 30
Mean of X̄	$\mu$ (unchanged by n)
Variance of X̄	$\sigma^2 / n$
Standard error	$\sigma / \sqrt{n}$ or $s / \sqrt{n}$
n × 4	standard error ÷ 2
n × 9	standard error ÷ 3

Memory hook The mean of n draws is calmer than one draw, by the square root of n. Four times the data, half the noise.

Know it cold The sample mean becomes normal, not the population. Divide by $\sqrt{n}$ , not by $n$ . Larger $n$ shrinks the spread of $\bar{X}$ , not its center.

Key terms

sampling distribution Use it
central limit theorem Memorize
standard error of the sample mean Memorize
sample mean Use it
population mean Use it

What the central limit theorem says about the sample mean

Sample mean, X̄large n, any population with finite variance

Shape Memorize approximately normal (n ≥ 30 as a guide)
Center Memorize equals the population mean μ
Spread Memorize variance σ² ÷ n
- Standard error Memorize σ ÷ √n, or s ÷ √n when σ is unknown

Which standard error formula?

Standard error of the sample mean

Population σ known rare
- σ ÷ √n Memorize
Population σ unknown the usual case
- s ÷ √n Memorize s is the sample standard deviation (n − 1 divisor)

Formulas

Distribution of the sample mean Memorize

\bar{X} \sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

Quick example

Returns have mean 8% and standard deviation 20%. With n = 100, the sample mean is approximately normal with mean 8% and variance 20² ÷ 100 = 4 (%²), so a standard deviation of 2%.

On your calculator

20x²÷100= 4

x² square

20ENTER×100÷ 4

ENTER enter (on the HP, puts the number on the stack)

Standard error (population σ known) Memorize

\sigma_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}}

Quick example

σ = 20%, n = 100. 20 ÷ √100 = 20 ÷ 10 = 2.0%.

On your calculator

20÷100√x= 2

√x square root

20ENTER100g√x÷ 2

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Standard error (estimated from the sample) Memorize

s_{\bar{X}} = \dfrac{s}{\sqrt{n}}

Quick example

A sample of 25 monthly returns has s = 15%. 15 ÷ √25 = 15 ÷ 5 = 3.0%.

On your calculator

15÷25√x= 3

√x square root

15ENTER25g√x÷ 3

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Sample size for a target standard error Use it

n = \left(\dfrac{s}{s_{\bar{X}}}\right)^2

Quick example

Returns have s = 12%. For a standard error of 1.5%, n = (12 ÷ 1.5)² = 8² = 64.

On your calculator

12÷1.5=x² 64

x² square

12ENTER1.5÷ENTER× 64

ENTER enter (on the HP, puts the number on the stack)

Bootstrap and jackknife: resampling to estimate a standard error

Known

MemorizeBootstrap versus jackknife

	Bootstrap	Jackknife
Draws	n, with replacement	n − 1, one left out, no replacement
Resamples	you choose B (1,000+)	exactly n
Random?	yes, results vary slightly	no, same result every run
Typical use	standard error, confidence intervals, statistics with no formula	standard error and bias, small samples

Memory hook Bootstrap = Bag with Replacement, shuffled at random. Jackknife = Just leave one out, in turn, so it cannot vary.

Know it cold Standard error from resamples: $\sqrt{\frac{1}{B-1}\sum(\hat\theta_b-\bar\theta)^2}$ . Divide by $B-1$ , then take the square root.

Key terms

resampling Use it
bootstrap Memorize
sampling with replacement Memorize
jackknife Memorize
empirical sampling distribution Use it

Resampling methods for inference

Resamplingnew samples built from the one you have

Bootstrap Memorize random, with replacement
- Resample size Memorize n, the same as the original
- Number of resamples B your choice, often 1,000 or more
- Result varies a little from run to run
Jackknife Memorize leave one out, no replacement
- Resample size n − 1
- Number of resamples Memorize exactly n
- Result is fixed Memorize same answer every run

Bootstrap standard error, step by step

Bootstrapfrom one sample of n observations

1. Resample Memorize draw n observations with replacement
2. Compute the statistic (mean, median, ratio) for that resample
3. Repeat B times for example 1,000
4. Take the spread Memorize the standard deviation of the B results is the standard error

Formulas

Bootstrap standard error Use it

s_{\bar{X}} = \sqrt{\dfrac{1}{B-1}\sum_{b=1}^{B}(\hat\theta_b - \bar\theta)^2}

Quick example

B = 500 resample means lie a total squared distance of 0.62 from their average. √(0.62 ÷ 499) = 0.0352, in the same units as the data: 3.52% if the returns are decimals, 0.0352% if they are already in percent.

On your calculator

0.62÷(500−1)=√x 0.0352

√x square root

0.62ENTER500ENTER1−÷g√x 0.0352

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Jackknife standard error Know the idea

SE = \sqrt{\dfrac{n-1}{n}\sum_{i=1}^{n}(\theta_i - \bar\theta)^2}

Quick example

Five leave-one-out means lie a total squared distance of 1.20 from their average. √(4 ÷ 5 × 1.20) = √0.96 = 0.98.

On your calculator

(5−1)÷5×1.2=√x 0.9798

√x square root

5ENTER1−5÷1.2×g√x 0.9798

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Cheat sheet · PDF · 3 A4 pages

Estimation and Inference on 3 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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