Statisticians, actuaries and operations research analysts use mathematics to answer practical questions: is this treatment effective, how much should this insurance cost, what is the cheapest way to route these deliveries? This beginner course shows what these careers involve and how people get into them.

Module 1 maps the math each role uses day to day, shows why pairing math with computing matters (you will run a small simulation in Python), and teaches you to state a model's assumptions and limits. Module 2 covers credentials honestly, from the SOA and CAS actuarial exam pathways to master's degrees in statistics and operations research, then helps you pick a public-data project and find internships and entry roles. Module 3 covers the communication and judgement that separate good analysts from merely accurate ones: explaining uncertainty, structuring an analysis so others can rerun it, and spotting bias and misuse in models.

Every lesson has a reading, free videos from reputable educators and a quiz, and most have a hands-on activity. This course does not replace an accredited degree or professional exams: actuaries must pass the Society of Actuaries or Casualty Actuarial Society exam series, and most statistician and operations research roles expect a master's degree. It builds on STEM Foundations.

How statisticians, actuaries and operations research analysts work: the math they use, pairing it with code, credentials, evidence, and explaining results honestly.

Foundations for Math, Statistics & Analytics Careers

## Why this matters

"Good at math" covers a lot of ground. A statistician, an actuary and an operations research analyst all have strong math backgrounds, but they lean on different branches every day. Knowing which ones helps you choose courses, read job postings and decide which career fits the kind of problems you enjoy.

## Three careers, three core questions

- **Statisticians** ask: what can we honestly conclude from data? They design surveys and experiments, analyse clinical trials, build models of data and report how confident we can be. They work in government statistics agencies, pharmaceutical companies, research institutes, tech companies and universities.
- **Actuaries** ask: how much financial risk does an uncertain future hold, and what should it cost? They price insurance, set reserves so insurers can pay future claims, and value pensions. Most work for insurers, consulting firms and governments.
- **Operations research (OR) analysts** ask: what is the best decision, given limits? They build optimisation and simulation models to schedule staff, route vehicles, plan inventory and design supply chains. They work in logistics, airlines, manufacturing, healthcare, defence and consulting.

## The math map

| Branch | What it does | Statistician | Actuary | OR analyst |
|---|---|---|---|---|
| Calculus | Rates of change, areas, optimisation | Deriving estimators, likelihoods | Continuous interest, survival models | Gradient-based optimisation |
| Linear algebra | Many variables at once | Regression, multivariate methods | Some modelling | Linear programming, networks |
| Probability | Reasoning about chance | Foundation of inference | Core: claims, lifetimes | Queues, simulation, risk |
| Statistics | Learning from data | Core | Experience studies, credibility | Estimating model inputs |
| Financial mathematics | Interest, annuities, present value | Occasionally | Core | Occasionally (cost models) |
| Optimisation | Finding the best feasible choice | Model fitting | Some | Core |

All three use probability and statistics. The differences are in the emphasis: statisticians go deepest into inference and study design, actuaries combine probability with financial mathematics, and OR analysts go deepest into optimisation and simulation.

## What a day looks like

**Statistician at a health agency.** Checks a survey's sampling plan, fits a model to estimate vaccination rates by region, and writes a short report with confidence intervals. Math used: probability, statistical inference, regression.

**Actuary at an insurer.** Updates the expected number and cost of car insurance claims using last year's data, then calculates a premium that covers expected claims, expenses and a margin. Math used: probability distributions, statistics, present value.

**OR analyst at a delivery company.** Builds a model that assigns 400 parcels to 12 vans to minimise total driving time, subject to van capacity and delivery time windows. Math used: linear and integer programming, graph algorithms, some simulation.

## Worked example: one problem, three lenses

A hospital wants to reduce emergency department waiting times.

- A **statistician** asks whether last year's change in triage actually reduced waits, or whether waits just fell because there were fewer patients. That needs careful comparison and a measure of uncertainty.
- An **actuary** (in a health insurer) asks how changes in admissions will affect the expected cost of claims next year.
- An **OR analyst** builds a queueing simulation of patient arrivals and staff shifts, then tests which roster gives the shortest average wait for the same budget.

Same hospital, same data, three different mathematical questions.

## How the math is usually learned

A typical path: calculus (two or three semesters), linear algebra, an introduction to proofs, probability, mathematical statistics, then specialised courses (regression, financial mathematics, optimisation, stochastic processes). Programming in Python or R runs alongside. The next lesson covers why that computing matters so much.

The US Bureau of Labor Statistics Occupational Outlook Handbook describes each career's typical duties, education and outlook at [bls.gov/ooh](https://www.bls.gov/ooh/). Checked on 2026-10-23.

## Common mistakes

- **Thinking these jobs are mostly arithmetic.** Computers do arithmetic; the job is choosing the right model and judging the result.
- **Ignoring probability.** It is the one branch all three careers depend on.
- **Assuming the careers are interchangeable.** Actuaries need professional exams; statisticians and OR analysts usually need a master's for senior roles (Module 2).
- **Skipping communication.** All three spend much of their time explaining results to non-specialists (Module 3).

## Quick recap

- Statisticians: honest conclusions from data. Actuaries: pricing financial risk. OR analysts: best decisions under constraints.
- Probability and statistics are shared; financial mathematics is central for actuaries, optimisation for OR analysts.
- The same real problem can be approached through all three lenses.

## Try it

This lesson has no activity. Watch the videos, then take the quiz in the **Do** tab.


Map which branches of math statisticians, actuaries and operations research analysts use day to day.

Calculus, probability, and statistics

What exactly do statisticians do?

What is Operational Research?

## Why this matters

Math on paper is powerful but slow, and real problems are messy. Employers in statistics, actuarial work and operations research want people who can turn math into working code: load real data, run a model, simulate thousands of scenarios and check the answer. Math plus computing plus knowledge of a real domain (insurance, health, logistics, finance) is the combination that gets people hired.

## What computing adds

- **Scale.** A spreadsheet struggles with millions of insurance policies or delivery records; code handles them routinely.
- **Simulation.** When a problem is too complicated to solve exactly, you can simulate it many times and measure what happens.
- **Repeatability.** A script can be rerun next month on new data, checked by a colleague and audited.
- **Checking.** Code lets you test a formula against a brute-force calculation, which catches mistakes.

The common tools are Python (with libraries such as pandas, NumPy and SciPy) and R for statistics; SQL for getting data out of databases; spreadsheets for quick work and communication; and specialised software for optimisation solvers and actuarial models. You do not need all of them. Python plus SQL plus a spreadsheet covers a lot of entry-level work.

## Monte Carlo simulation

A **Monte Carlo simulation** estimates a probability or an average by running a random experiment many times and counting results. The name comes from the Monte Carlo casino, and the method became widely used in the 1940s during early computer work at Los Alamos.

The recipe:

1. Write code that runs one random trial of the situation.
2. Repeat it many times (thousands or millions).
3. Count how often the outcome you care about happens, and divide by the number of trials.
4. Check that the answer settles down as trials increase.

The error of a simulated probability shrinks roughly in proportion to one over the square root of the number of trials. So 100 times more trials gives about 10 times more precision.

## Worked example: at least one six in four rolls

What is the chance of getting at least one six when you roll a fair die four times? This is a famous puzzle from the 1600s, discussed by Pascal and Fermat.

**Exact answer.** It is easier to find the chance of *no* sixes. Each roll misses a six with probability 5/6, and the rolls are independent, so the chance of no sixes in four rolls is (5/6)⁴ ≈ 0.4823. The chance of at least one six is 1 − 0.4823 = **0.5177**, slightly better than even.

**Simulated answer.** In Python:

```python
import random

rng = random.Random(42)
trials = 20000
hits = 0
for _ in range(trials):
    rolls = [rng.randint(1, 6) for _ in range(4)]
    if 6 in rolls:
        hits += 1
print(hits / trials)   # about 0.52
```

With 100 trials the estimate might be anywhere from about 0.42 to 0.62. With 20,000 trials it is usually within about 0.01 of 0.5177. The exact formula and the simulation agree, and each checks the other.

Why simulate when the exact answer is available? Because real problems often have no tidy formula: an insurer's total claims next year, with thousands of policies and different claim sizes; a queue with breaks, shift changes and rush hours; a portfolio with correlated assets. Analysts use the same four-step recipe on those problems, and they test their code first on a case like this one where the exact answer is known.

## Seeds and repeatability

Computers use **pseudo-random** number generators. Setting a **seed** (here 42) makes the "random" sequence the same every run, so a colleague can reproduce your exact numbers. Use a seed for reports and testing; change it to check that your conclusion does not depend on one lucky sequence.

## Common mistakes

- **Too few trials.** A few hundred trials can be badly off. Increase the count until the answer settles.
- **Not checking against a known case.** Test your simulation where an exact answer exists before trusting it elsewhere.
- **Hidden dependence.** The exact formula assumed independent rolls. If your simulation or real system breaks that assumption, the answers differ.
- **No seed.** Results that change every run are hard to review and debug.
- **Learning tools without math.** Code runs a wrong model just as happily as a right one.

## Quick recap

- Employers value math combined with programming and domain knowledge.
- Monte Carlo: simulate one trial, repeat many times, count, check convergence.
- More trials means more precision, roughly by the square root; seeds make results repeatable.

## Try it

Open the **Do** tab and finish the simulation by writing one line of Python, then watch the estimate approach 0.5177.


Run a quick Monte Carlo simulation in Python and compare it with an exact probability.

Pairing math with computing

What is Monte Carlo Simulation?

Monte Carlo Simulation

## Why this matters

Every number an analyst produces comes from a **model**: a simplified mathematical description of something real. Simplification is the point, because a model that includes everything is as complicated as the world. But every simplification is an **assumption**, and when an assumption stops being true the model can fail badly. The statistician George Box put it memorably: "All models are wrong, but some are useful." The professional skill is knowing which assumptions you made and where the model stops being useful.

## What a model is

A model has three parts:

1. **Inputs** (data and parameters), such as last year's claim counts or the average time to serve a customer.
2. **Structure**: the mathematical relationship, such as "demand falls in a straight line as price rises" or "customers arrive at random at a steady average rate".
3. **Outputs**: the prediction, price, schedule or estimate you use to make a decision.

Assumptions live in all three. Inputs assume the data is accurate and relevant. Structure assumes the relationship has a particular shape. Outputs assume the future resembles the period the model was built on.

## Common assumptions to look for

| Assumption | What it means | When it breaks |
|---|---|---|
| Independence | One event does not affect another | Floods damage many houses at once; infections spread |
| Stationarity | The process does not change over time | New laws, new products, a pandemic |
| Distribution shape | Data follows, say, a normal (bell-shaped) distribution | Rare large losses ("fat tails") are more common than a normal curve predicts |
| Linearity | Doubling an input doubles the effect | Effects saturate or accelerate |
| Range of validity | The model was fitted on certain values | Predicting far outside them (extrapolation) |
| Representative data | The sample reflects the population | Survey only reaches people with landlines |

## Worked example 1: a simple claims model

An insurer models the number of home insurance claims per month as random, independent events with an average of 50. Under this model, months with more than about 65 claims are rare.

**Stated assumptions:** claims are independent; the average rate is stable; the policy count does not change much.

**Where it breaks:** a major storm causes 2,000 claims in one week. The claims were not independent: one event caused all of them. Insurers handle this by modelling catastrophes separately, buying reinsurance and holding capital. The everyday model is still useful for ordinary months, as long as everyone knows it does not cover catastrophes.

## Worked example 2: a straight-line forecast

An analyst fits a straight line to five years of a café's monthly sales, which rose steadily from 2,000 to 3,000 cups. The line predicts 6,000 cups in ten years.

**Assumptions:** growth continues at the same rate; nothing limits demand; the past five years are typical.

**Limits:** the café has 40 seats and fixed opening hours, so physical capacity caps sales. Extrapolating a trend far beyond the data ignores that cap. A better statement is: "If recent growth continues, sales could reach about 3,200 cups next year. Beyond that, seating capacity becomes the limit."

## Worked example 3: an optimisation model

An OR analyst's staffing model assumes each nurse can handle the same number of patients per hour. It recommends cutting one evening shift. In reality, evening patients are sicker and take longer. The model's output was correct for its assumptions, and wrong for the hospital. Checking assumptions with the people who do the work, here the nurses, would have caught it.

## How to state assumptions and limits

Good analysts write them down next to the result, in plain language:

- **What we assumed:** the main assumptions, ideally three to five.
- **Why it is reasonable:** evidence such as past data or expert input.
- **What would change the answer:** which assumption matters most.
- **Where not to use it:** conditions outside the model's valid range.

**Sensitivity analysis** makes this concrete: change one input or assumption at a time and see how much the output moves. If a small change in an uncertain input flips the decision, say so.

## Common mistakes

- **Hiding assumptions.** Decision-makers cannot judge a result whose assumptions they never saw.
- **Extrapolating.** A model fitted on one range of data is weakest outside that range.
- **Confusing fit with truth.** A model can match past data well and still predict poorly.
- **Assuming independence by default.** Many real risks move together, which matters most in bad times.
- **Never revisiting.** Conditions change; models need regular checks against new data.

## Quick recap

- A model is a deliberate simplification; every simplification is an assumption.
- Check independence, stationarity, distribution shape, linearity, range and representativeness.
- State assumptions, their justification, what would change the answer and where not to use the model; test sensitivity.

## Try it

This lesson has no activity. Watch the video, then take the quiz in the **Do** tab.


State a model's assumptions and the limits of where it can be trusted.

Models and assumptions

What is a (mathematical) model?

The math statisticians, actuaries and operations research analysts use, pairing it with computing, and stating a model's assumptions and limits.

Thinking in Numbers

## Why this matters

The three careers in this course have very different credential paths. Actuaries qualify mainly through a long series of professional exams. Statisticians and operations research analysts qualify mainly through degrees, often a master's. Knowing the path early helps you choose courses, plan finances and avoid surprises.

Credential rules, exam lists and fees change, so always check the official body. This lesson gives the shape of each path, not a substitute for the official pages.

## Actuaries: the exam pathway

In the United States and Canada, actuarial credentials are awarded by two main societies:

- **Society of Actuaries (SOA)** for life insurance, health, retirement and investment work: [soa.org](https://www.soa.org/).
- **Casualty Actuarial Society (CAS)** for property and casualty insurance, such as car, home and business insurance: [casact.org](https://www.casact.org/).

Both are introduced together at [beanactuary.org](https://www.beanactuary.org/). Checked on 2026-10-23.

The general shape:

1. **Preliminary exams.** Two early exams are shared by both societies: **Exam P (Probability)** and **Exam FM (Financial Mathematics)**. Many students pass one or both during university.
2. **More exams and modules** specific to each society's track, covering topics such as long-term and short-term actuarial mathematics, statistics for risk modelling, and predictive analytics.
3. **VEE credits** (Validation by Educational Experience): college courses or approved alternatives in subjects such as economics and accounting and finance, recognised instead of exams. Check the current list of VEE subjects on each society's site.
4. **Associate level:** ASA (Associate of the Society of Actuaries) or ACAS (Associate of the Casualty Actuarial Society), usually reached after several years.
5. **Fellow level:** FSA or FCAS, after further exams and modules, often several more years.

Most actuaries earn these while working. Many employers pay exam fees, give paid study time and raise salaries as exams are passed. A bachelor's degree is typical; majors include actuarial science, math, statistics, economics and finance. Passing exams before graduating is a strong signal for internships (Lesson 3).

Outside North America, other bodies award credentials, such as the Institute and Faculty of Actuaries in the UK.

## Statisticians: degrees first

Most statistician jobs ask for a **master's degree in statistics** or a closely related field such as biostatistics; research and senior methodology roles often want a **PhD**. A bachelor's in statistics or math can lead to analyst, data analyst or statistical programmer roles.

A typical MS covers mathematical statistics, regression and linear models, experimental design, Bayesian methods, statistical computing in R or Python, and often a consulting or capstone project.

The American Statistical Association (ASA) offers voluntary accreditation (PStat for experienced statisticians, GStat for recent graduates) and publishes ethical guidelines: [amstat.org](https://www.amstat.org/). Accreditation is optional; it is not a licence to practise. Checked on 2026-10-23.

## Operations research analysts: degrees and projects

Entry-level OR analyst roles sometimes hire bachelor's graduates in math, industrial engineering, computer science or analytics, but many employers prefer a **master's in operations research, industrial engineering, analytics or business analytics**. Core courses include linear and integer programming, stochastic models, simulation, and statistics.

INFORMS (the Institute for Operations Research and the Management Sciences) is the main professional society: [informs.org](https://www.informs.org/). It has offered analytics certification; check its site for the current status before planning around one. Checked on 2026-10-23.

## Worked example: planning a first year after graduating

**Priya** wants to be an actuary. She passes Exam P in her third year of a math degree and Exam FM before graduating, takes an economics course that counts toward a VEE, and joins an insurer's actuarial programme with paid study time. She plans for several years of exams while working.

**Luis** wants to work in clinical trials. After a math BS with a statistics minor he works for a year as a statistical programmer, then starts a funded MS in biostatistics.

**Ana** enjoys logistics problems. She takes an optimisation elective and a simulation project in her industrial engineering BS, gets an entry analyst role, and her employer supports a part-time MS in analytics.

## Common mistakes

- **Treating actuarial exams like university finals.** Each exam needs months of study, and pass rates are often well below what students expect. Plan for it.
- **Copying fees and syllabi from forums.** Check the SOA, CAS or other official site.
- **Skipping programming.** Every path expects Python, R, SQL or spreadsheets.
- **Assuming a credential replaces experience.** Employers still want internships and projects.

## Quick recap

- Actuaries: SOA or CAS exams (P and FM first), VEE credits, then ASA/ACAS and FSA/FCAS, usually while working.
- Statisticians: a master's is typical, PhD for research; ASA PStat/GStat accreditation is optional.
- OR analysts: a master's in OR, industrial engineering or analytics is common; INFORMS is the professional society.

## Try it

This lesson has no activity. Watch the videos, then take the quiz in the **Do** tab.


Map the SOA and CAS actuarial exams, master's degrees in statistics and operations research, and optional accreditations.

Degrees, exams, and credentials

What is an Actuary?

A Visual Guide to Actuarial Exams | SOA Preliminary Exams

Pick the career you are aiming at (statistician, actuary or operations research analyst) and write the kind of question that career answers.

Browse one of the free sources and shortlist three datasets related to that kind of question.

For each, note who published it, when it was last updated, how many rows and columns it has, and its licence or terms of use.

Choose one dataset and write a specific question it can answer, in one sentence.

Download it and open it in a spreadsheet or Python. Record any missing values, odd codes or units you need to understand.

Write a plan: the method you will use (for example a comparison of averages with a confidence interval, a claims frequency table, or a simple scheduling model), and what the final output will be (a short report, a chart or a small dashboard).

List the limitations you already expect, such as missing groups or old data.

Set a deadline no more than two weeks away for a first complete version.

## Why this matters

Employers in analytics hire people who can take a messy question and real data and produce an answer others can trust. Grades and exam passes show you know the math. A finished project shows you can **use** it: clean data, choose a method, interpret the result and explain the limitations. For internships especially, one or two good projects can be the difference between an interview and silence.

## What counts as evidence

- **Analysis write-ups:** a short report that states a question, describes the data and method, shows the results and discusses limitations.
- **Models:** a regression, a forecast, a pricing model, or an optimisation model, with code others can run.
- **Dashboards:** an interactive or static summary that helps someone monitor something, built in a spreadsheet, Python or a free tool.
- **Competitions:** platforms such as Kaggle host data challenges with public leaderboards and shared notebooks.
- **Internships, research and case competitions:** the strongest evidence, covered in the next lesson.

## Choosing a dataset

Good public sources include government open-data portals such as [data.gov](https://data.gov/), national statistics agencies, the [UCI Machine Learning Repository](https://archive.ics.uci.edu/) and [Kaggle](https://www.kaggle.com/) datasets. Checked on 2026-10-23.

Choose data that:

1. **Matches your target career.** Health or survey data for statistics; mortality, claims, accident or weather-loss data for actuarial work; transport, scheduling or supply chain data for operations research.
2. **Has documentation.** You need to know what each column means and how the data was collected.
3. **Is the right size.** Thousands to a few hundred thousand rows is plenty for a first project.
4. **Allows reuse.** Check the licence or terms, and avoid personal data about identifiable people.
5. **Is not overused.** The Titanic and Iris datasets are fine for practice, but employers have seen them many times.

## Worked example: three project ideas

**Statistics:** "Did average commute times change between two years in my state?" Data: a national household survey's published tables. Method: compare means with confidence intervals, check sample sizes. Output: a two-page report with one chart.

**Actuarial:** "How do claim frequency and severity differ by vehicle age?" Data: a public insurance claims dataset from a teaching or research repository. Method: frequency (claims per policy-year) and severity (average cost per claim) tables, with a short note on what an insurer would need to add for pricing. Output: a spreadsheet model and a one-page summary.

**Operations research:** "How many checkout lanes does a shop need to keep the average wait under three minutes?" Data: arrival counts published for a public service, or a clearly stated assumption. Method: a queueing simulation in Python. Output: a chart of average wait against number of lanes, with assumptions listed.

Each idea has a specific question, a named data source, a method that matches the career and a defined output.

## Making the project convincing

- **Lead with the question and the answer**, not with the code.
- **Show the data cleaning.** Real data has missing values and errors; explaining how you handled them shows judgement.
- **State limitations honestly.** "The survey excludes people without internet access" is a strength, not a weakness, in a write-up.
- **Make it reproducible.** Share code and a README so others can rerun it (Module 3 covers this).
- **Keep it short.** A clear two-page report beats a fifty-cell notebook without explanation.

## About competitions

Competitions are a good way to practise and to read other people's code. But a high leaderboard score on a prepared dataset is only part of the job. Employers also want to see that you can define a problem, find data and explain results. Treat competitions as practice and pair them with at least one project you scoped yourself.

## Common mistakes

- **Starting with a technique instead of a question.** "I want to use a neural network" is not a project; "Which factors predict claim size?" is.
- **Choosing data that is too large or undocumented.** You will spend the whole project guessing what columns mean.
- **No conclusion.** Every project needs a plain-language answer, even if the answer is "the data cannot tell us".
- **Never finishing.** A small finished project is better than an ambitious abandoned one.

## Quick recap

- Projects, models, dashboards and competitions show you can apply the math.
- Pick documented, reusable data that matches your target career and is not overused.
- Lead with question and answer, show cleaning, state limits, make it reproducible.

## Try it

Open the **Do** tab and write a one-page plan for a project on a public dataset.


Pick a public dataset and plan a small analysis project that shows your skills.

Evidence: models, dashboards, and competitions

How to Enter a Kaggle Competition (using Kernels) | Kaggle

Write your target career and two or three job titles to search for (see the table in the lesson).

Search one general job site and save three postings. Include at least one government or public-sector posting (for example from usajobs.gov if you are in the US).

If you are aiming at actuarial work, read the internship and exam advice on beanactuary.org and note what exams employers mention in your saved postings.

For each posting, list the required and preferred qualifications in a table with columns: requirement, I have it (yes/partly/no), evidence or plan.

Write a two-line summary of your strongest project or course that matches the postings.

Find one person in your target role (a speaker at a careers event, an alumnus, or a professional society member) and draft a short, polite message asking one specific question.

Record the application deadline for each posting and set reminders one week before.

## Why this matters

Most people in these careers get their first job through an internship or an entry-level programme. Internships give you real data, real deadlines and a reference, and many employers hire interns into permanent roles. Knowing where to look, what employers screen for and when to apply makes the search much more effective.

## Where the jobs are

| Career | Typical employers | Entry titles to search |
|---|---|---|
| Statistician | Government statistical agencies, pharmaceutical and clinical research companies, health systems, research institutes, tech companies, universities | Statistical analyst, research assistant, statistical programmer, biostatistician I, survey statistician, data analyst |
| Actuary | Life, health and property and casualty insurers, reinsurers, consulting firms, pension administrators, regulators | Actuarial intern, actuarial analyst, actuarial assistant, pricing analyst |
| Operations research analyst | Airlines, logistics and delivery companies, manufacturers, retailers, hospitals, defence and government, consultancies | Operations research analyst, operations analyst, supply chain analyst, decision science analyst, analytics associate |

In the US, federal agencies post jobs and student programmes at [usajobs.gov](https://www.usajobs.gov/). Government statistical agencies and defence organisations hire many statisticians and OR analysts. Checked on 2026-10-23.

## What employers screen for

**For actuarial internships**, exam progress is a major factor. Many employers look for at least one passed preliminary exam (often Exam P or FM), along with strong grades in math courses, spreadsheet skills and some programming. The SOA and CAS student site [beanactuary.org](https://www.beanactuary.org/) has internship and career guidance. Checked on 2026-10-23.

**For statistics roles**, employers look for coursework in probability, statistics and regression, experience with R, Python or SAS, and evidence of analysis such as a project, research assistantship or consulting work.

**For OR and analyst roles**, employers look for optimisation or simulation coursework, Python or a modelling language, SQL, spreadsheet skills and a project that solved a practical decision problem.

All three value clear communication. Many interviews include explaining a past project to a non-specialist.

## Timing

Large employers often recruit for summer internships many months ahead, sometimes starting in late summer or autumn of the previous year. Smaller companies and public agencies may post later. Check postings and university careers services early, and keep a calendar of deadlines.

## Worked example: a focused search

Kai is a third-year math student who has passed Exam P and wants an actuarial internship.

1. **Titles:** actuarial intern, actuarial analyst intern, pricing analyst intern.
2. **Sources:** the careers pages of a few insurers and consultancies, the university careers portal, beanactuary.org, and a state insurance regulator's student programme.
3. **Gap check:** most postings mention Excel and "exam progress"; several prefer SQL or Python. Kai has strong Excel and Exam P, some Python, no SQL.
4. **Plan:** take a free SQL course this month, add a small claims-frequency project to the CV, and register for Exam FM.
5. **Networking:** attend the university's actuarial club meeting and ask a visiting actuary what their team looks for in interns.
6. **Applications:** submit to eight internships, each with a CV adjusted to the posting's top three requirements.

## Applying well

- **One-page CV** with relevant courses, exams passed, technical skills, projects (with a one-line result each) and work experience.
- **Tailor** the CV and cover note to each posting's main requirements.
- **Prepare stories** about a project: the question, your method, the result and what you would do differently.
- **Expect technical questions**: probability puzzles, interpreting a regression, a spreadsheet exercise or a simple SQL query.
- **Follow up** politely after interviews.

## If you do not land an internship

Other strong options: a research assistant role with a professor, a statistics or analytics job in a non-profit or campus office, a case competition, or an independent project using public data (previous lesson). Entry analyst roles in related areas such as data analysis, underwriting support or operations can also lead into these careers.

## Common mistakes

- **Applying too late.** Large programmes can fill months ahead.
- **Sending the same CV everywhere.** Match it to each posting.
- **Listing tools without evidence.** "Python" means more with a project behind it.
- **Ignoring the public sector.** Government agencies are major employers of statisticians and OR analysts.
- **Neglecting exams** if you are aiming at actuarial work.

## Quick recap

- Search several titles at insurers, agencies, health and research organisations, logistics and consulting firms.
- Actuarial internships weigh exam progress; statistics and OR roles weigh coursework, programming and projects.
- Apply early, tailor each application and prepare to explain a project clearly.

## Try it

Open the **Do** tab and build a target list with a gap table and an application plan.


Find analyst, statistical and actuarial internships and entry roles, and prepare to apply.

Internships and entry roles

If you want to land a data analyst internship job, here's what to consider

Actuarial Internships Are Competitive. Do This Instead.

Degrees and professional exams, a public-data project as evidence, and finding analyst and actuarial internships.

Proving Your Skills

Write the result the way an analyst might first write it, with the point estimate, the interval and any technical terms.

Rewrite the headline as one sentence that starts with the decision it affects, for example: 'We should plan for between X and Y.'

Express one probability as a natural frequency (for example '3 in 10' instead of '30%'), and any risk change in absolute terms as well as relative terms.

Add one sentence on what drives the uncertainty and one on what would narrow it.

Replace every vague word (likely, significant, probably) with a number or remove it.

Make a simple chart or sketch that shows the range, not just a single number.

Explain it out loud to your listener in under one minute, then ask them to repeat back what they would do.

Note anything they misunderstood and revise your wording.

## Why this matters

Every analysis carries uncertainty: the sample might be unrepresentative, the future might differ from the past, the model is a simplification. Managers and clients still need to make decisions. If you hide the uncertainty, they will be overconfident; if you bury them in caveats, they will ignore you. Explaining uncertainty clearly, in terms of the decision, is one of the most valued skills in statistics, actuarial work and operations research.

## Say a range, not just a number

A **point estimate** is a single best guess: "We expect 1,200 claims next quarter." It sounds precise but hides how wrong it could be. A **range** shows the uncertainty: "We expect about 1,200 claims, very likely between 1,050 and 1,350."

Statisticians often report a **confidence interval**, for example a 95% interval. Technically, a 95% confidence interval comes from a procedure that captures the true value in about 95% of repeated samples. In plain language for a non-specialist you can say: "Our best estimate is 1,200; the data are consistent with anything from about 1,050 to 1,350."

For forecasts, a **prediction interval** describes where a future value is likely to fall. It is usually wider than a confidence interval for an average, because individual outcomes vary more than averages.

## Use numbers, not vague words

Words like "likely", "possible" or "significant" mean different things to different people. In studies of how people interpret these words, "likely" has been read as anything from about 50% to over 90%. Some organisations fix the meanings: the Intergovernmental Panel on Climate Change, for example, uses "likely" to mean a 66 to 100% probability. Unless your audience shares a defined scale, give the number.

The word "significant" is especially risky. In statistics it means a result is unlikely to be explained by chance alone under a particular test; to a manager it means "large and important". A tiny effect can be statistically significant in a big sample. Say how big the effect is.

## Natural frequencies and absolute risk

People understand counts better than percentages and conditional probabilities.

- **Natural frequencies:** "3 out of every 10 similar projects ran over budget" is easier to grasp than "a 30% probability of overrun".
- **Absolute versus relative risk:** "The new process cuts defect risk by 50%" sounds dramatic. If defects fall from 2 in 1,000 to 1 in 1,000, say both: "It halves defects, from 2 per 1,000 parts to 1 per 1,000."

## Worked example: a staffing forecast

An analyst forecasts call-centre volume for next month.

**Technical version:** "The model's point forecast is 42,000 calls, with an 80% prediction interval of 38,000 to 47,000. The upper tail is heavier due to weather-related surges."

**Manager version:** "Plan for about 42,000 calls. In 8 out of 10 months like this, volume falls between 38,000 and 47,000. The bigger risk is on the high side, mostly from storms. If we staff for 42,000, we have roughly a 1 in 10 chance of more than 47,000 calls and long waits. Staffing for 45,000 would cost about 6% more and cut that risk substantially. The forecast will tighten once we have the first week's data."

The manager version starts from the decision, gives a range, uses natural frequencies, says which direction the risk lies, offers options with costs and says what would reduce the uncertainty.

## Show the uncertainty in charts

- Draw intervals as error bars or shaded bands, not just a line.
- For forecasts, a "fan chart" with darker bands for more likely ranges works well.
- Label what the band means ("80% of outcomes fall in the shaded area").
- Avoid chart styles that imply false precision, such as many decimal places.

## Common mistakes

- **Reporting a single number** as if it were certain.
- **Using "significant" without saying how big.** Give the size of the effect.
- **Relative risk alone.** Always give the baseline.
- **Too many caveats.** Keep the main uncertainty in the headline; put the technical details in an appendix.
- **Implying certainty you do not have**, or false precision such as "1,203.6 claims".

## Quick recap

- Give a best estimate and a range, and say what the range means in plain words.
- Replace vague words with numbers; distinguish statistical significance from importance.
- Use natural frequencies and absolute risk; start from the decision and show uncertainty in charts.

## Try it

Open the **Do** tab and rewrite an uncertain result for a non-specialist, then test it on a listener.


Explain a range and a probability clearly to a manager who is not a specialist.

Explaining uncertainty

Communicating uncertainty

Confidence Intervals, Clearly Explained!!!

Create a project folder with subfolders: data/raw, data/processed, src (or notebooks), output, and a README file.

Put the downloaded dataset in data/raw, and write in the README where it came from, the download date and its licence.

Write a short script or notebook that reads only from data/raw, does one cleaning step, and saves the cleaned file to data/processed.

Add one analysis step that produces a table or chart in output/. If anything is random, set a seed.

Record the software and library versions you used in the README or a requirements file.

Restart your Python or R session completely and run everything from the top. Fix anything that fails.

Write the exact commands or steps to reproduce the results at the top of the README.

Optional: put the folder under version control with Git and make a first commit (do not commit private or very large data files).

## Why this matters

An analysis that only works on your laptop, in your head, on the day you ran it, is fragile. Next month the data updates and you cannot remember which cells you ran. A colleague cannot check your numbers. An auditor or regulator asks how a reserve figure was calculated and nobody can say. **Reproducible analysis** means someone else, or future you, can rerun the work from the raw data and get the same results. In actuarial work and regulated industries this is often required; everywhere else it is simply good practice.

## Reproducible versus replicable

- **Reproducible:** same data, same code, same results.
- **Replicable:** new data, same method, consistent conclusions.

This lesson is about the first. It is the foundation for the second.

## The core habits

**1. Never edit raw data.** Keep the original files exactly as downloaded or delivered. Every cleaning step happens in code, and the output is saved as a new file. Anyone can then see and rerun every change.

**2. Use a clear folder structure.**

```text
claims-analysis/
  README.md          what this is, where the data came from, how to run it
  requirements.txt   software and versions
  data/
    raw/             original files, never edited
    processed/       outputs of cleaning scripts
  src/               scripts or notebooks, numbered in run order
  output/            tables and charts for the report
```

**3. Script everything.** Avoid manual copy-paste steps between tools. If a step must be manual, write it down in the README.

**4. Make notebooks run from top to bottom.** Notebooks such as Jupyter are excellent for exploring, but cells run out of order can produce results that cannot be recreated. Before you share, restart the kernel and run all cells in order.

**5. Set seeds for randomness.** Simulations, train/test splits and random samples should use a fixed seed so results repeat.

**6. Record versions.** Library updates can change results. List the versions you used, for example in a `requirements.txt` file.

**7. Use version control.** Git records every change to your code with a message explaining why. It also lets you go back to the exact code behind last quarter's report.

**8. Write a README.** Explain the question, data sources (with dates and licences), how to run the analysis and where the outputs go.

## Worked example: turning a messy analysis around

**Before:** An analyst downloads a claims spreadsheet, deletes some rows that look wrong, adds a column by hand, copies a pivot table into a slide and emails it. A month later the manager asks for the same chart with new data. The analyst cannot remember which rows were deleted.

**After:**

1. The original file is saved in `data/raw/claims_2026-09.csv`.
2. `src/01_clean.py` removes rows with negative claim amounts (and logs how many), adds the new column and writes `data/processed/claims_clean.csv`.
3. `src/02_summary.py` builds the summary table and chart in `output/`.
4. The README says: "Run `python src/01_clean.py` then `python src/02_summary.py`."

Next month the analyst drops the new file in `data/raw`, updates one file name and reruns. The manager also asks why 37 rows were excluded; the cleaning log answers immediately.

## Make it checkable

- Add simple **sanity checks** in code: totals match the source, no negative ages, row counts are as expected.
- Print or save a short **log** of what each step did.
- Ask a colleague to run your project from the README before you rely on it.

## Privacy and data handling

Reproducibility does not mean publishing everything. Do not put personal, confidential or licensed data in public repositories. Share code and documentation, and describe how authorised people can obtain the data.

## Common mistakes

- **Editing raw files by hand.** Changes become invisible and unrepeatable.
- **Out-of-order notebooks.** Always restart and run all before sharing.
- **Hard-coded personal file paths** such as `C:/Users/me/Desktop`. Use paths relative to the project folder.
- **No record of versions or seeds.**
- **Committing secrets or private data** to version control.

## Quick recap

- Reproducible means same data plus same code gives the same results.
- Keep raw data untouched, script every step, use a standard folder layout, set seeds, record versions, use Git and write a README.
- Test by rerunning from a clean start, and have someone else try.

## Try it

Open the **Do** tab and set up a small reproducible project on your own computer.


Structure an analysis project so that someone else can rerun it and get the same results.

Reproducible analysis

Reproducible Data Analysis in Jupyter, Part 1/10: Loading and Visualizing Data

## Why this matters

Models decide who gets a loan, what an insurance policy costs, which patients get extra care and where police patrol. A model can be mathematically correct and still treat people unfairly, mislead decision-makers or be used for something it was never designed to do. Statisticians, actuaries and operations research analysts are often the people best placed to notice. Professional codes expect them to speak up.

## Where bias comes from

**Biased data.** If the historical data reflects past discrimination, a model trained on it learns that pattern. A well-known example: in 2018, Reuters reported that Amazon had abandoned an experimental recruiting tool after finding it penalised CVs that mentioned women's organisations, because it had learned from a decade of mostly male hires.

**Unrepresentative samples.** A survey or training set that under-represents some groups produces estimates that are less accurate for them.

**Proxy variables.** A model may not use a protected characteristic such as race directly, but other variables (postcode, school, shopping patterns) can stand in for it. This is called **proxy discrimination**.

**Wrong target.** A model can optimise the wrong thing. A widely cited 2019 study in *Science* by Obermeyer and colleagues found that a health-care algorithm used past health spending as a stand-in for health need. Because less money had historically been spent on Black patients with the same level of illness, the algorithm underestimated their need for extra care.

**Feedback loops.** If a model sends more inspections to one area, more problems are found there, which makes the model send even more inspections, regardless of where problems really are.

## Different ideas of fairness

There is no single mathematical definition of fairness. Common ones include:

- **Equal accuracy:** the model is equally accurate for each group.
- **Equal error rates:** for example, the same false positive rate across groups.
- **Calibration:** a predicted 30% risk means 30% for every group.

In 2016 ProPublica analysed a criminal risk-scoring tool (COMPAS) and reported that Black defendants who did not reoffend were more likely to be wrongly labelled high risk. The tool's maker argued it was equally calibrated across groups. Researchers later showed that, when groups have different underlying rates, it is mathematically impossible to satisfy all these fairness criteria at once. Choosing between them is an ethical and legal decision, not only a technical one, and it should be made openly.

## Misuse beyond bias

- **Using a model outside its purpose or range.** A model built to rank applicants for a scholarship is not valid for deciding who gets a job.
- **Hiding uncertainty** so a decision looks more certain than it is (lesson 1 of this module).
- **Data dredging (p-hacking):** testing many variations until one gives a "significant" result, then reporting only that one.
- **Goodhart's law:** when a measure becomes a target, people change their behaviour to hit the number, and it stops measuring what it used to.
- **Pressure to deliver a number.** Being asked to adjust assumptions until a project looks profitable or a reserve looks smaller.

## Worked example: reviewing an insurance pricing model

An actuary reviews a new car insurance model that uses 40 variables. Questions to ask:

1. **Legal:** which characteristics are prohibited or restricted by insurance regulation in this jurisdiction? Is any variable a close proxy for one?
2. **Data:** does the training data under-represent any group or region?
3. **Outcomes:** compare prices and error rates across groups and regions. Are differences explained by genuine differences in risk?
4. **Explainability:** can the insurer explain to a customer and a regulator why their price is what it is?
5. **Monitoring:** who checks the model after launch, and how often?

If something looks wrong, the actuary documents it, raises it with the responsible people and, if needed, follows the professional code's guidance on escalation.

## Professional codes

Each profession has written standards. The American Statistical Association publishes Ethical Guidelines for Statistical Practice at [amstat.org](https://www.amstat.org/your-career/ethical-guidelines-for-statistical-practice). Actuaries follow their society's Code of Professional Conduct (see [soa.org](https://www.soa.org/) and [casact.org](https://www.casact.org/)) and, in the US, the Actuarial Standards of Practice issued by the [Actuarial Standards Board](https://www.actuarialstandardsboard.org/). INFORMS publishes ethics guidelines for operations research and analytics at [informs.org](https://www.informs.org/). Checked on 2026-10-23.

Common themes: be honest about methods and limitations, respect privacy, avoid conflicts of interest, do not let clients or employers pressure you into misleading results, and consider who is affected by your work.

## Common mistakes

- **"The model is just math, so it is neutral."** Data, targets and choices carry values.
- **Removing a protected variable and assuming the problem is solved.** Check for proxies.
- **Checking only overall accuracy.** Look at results for each group.
- **Staying silent.** Professional codes expect you to raise concerns.

## Quick recap

- Bias enters through data, samples, proxies, wrong targets and feedback loops.
- Fairness has several definitions that cannot all hold at once; choose openly.
- Watch for misuse: wrong purpose, hidden uncertainty, p-hacking, Goodhart's law, pressure.
- Know your profession's code: ASA guidelines, actuarial codes and ASOPs, INFORMS ethics guidelines.

## Try it

This lesson has no activity. Watch the videos, then take the quiz in the **Do** tab.


Spot bias, proxies and misuse in statistical and decision models, and know the professional codes that apply.

Ethics of models

Algorithmic Bias and Fairness: Crash Course AI #18

The Truth About Algorithms | Cathy O'Neil

Explain uncertainty to non-specialists, make analyses reproducible, and recognise bias and misuse in models.

Foundations for Math, Statistics & Analytics Careers

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Thinking in Numbers

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Communicating Quantitative Work

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