Lesson 1 of 8 · All lessons in this module

Free lesson: the first lesson of every module is open to everyone.

Lesson 1 of 8 · 13 min

Populations, samples and sampling error

We study a sample because examining the whole population is impossible or too costly, and the price we pay is sampling error.

In short

A population is every member of the group we care about; a parameter (e.g. $\mu$ ) describes it. A sample is a subset; a statistic (e.g. $\bar X$ ) describes the sample and estimates the parameter.
Probability sampling gives every member an equal chance of selection and tends to produce a representative sample. Non-probability sampling relies on judgment or convenience and risks a non-representative one.
Simple random sampling suits a homogeneous population. Systematic sampling (every $k$ th member of a list) is a practical way to get an approximately random sample.
Sampling error = statistic − parameter. It exists because only part of the population is observed, even when the sample is drawn perfectly.
A statistic is a random variable. Its sampling distribution is the distribution of all the values it can take across samples of the same size from the same population.

Population, sample, parameter, statistic

A population is every member of a group you want to learn about: all shares listed in Oslo, all euro investment-grade bonds, all clients of a bank. A sample is any subset of that population.

A parameter is a number that describes the population, such as the population mean $\mu$ or standard deviation $\sigma$ . A statistic is the same kind of number computed from a sample, such as the sample mean $\bar X$ or the sample standard deviation $s$ . We use statistics to estimate parameters.

Why not measure the whole population? Sometimes it is impossible: you cannot observe every possible future return. More often it is possible but wasteful: surveying 4,000 companies costs far more time and money than surveying 200. Sampling trades time and money for some loss of accuracy.

Before collecting data, the analyst sets a sampling plan: the set of rules that decides which members end up in the sample. The plan determines whether the sample can support sound conclusions about the population.

The two sides of every estimation problem
	Population	Sample
What it is	Every member of the group	A subset of the group
A summary number is called	Parameter	Statistic
Mean	$\mu$	$\bar X$
Standard deviation	$\sigma$	$s$
Fixed or random?	Fixed: one true value, usually unknown	Random: changes from sample to sample

Two families: probability and non-probability sampling

In probability sampling, chance decides who is selected, and every member of the population has an equal probability of being picked. Because nobody's preferences steer the choice, the sample tends to mirror the population.

In non-probability sampling, something other than chance decides: the researcher's judgment, or simply how easy the data are to reach. That can be fast and cheap, but there is a significant risk that the sample does not represent the population.

All else equal, probability sampling gives more accurate and more reliable estimates. This module covers simple random, systematic, stratified random and cluster sampling (probability methods) and convenience and judgmental sampling (non-probability methods).

Exam shortcut: words like 'at random', 'equal chance' or 'every kth' point to a probability method. Words like 'already available', 'easiest to access' or 'an expert selects' point to a non-probability method.

Simple random and systematic sampling

A simple random sample is a subset chosen so that every member of the population has the same probability of being selected. Think of every name going into one hat. The procedure is called simple random sampling, and it is the basic design from which statistically sound conclusions about a population can be drawn.

It works best when the population is homogeneous, meaning its members are broadly similar in characteristics such as size or region. If the population is made up of very different groups, a purely random draw can over- or under-represent some groups by chance, and other designs (next lesson) do better.

Systematic sampling is a practical route to an approximately random sample when you cannot number, or even identify, every member in advance. You select every $k$ th member of a list until you reach the sample size you want, where $k$ = population size ÷ sample size. Begin at a random point within the first $k$ members.

Sampling error and the sampling distribution

Even a perfectly drawn random sample will not reproduce the population exactly, because it contains only some of the members. Sampling error is the difference between the observed value of a statistic and the parameter it is meant to estimate. For the mean, sampling error = $\bar X - \mu$ .

Sampling error is not a calculation mistake or a data problem. It comes from sampling variation: a different random sample would contain different members and give a different statistic. An analyst who samples accepts it as the cost of saving time and money.

Because a statistic changes from sample to sample, it is itself a random variable with its own probability distribution. The sampling distribution of a statistic is the distribution of all the distinct values it can take when computed from samples of the same size drawn randomly from the same population.

A random sample reflects the population without systematic distortion, so statistics computed from it are valid estimates of the parameters. The sampling distribution then tells us how far, typically, an estimate lands from the truth. The rest of this module is about measuring and shrinking that distance.

Two random samples of 40 from the same 500 stocks (true mean yield 2.85%) miss in opposite directions: +0.25 and −0.15 points. Neither is a mistake; the gap is sampling variation, and it changes with every new sample.

Keep two distributions apart: the distribution of the data (individual returns) and the sampling distribution of a statistic (the mean of n returns). They usually differ in shape and always differ in spread.

Formulas

Sampling error of the mean: $\text{Sampling error} = \bar X - \mu$
Statistic minus parameter. Positive means the sample overestimates the population mean.
Systematic sampling interval: $k = \dfrac{N}{n}$
N = population size, n = desired sample size. Select every kth member, starting at a random point among the first k.

Worked examples

Example 1 Setting up a systematic sample

A data vendor lists 2,400 euro corporate bonds. You want a systematic sample of 80 bonds, and your random starting point is bond number 17. What is the sampling interval, and which bonds are selected?

Example 2 Measuring sampling error

An index of 500 stocks has a mean dividend yield of 2.85%. Analyst A's random sample of 40 stocks has a mean yield of 3.10%; Analyst B's random sample of 40 has a mean of 2.70%. What is each sampling error, and did either analyst make a mistake?

Example 3 Parameter or statistic? (exam style)

A researcher wants the average expense ratio of all 3,200 equity funds sold in Norway. She draws 150 funds at random and finds an average of 1.12% with a standard deviation of 0.41%. Identify the population, parameter and statistics, and say what a colleague repeating the exercise would find.

On the exam

Expect definition questions (sampling error, sampling distribution, parameter vs statistic) and quick classification of a described method as simple random or systematic. Occasionally a one-line calculation of the interval k or of a sampling error.

Sampling error is not a calculation mistake and does not vanish with a perfectly random sample; it shrinks with larger samples and disappears only if the whole population is examined.
The sampling distribution is the distribution of a statistic across many samples, not the distribution of the observations within one sample.
Systematic sampling (every kth member) is a probability method that approximates random sampling, not a non-probability method.
Sign of sampling error: statistic minus parameter, not the reverse.

Check yourself

Q1An analyst computes the average P/B ratio of 60 randomly chosen stocks from a 1,500-stock index. The average P/B of all 1,500 stocks is best described as a:
Q2Using systematic sampling, an analyst wants a sample of 120 funds from a database of 3,000 funds. She should select every:
Q3The population mean return of a fund universe is 5.80%. A random sample of 50 funds has a mean return of 6.20%. The sampling error is closest to:
Q4The sampling distribution of the sample mean is best described as the distribution of:
Q5Simple random sampling is most appropriate when the population is:
Q6Which statement about sampling error is most accurate? Sampling error:

0/6Check yourself

Answer the 6 questions: +10 XP for each one you get right first time.

Liked this lesson? Get the rest free for 7 days

Create a free account to open the other 7 lessons in this module, with videos, flashcards that come back before you forget them, the question bank and full mocks. No card needed.

Start your 7-day free trial Log in