Populations, Samples, and Variables | Malone Math

The Central Idea

Most statistical studies involve three basic questions:

What group do we want to understand?
Which members of that group do we actually observe?
What characteristics do we record for each observed case?

These questions lead to the concepts of population, sample, and variable.

What Is a Population?

A population is the complete collection of individuals, objects, events, measurements, or cases about which a statistical question is being asked.

Examples include:

all registered voters in a state;
all light bulbs produced by a factory this month;
all transactions processed by a bank during one year;
all students enrolled at a university;
all possible outcomes generated by a defined process.

The population is determined by the question being investigated.

The Population Must Be Defined Precisely

A statement such as "we want to study customers" is incomplete.

We must ask:

Which customers?
From which business?
During what time period?
In what geographic area?
Under what eligibility rules?

A precise statistical population might be:

all customers who completed an online purchase from the company between January 1 and March 31.

What Is a Sample?

A sample is a subset of the population that is actually observed or measured.

If a university has $18{,}000$ students and researchers survey $600$ of them, then

\[ \text{population size}=18{,}000 \]

while

\[ \text{sample size}=600. \]

The purpose of the sample is usually to provide information about the larger population.

Why Use Samples?

Studying every member of a population may be impractical, expensive, destructive, or impossible.

Sampling can reduce:

cost;
time;
labor;
data-storage requirements;
physical disruption to the population.

A carefully selected sample can often provide useful information about the whole population without measuring every case.

What Is a Census?

A census attempts to collect information from every member of the population.

If every employee in a company is asked to complete a workplace survey, the intended design is a census of that company's employees.

A census does not automatically eliminate all error. Missing responses, inaccurate measurements, changing membership, or flawed questions can still affect the results.

Cases and Observational Units

A case, or observational unit, is one individual object, event, or unit about which data are recorded.

Depending on the study, a case might be:

a person;
a household;
a school;
a machine;
a city;
a transaction;
a day;
a product shipment.

Rows Usually Represent Cases

In a rectangular dataset, rows commonly represent cases and columns represent variables.

Customer	Region	Orders	Annual Spending
001	West	8	$1,240
002	South	5	$860
003	West	11	$1,930

Each row describes one customer. Each column records one characteristic.

What Is a Variable?

A variable is a characteristic that can take different values across cases.

Examples include:

age;
income;
department;
number of purchases;
temperature;
response category;
delivery time.

A variable tells us what is being recorded for each case.

Variables and Values

A variable is the characteristic being measured. A value is the particular recorded result for one case.

If the variable is number of purchases, possible observed values might be

\[ 0,\qquad 3,\qquad 7,\qquad 12. \]

The variable is the general attribute; the values are its observed realizations.

Several Variables Can Be Recorded for the Same Case

A study of employees might record:

department;
years of service;
salary;
training hours;
performance category.

Each employee is one case, while the dataset contains several variables describing that case.

Population Variables and Sample Variables

The same variable can be conceptually defined for every member of a population even when it is observed only for the sample.

Suppose we want to study annual household electricity use.

The variable exists conceptually for every household in the target population, but researchers may measure it for only a selected sample of households.

Parameters Describe Populations

A parameter is a numerical characteristic of a population.

Examples include:

the true mean income of all households in a city;
the true proportion of all voters supporting a proposal;
the true standard deviation of product weights produced by a machine.

Parameters are usually fixed for a clearly defined population, even when their values are unknown.

Statistics Describe Samples

A statistic is a numerical characteristic calculated from sample data.

Common sample statistics include:

sample mean;
sample proportion;
sample median;
sample standard deviation.

Statistics are often used to estimate unknown population parameters.

Parameter Versus Statistic

Suppose a population contains $N$ cases with values $x_1,x_2,\ldots,x_N$.

The population mean is commonly written

\[ \mu=\frac{1}{N}\sum_{i=1}^{N}x_i. \]

If a sample contains $n$ observed cases, its sample mean is commonly written

\[ \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i. \]

The population quantity $\mu$ is a parameter, while the sample quantity $\bar{x}$ is a statistic.

Population Proportion and Sample Proportion

If a proportion in the full population is denoted by $p$, then a sample proportion is often denoted by

\[ \hat{p}. \]

If $84$ of $120$ sampled customers report being satisfied, then

\[ \hat{p} = \frac{84}{120} = 0.70. \]

The sample statistic $0.70$ may be used to estimate the unknown population proportion $p$.

Descriptive and Inferential Roles

Statistics can play two broad roles.

Descriptive statistics organize and summarize the data actually observed.

Statistical inference uses sample information to reason about a broader population.

This chapter begins with descriptive methods, but the population-sample distinction prepares the foundation for later inference.

Target Population

The target population is the full group to which the study intends its conclusions to apply.

For example, a manufacturer may want conclusions about

all units produced on a particular assembly line during the next quarter.

The target population must match the scope of the claim being made.

Accessible Population

The accessible population is the portion of the target population that researchers can practically reach or identify.

The accessible population may differ from the target population because of:

geographic limitations;
missing records;
eligibility restrictions;
technology access;
institutional boundaries.

This distinction can matter when assessing how broadly results should be generalized.

Sampling Frame

A sampling frame is the operational list, database, or mechanism used to identify members from which a sample is selected.

Examples include:

a voter registration list;
a customer database;
a class roster;
a list of manufactured serial numbers;
a directory of licensed businesses.

Ideally, the sampling frame closely matches the target population.

Coverage Problems

A sampling frame may fail to include some members of the target population or may include cases that do not belong.

These problems are forms of coverage error.

For example, an online-only survey may poorly cover members of a population who lack reliable internet access.

Sample Size

The number of cases in a sample is called the sample size and is commonly denoted by

\[ n. \]

If $n=500$, then the sample contains $500$ observed cases.

A larger sample can provide more information, but sample size alone does not guarantee quality.

A Large Biased Sample Can Still Be Misleading

Suppose a university wants to measure student satisfaction but surveys only students attending an optional honors reception.

Even if thousands of students respond, the sample may systematically differ from the full student population.

Representativeness depends on how the sample is obtained, not merely on how many observations are collected.

Representative Samples

A representative sample reflects relevant characteristics of the population well enough to support the intended analysis.

Representativeness does not mean that every sample characteristic must exactly equal the corresponding population characteristic.

Random variation is expected. The goal is to avoid systematic distortion that consistently favors some kinds of cases over others.

Sampling Variability

Different samples drawn from the same population will usually produce somewhat different statistics.

This natural sample-to-sample change is called sampling variability.

For example, two samples may produce

\[ \hat{p}_1=0.61 \]

and

\[ \hat{p}_2=0.64. \]

Neither difference automatically indicates an error. Sample statistics naturally vary.

Sampling Error

The difference between a sample statistic and its corresponding population parameter is often called sampling error.

For a sample mean,

\[ \text{sampling error}=\bar{x}-\mu. \]

This difference arises because only part of the population was observed.

Sampling error is distinct from mistakes in measurement, data entry, or study design.

Sampling Error Is Not the Same as Bias

Sampling error is ordinary variation caused by observing a sample rather than the entire population.

Bias is a systematic tendency for a method to favor some outcomes or misrepresent the population.

A well-designed random sample can still have sampling error, while a poorly designed sample can have both sampling error and bias.

Common Sources of Bias

Bias can arise from many sources, including:

undercoverage;
voluntary response;
nonresponse;
leading questions;
measurement procedures;
selection rules that favor certain cases.

Later statistical methods depend on understanding how the data were obtained.

Observational Studies and Experiments

In an observational study, researchers record variables without assigning treatments or interventions.

In an experiment, researchers deliberately assign treatments or conditions and observe the resulting outcomes.

Both designs contain populations, samples, cases, and variables, but the design affects what conclusions can reasonably be drawn.

Explanatory and Response Variables

When studying relationships between variables, one variable may be treated as an explanatory variable and another as a response variable.

The explanatory variable is used to help explain or predict changes in the response variable.

For example, a study might investigate whether hours of training help explain variation in task completion time.

Variables Do Not Automatically Establish Causation

If two variables are associated, that relationship does not by itself prove that one causes the other.

Statistical conclusions depend on:

how the sample was selected;
how variables were measured;
whether treatments were assigned;
whether important alternative explanations were controlled.

The structure of the study matters as much as the numerical summaries.

Primary and Derived Variables

A primary variable is recorded directly from the case.

A derived variable is calculated from one or more recorded variables.

If a dataset records revenue and cost, then profit may be derived as

\[ \text{profit} = \text{revenue} - \text{cost}. \]

Derived variables can be useful, but their meaning depends on the quality and definitions of the variables from which they were constructed.

Units Matter

A quantitative variable should usually be recorded with its unit of measurement.

For example:

\[ 12 \]

is ambiguous, while

\[ 12\text{ minutes} \]

clearly identifies a duration.

Units help distinguish variables, interpret calculations, and prevent accidental comparisons of incompatible quantities.

Example: Customer Satisfaction Study

Suppose a retailer wants to estimate satisfaction among all customers who purchased during August.

The study might be organized as follows:

population: all August customers;
sample: 1,000 selected August customers;
case: one selected customer;
variable: satisfaction rating;
statistic: the sample proportion reporting high satisfaction;
parameter: the true population proportion reporting high satisfaction.

Example: Manufacturing Quality

A factory produces $40{,}000$ components during one week. Inspectors randomly select $400$ components and record whether each component passes a tolerance test.

population: all $40{,}000$ components;
sample: the $400$ inspected components;
case: one component;
variable: pass/fail status;
sample statistic: the proportion passing in the sample;
population parameter: the proportion passing among all $40{,}000$ components.

Example: University Commute Times

Researchers want to understand commute times for all students at a university.

They survey $750$ students and record each commute time in minutes.

population: all university students;
sample: the $750$ surveyed students;
variable: commute time;
statistic: the mean commute time of the $750$ students;
parameter: the mean commute time of all university students.

Common Error: Confusing the Population with the Sample

If $300$ voters are surveyed from a city with $80{,}000$ registered voters, the $300$ voters are not the population.

They are the sample.

The population is the full group about which the statistical claim is intended.

Common Error: Defining the Population Too Broadly

A survey conducted among customers of one store cannot automatically support conclusions about all consumers everywhere.

The target population must match the design and reach of the study.

Statistical conclusions should not extend beyond the population that the data collection process can reasonably represent.

Common Error: Treating the Sample Statistic as the Exact Parameter

Suppose

\[ \hat{p}=0.58. \]

This means $58\%$ of the sampled cases had the characteristic.

It does not automatically mean that exactly $58\%$ of the entire population has the characteristic.

The statistic is evidence about the parameter, not ordinarily an exact measurement of it.

Common Error: Assuming a Census Has No Problems

Even when every intended case is contacted, data quality can still be affected by:

nonresponse;
misunderstood questions;
incorrect measurements;
duplicate records;
data-entry mistakes;
changes in the population during collection.

Complete intended coverage does not guarantee perfect data.

Common Error: Calling Every Column a Measurement

A dataset column may be:

a categorical label;
a numerical measurement;
an identifier;
a date;
a derived value;
a control field used only for data management.

The statistical role of a variable depends on what the column represents.

A Reliable Study-Identification Procedure

State the statistical question.
Identify the full population relevant to that question.
Identify the sample actually observed.
Identify one observational unit or case.
List the variables recorded for each case.
Classify those variables by data type and measurement level when useful.
Identify any sample statistics being calculated.
Identify the corresponding population parameters when inference is intended.
Check whether the sampling method reasonably connects the sample to the target population.

Practice 1: Population and Sample

A city wants to estimate the average monthly water use of its $52{,}000$ households. It randomly selects $800$ households and records their water use.

Identify the population and sample.

Practice 2: Cases and Variables

A researcher records the following information for $250$ restaurants:

number of employees;
average meal price;
cuisine category;
weekly customer count.

Identify the cases and variables.

Practice 3: Parameter or Statistic

Classify each quantity as a parameter or statistic.

the mean age of all employees in a company;
the mean age of $120$ sampled employees;
the proportion of all manufactured units that are defective;
the proportion defective among $500$ inspected units.

Practice 4: Identify the Sample Size

A poll contacts $1{,}250$ adults selected from a national population.

What is the sample size?

Practice 5: Census or Sample?

Classify each data collection effort.

Every employee in a 40-person company completes a survey.
200 customers are selected from a database of 25,000 customers.
Every item in a shipment of 600 items is inspected.
50 students are selected from a school of 1,400 students.

Practice 6: Sampling Frame

A county wants to survey all households but selects addresses from a property-tax database.

What is the sampling frame? Give one possible coverage concern.

Practice 7: Population Definition

A business wants to study "customer spending."

Explain why this description is not yet a sufficiently precise population definition.

Practice 8: Statistic and Parameter

In a sample of $300$ shipments, the mean delivery time is $2.8$ days.

Identify the statistic and the corresponding population parameter.

Practice 9: Sampling Error

Suppose the true population mean is

\[ \mu=50 \]

and a sample produces

\[ \bar{x}=52. \]

Compute the sampling error using

\[ \bar{x}-\mu. \]

Practice 10: Full Study Description

A manufacturer wants to estimate the mean lifetime of all batteries produced during one production run. It randomly selects $100$ batteries, measures each lifetime in hours, and computes a sample mean of $1{,}240$ hours.

Identify:

the population;
the sample;
the case;
the variable;
the sample statistic;
the corresponding population parameter.

Practice Answers

Practice 1

The population is all $52{,}000$ households in the city. The sample is the $800$ selected households.

Practice 2

Each restaurant is one case.

The variables are number of employees, average meal price, cuisine category, and weekly customer count.

Practice 3

parameter;
statistic;
parameter;
statistic.

Practice 4

The sample size is

\[ \boxed{n=1{,}250}. \]

Practice 5

census;
sample;
census;
sample.

Practice 6

The property-tax database is the sampling frame. One possible coverage concern is that households not represented correctly in the database may be omitted.

Practice 7

The description does not specify which customers, which business, what time period, or what eligibility conditions define the population.

Practice 8

The statistic is the sample mean

\[ \bar{x}=2.8\text{ days}. \]

The corresponding parameter is the true mean delivery time $\mu$ for the target population of shipments.

Practice 9

\[ \bar{x}-\mu = 52-50 = \boxed{2}. \]

Practice 10

population: all batteries produced during the production run;
sample: the $100$ selected batteries;
case: one battery;
variable: battery lifetime in hours;
sample statistic: $\bar{x}=1{,}240$ hours;
population parameter: the true mean lifetime $\mu$ of all batteries in the run.

What to Remember

A population is the full collection about which a statistical question is asked.
A sample is the subset actually observed.
A case is one observational unit.
A variable is a characteristic recorded for each case.
A parameter describes a population.
A statistic describes a sample.
Samples are often used because censuses are costly, slow, or impractical.
Large sample size does not compensate for a systematically biased sampling method.
Sampling variability is expected when different samples are drawn from the same population.
Good statistical reasoning begins by matching the population, sample, variables, and claims.

Where We Go Next

Once the population, sample, cases, and variables have been identified, the next task is to organize observed values so that patterns become visible.

The next lesson, Frequency Tables, develops methods for counting how often values or categories occur and organizing those counts into useful summaries.