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1.3 · Distinguish populations, samples, experiments, and summation notation

Learn to distinguish populations, samples, experiments, and summation notation through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Descriptive Statistics

MATH 215 study topic 1.3

Statistics begins by asking who or what the data describe and how those data were gathered. A population is the full group a question is about; a sample is the part of that group for which data are collected. The way data are gathered also matters: researchers may observe what already happens, or they may assign a condition in an experiment. Finally, summation notation is a compact way to write the addition of several values. This lesson develops these ideas using simple examples and arithmetic. The notation relies on familiar algebra: a variable stands for a value, an index labels positions in a list, and a summation sign tells us to add specified terms.

What you will learn

Distinguish a population from a sample and explain how they are related.
Identify a variable, a population parameter, and a sample statistic in context.
Distinguish an experiment from a study that only observes existing information.
Read and evaluate summation notation using a list of data values.

1. Populations, samples, variables, and summaries

A population is the complete group of people, objects, or other units that a statistical question concerns. The units might be students, packages, or days. State the population precisely: “students enrolled in a particular course this term” is clearer than “students.” A sample is the subset of that population from which data are actually collected. A census collects data from every member of the population; a sample study collects data from only some members.

A variable is a characteristic recorded for each unit. For example, for each student the variable could be the number of hours spent studying during a particular week. A variable may be recorded as a number or as a category, such as a student’s chosen program. A single recorded value is one observation.

A parameter is a numerical description of a population, such as the population’s average study time. A statistic is a numerical description calculated from sample data, such as the average study time in the sample. Parameters describe the whole population; statistics describe the observed sample. A statistic may be used to learn about a population, but it is not automatically equal to the parameter.

Before interpreting a reported number, ask what group it describes and how it was obtained. A value calculated from a complete population is a population summary; one calculated from only the collected cases is a sample summary. These labels depend on the group being discussed, not on whether the number looks precise.

Population: the complete group of interest.
Sample: the units from that population that provide observed data.
Variable: the characteristic recorded for each unit.
Parameter describes a population; statistic describes a sample.

2. Observational studies and experiments

In an observational study, researchers record information about units without assigning them a condition or treatment. For instance, they might ask students how many hours they studied and record their exam scores. Researchers observe existing choices and circumstances rather than deciding who studies for a particular number of hours.

In an experiment, researchers deliberately assign a condition to units and then record a response. The assigned condition is often called a treatment. For example, a researcher might assign some participating students to receive a study-planning session and others to receive no session, then record a later quiz score. The response is the outcome being measured, such as the quiz score.

To classify a study, look for deliberate assignment by the researcher. Merely comparing people who already chose different study habits is observational, even if the researcher compares their scores. Asking a group to follow assigned study plans makes the data collection experimental. The word “experiment” does not mean that every aspect of the study is controlled; the defining feature here is assignment of a condition.

Also identify the population and sample separately from the study type. A study can be observational or experimental whether its participants are a sample or, in principle, the entire population. The words answer different questions: population and sample concern which units are included, while observational study and experiment concern how information or conditions are obtained.

No researcher-assigned condition: observational study.
Researcher assigns a condition: experiment.
Population/sample describes the group; observational/experimental describes the data-gathering approach.

3. Reading and using summation notation

When data are listed, summation notation gives a compact instruction to add selected values. Suppose the observations are labelled

x_1, x_2, \ldots, x_n

. The letter

x

names the variable, and each subscript identifies a position:

x_1

is the first observation and

x_n

is the last. The symbol

n

is the number of observations in the list.

The Greek capital sigma,

\sum

, means “add.” In

\sum_{i=1}^{n} x_i

, the lower limit says to begin with index

i=1

, the upper limit says to stop at

i=n

, and

x_i

is the term to include at each index. The index

i

is a counting label; it is not an additional measured variable. Expand the notation by replacing

i

successively with each integer from the lower limit through the upper limit.

The same idea works when the upper limit is a particular number, such as

\sum_{i=1}^{4} x_i

. This means

x_1+x_2+x_3+x_4

. If a value is repeated in the list, each occurrence is included. A common arithmetic mistake is to stop one index too soon or to include an index outside the stated limits.

A basic algebra reminder helps: addition can be written in any order without changing its total, but the terms must still be included exactly as specified. If the observations are CAD 3, 5, 4, their sum is

3+5+4

. Summation notation does not change the data; it only abbreviates the addition.

\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n

$\sum$ instructs you to add.
The limits determine which index values to use, including both endpoints.
The subscript identifies an observation’s position.

4. A careful classification routine

When you read a study description, first name the units and the full group the question concerns. Next identify which units actually provided data; that group is the sample. Name the variable recorded for each unit. If a numerical value describes the full population, call it a parameter; if it is calculated from the sample, call it a statistic.

Then check how the data were obtained. Ask whether the researcher assigned a condition. If yes, classify the study as an experiment; if no, classify it as observational. Do not infer assignment merely because two groups are compared. Finally, if a total is requested, translate the given summation limits into a list of terms and add those terms once each.

This sequence prevents common mix-ups. A sample is not the same thing as a statistic: one is a group of units, and the other is a numerical summary. Similarly, a variable is the recorded characteristic, not the value of that characteristic for one person. These distinctions make a study description easier to interpret accurately.

Define the group before interpreting its numerical summaries.
Look for assignment to distinguish an experiment from observation.
Expand the summation limits before doing arithmetic.

Worked example

Classifying a study and evaluating a sum

A researcher wants to study the effect of a planning session on quiz scores for students enrolled in an introductory course this term. The researcher recruits five students, assigns each one a planning session, and records their next quiz scores: 7, 9, 8, 6, and 10 points. Identify the population, sample, variable, study type, and sample statistic given by the total score. Write the total with summation notation and calculate it.

Identify the population and sample
The population is all students enrolled in the course this term, because that is the group the researcher wants to study. The sample is the five recruited students, because they are the units whose scores were recorded.
Name the variable and study type
The variable is each student’s next quiz score, measured in points. This is an experiment because the researcher assigned the planning session rather than simply recording which students had chosen to attend one.
Represent the sample total
Let $x_i$ denote the quiz score for the student in position $i$ . There are five scores, so the sample total adds the observations from index 1 through index 5.
$\sum_{i=1}^{5} x_i$
Substitute and calculate
Replace each indexed value with its score and add the five values. Each score is included once.
$7+9+8+6+10=40$
Interpret the result
The sample statistic calculated here is a total, not an average: the five recruited students earned 40 quiz points combined. The result describes these five students’ recorded scores; by itself, it does not give the total for every student in the course.

Answer: The population is all students enrolled in the course this term; the sample is the five recruited students. The variable is next quiz score in points. The study is an experiment because the researcher assigned the planning session. The sample total is 40 points.

Check: There are five observations and the summation limits run from 1 through 5, so all five scores are included.

Common mistakes and how to avoid them

Calling the five observed students the population even though the question concerns all students enrolled in the course.

Correction: Name the full group of interest as the population and the observed subset as the sample.

Calling a study experimental because it compares two groups.

Correction: Check whether the researcher assigned a condition. A comparison without assignment is observational.

Treating a statistic as a group of people, or a sample as a numerical result.

Correction: A sample is a set of units; a statistic is a numerical summary calculated from sample data.

Reading

\sum_{i=1}^{4} x_i

as stopping at

x_3

or including

x_5

Correction: Use every integer index from 1 through 4, including both limits.

Lesson summary

A population is the complete group of interest; a sample is the observed subset.
A variable records a characteristic of each unit. Parameters describe populations, and statistics describe samples.
Observational studies record without assigning conditions; experiments include researcher assignment.
Summation notation uses $\sum$ and its limits to specify exactly which values to add.

Check your understanding

Question 1

A researcher asks 30 selected commuters how many minutes they travelled to work. The question concerns all commuters in a city. What are the population and sample?

Population: the 30 selected commuters; sample: all city commuters
Population: all commuters in the city; sample: the 30 selected commuters
Population: travel minutes; sample: all commuters in the city
correctIndex":1,"explanation":"The population is the full group the question concerns. The 30 selected commuters are the observed subset, so they form the sample.

Show answer and explanation

Population: all commuters in the city; sample: the 30 selected commuters

The population is the full group the question concerns. The 30 selected commuters are the observed subset, so they form the sample.

Question 2

A researcher records the existing sleep habits and quiz scores of students without assigning anyone a sleep schedule. How is the study classified?

An experiment, because quiz scores are measured
An observational study, because no sleep condition was assigned
A census, because two variables were recorded
correctIndex":1,"explanation":"The researcher recorded existing habits and did not assign a condition, so this is observational.

Show answer and explanation

An observational study, because no sleep condition was assigned

The researcher recorded existing habits and did not assign a condition, so this is observational.

Question 3

x_1=4

x_2=2

, and

x_3=7

, what is

\sum_{i=1}^{3}x_i

9
13
14
correctIndex":1,"explanation":"The limits include indices 1, 2, and 3, so add the three values: 4+2+7=13.

Show answer and explanation

The limits include indices 1, 2, and 3, so add the three values: 4+2+7=13.

Key terms

Population: The complete group of units that a statistical question concerns.
Sample: The subset of the population from which data are collected.
Variable: A characteristic recorded for each unit.
Parameter: A numerical description of a population.
Statistic: A numerical description calculated from sample data.
Observational study: A study in which researchers record information without assigning a condition.
Experiment: A study in which researchers assign a condition and record a response.
Summation notation: A compact notation that uses a summation sign and index limits to specify terms to add.

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Published by DoAssignment. This AI-assisted lesson follows Athabasca University MATH 215: Introduction to Statistics, study topic 1.3. It is a study resource, not an official curriculum publication.

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