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1.1 · Use basic statistical terms and notation

Learn to use basic statistical terms and notation through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Descriptive Statistics

Population, sample, variable types, parameters, statistics, and standard notation

Every statistics course begins with a shared vocabulary, because the same word can mean something quite specific in statistics even though it sounds like everyday English. Before any calculation can be trusted, we need to know exactly which group of people or objects we are describing, whether we measured all of them or only some, and whether the characteristic we recorded is a number we can do arithmetic on or simply a label. This lesson builds that vocabulary from the ground up. We will define population, sample, individual, variable, parameter, and statistic; distinguish quantitative from qualitative variables; introduce the standard notation used throughout MATH 215; and practice identifying each term in a realistic scenario. No formulas from later chapters are needed here, only careful reading and correct labeling, which is itself a skill that is graded and tested in this course."

What you will learn

Define population, sample, individual, and variable, and correctly identify each one in a written scenario.
Distinguish a parameter from a statistic and explain why a statistic is only an estimate of a parameter.
Classify a variable as quantitative or qualitative and justify the classification.
Read and correctly interpret basic statistical notation, including $N$ , $n$ , $\mu$ , $\bar{x}$ , $\sigma$ , and $s$ .
Apply these terms and notation together to summarize a data-collection scenario in correct statistical language.

Population, Sample, and Individual

Before we can compute anything in statistics, we must identify exactly who or what is being studied. The population is the entire collection of individuals or objects that the researcher wants to know about. An individual is one single member of that population, such as one person, one plant, or one manufactured part. In most real studies it is too slow, too costly, or physically impossible to measure every individual in the population, so researchers instead measure a smaller, manageable group called a sample. A sample is a subset of the population, meaning it is made up entirely of individuals who also belong to the population, but it does not include everyone.

The size of the population is usually written with a capital letter

N

, and the size of the sample is usually written with a lowercase letter

n

. This distinction in letter case is not decorative; it tells the reader immediately whether a count refers to the whole population or to only the observed sample. For example, if a college has

N = 8000

registered students and a survey collects responses from

n = 250

of them, the 250 respondents form the sample and the 8000 students form the population.

Choosing the sample carefully matters because conclusions drawn from the sample are meant to describe the population. If the sample is not representative, for example if it includes only students from one program, then conclusions about all 8000 students could be misleading. This course will return to sampling methods later, but for now the key skill is simply recognizing, in any description of a study, which group is the population and which smaller group is the sample.

Variables: Quantitative and Qualitative

A variable is a characteristic recorded on each individual that can differ from one individual to the next. If every individual had exactly the same value for a characteristic, there would be nothing to study, so a true variable must be able to vary across individuals. For instance, the height of a plant, the color of a car, and the number of children in a household are all variables because different individuals can show different values.

Variables are sorted into two broad types. A quantitative variable is recorded as a number that expresses a measurable amount, and it makes sense to perform arithmetic on it, such as adding, averaging, or finding a difference. Height in centimeters, exam score out of 100, and number of children are quantitative because averaging them produces a meaningful result. A qualitative variable, also called a categorical variable, is recorded as a label or category name rather than a measurable amount. Car color, blood type, and yes-or-no survey answers are qualitative because it makes no sense to average them.

A common source of confusion is a variable that looks numeric but is really a label. A student identification number or a postal code is written using digits, but adding two ID numbers together produces a meaningless result, so these are qualitative variables even though they contain numerals. The test to apply is always the same: ask whether performing arithmetic, such as computing an average, produces a number that has real meaning for that characteristic. If yes, the variable is quantitative; if no, it is qualitative.

Parameters, Statistics, and Standard Notation

Once we know the population, the sample, and the variable of interest, we often want a single number that summarizes that variable, such as an average or a measure of spread. A parameter is a numerical summary describing the entire population. Because parameters describe the population, and populations are usually large or impossible to measure completely, parameters are almost always unknown constants that researchers try to estimate. A statistic is a numerical summary computed from the sample data only. Because a statistic uses observed data, it can always be calculated directly, and it is then used as an estimate of the corresponding unknown parameter.

Statistics notation is built to keep parameters and statistics visually distinct. The population mean, meaning the average of the variable across every individual in the population, is written

\mu

, the lowercase Greek letter mu. The sample mean, meaning the average of the variable across only the individuals in the sample, is written

\bar{x}

, read as x-bar. Likewise the population standard deviation, a number describing how spread out the population values are, is written

\sigma

, the lowercase Greek letter sigma, while the sample standard deviation is written

s

. Greek letters are reserved for parameters, and ordinary Latin letters or Latin letters with a bar are reserved for statistics.

The sample mean itself is calculated by adding every observed value together and dividing by the number of observations in the sample. In symbols, this is

\bar{x} = \frac{\sum x_i}{n}

, where

\sum x_i

means the sum of all the individual data values

x_1, x_2, \dots, x_n

, and

n

is the sample size. This formula only requires basic arithmetic: addition and division, both of which are prerequisite algebra skills. The corresponding population mean formula looks identical in structure but uses

N

and covers every individual in the population,

\mu = \frac{\sum x_i}{N}

. Because a population is usually too large to measure completely,

\mu

is normally unknown, and

\bar{x}

is calculated from the sample and used as its estimate.

Parameter versus Statistic Notation

Concept	Population version	Sample version
Size of group	$N$	$n$
Mean (average)	$\mu$	$\bar{x}$
Standard deviation	$\sigma$	$s$
Type of number	Parameter (usually unknown)	Statistic (calculated from data)

Worked example

Identifying Terms and Computing a Sample Mean

A college has

N = 8000

registered students. A researcher wants to estimate the average number of hours students spend studying per week. She randomly selects

n = 6

students and records the following weekly study hours: CAD 12, 15, 9, 20, 14, 10. Identify the population, sample, individual, variable, variable type, and the relevant parameter and statistic, then calculate the sample mean.

Identify the population and sample
The population is the entire group the researcher ultimately wants to describe, which is all 8000 registered students at the college. The sample is the smaller group actually measured, namely the 6 students whose study hours were recorded. Every one of those 6 students is also part of the 8000, which is required for a valid sample.
Identify the individual and the variable
Each single student in the study is an individual. The characteristic being recorded on each individual is weekly study hours, and this is the variable, because different students report different numbers of hours.
Classify the variable
Weekly study hours are expressed as numbers, and it is meaningful to add them together or average them, since the average would represent a genuine typical study time. Therefore this variable is quantitative.
Identify the parameter and statistic in context
The unknown average study time across all 8000 students is a parameter, written $\mu$ , since it describes the whole population. The average computed from only the 6 sampled students is a statistic, written $\bar{x}$ , since it is computed from sample data and will be used to estimate $\mu$ .
Apply the sample mean formula
The sample mean is found by summing all observed values and dividing by the sample size $n$ . Here the data values are $x_1=12,\ x_2=15,\ x_3=9,\ x_4=20,\ x_5=14,\ x_6=10$ , and $n = 6$ .
$\bar{x} = \frac{\sum x_i}{n}$
Substitute the values
Add the six study-hour values together first, then divide by the sample size of 6.
$\bar{x} = \frac{12+15+9+20+14+10}{6}$
Calculate
The sum of the six values is $12+15+9+20+14+10 = 80$ . Dividing this sum by the sample size 6 gives the sample mean, kept to six decimal places before final rounding.
$\bar{x} = \frac{80}{6} = 13.333333$
Round and interpret in context
Rounding to two decimal places, since study hours are typically reported that precisely, the sample mean weekly study time is 13.33 hours. This value is a statistic: it describes only the 6 sampled students and serves as an estimate of the unknown population mean $\mu$ , the true average weekly study time for all 8000 students at the college.
$\bar{x} \approx 13.33 \text{ hours}$

Answer: The sample mean is

\bar{x} \approx 13.33

hours per week, a statistic estimating the unknown population parameter

\mu

Check: Recompute the sum a second time in a different grouping order to confirm accuracy:

(12+10)+(15+14)+(9+20) = 22+29+29 = 80

, which matches the original sum, confirming

\bar{x} = 80/6 \approx 13.33

hours is correct.

Common mistakes and how to avoid them

Using the symbol

\mu

for a sample mean or

\bar{x}

for a population mean.

Correction: Match the symbol to the group described: use

\mu

and

N

only when every member of the population was measured; use

\bar{x}

and

n

when only a sample was measured.

Calling every number that describes data a statistic.

Correction: Reserve the word statistic for a number computed from a sample. A number computed from the whole population is a parameter.

Treating a numerically coded categorical variable, such as postal codes or team jersey numbers, as quantitative.

Correction: Check what the number represents. If it only labels a category and arithmetic on it is meaningless, it is qualitative even though it looks numeric.

Confusing the individual (the object measured) with the variable (the characteristic measured on it).

Correction: Ask: is this the thing being studied, or the property recorded about that thing? A student is an individual; the student's exam score is a variable.

Assuming a sample statistic and the population parameter it estimates must be identical.

Correction: Remember that a statistic is an estimate. It usually differs somewhat from the true parameter because it is based on only part of the population.

Lesson summary

A population is the entire group of interest; a sample is a subset actually observed. Population size is $N$ and sample size is $n$ .
An individual is a single member of the population or sample; a variable is a characteristic recorded on each individual that can differ across individuals.
A quantitative variable is a measurable number where arithmetic is meaningful; a qualitative variable is a category label, even if it is written using digits.
A parameter, such as $\mu$ or $\sigma$ , describes the whole population and is usually unknown; a statistic, such as $\bar{x}$ or $s$ , is computed from sample data and estimates the parameter.
Greek letters denote parameters and Latin letters (often with a bar) denote statistics, keeping population summaries and sample summaries notationally distinct.
The sample mean formula $\bar{x} = \frac{\sum x_i}{n}$ uses only addition and division, applying prerequisite algebra to a new statistical purpose.

Check your understanding

Question 1

A veterinary researcher wants to know the average weight of all 500 dogs registered at a city shelter. She weighs 40 of those dogs and calculates their average weight. What does the number

500

represent in correct statistical notation?

$n$ , the sample size
$N$ , the population size
$\bar{x}$ , the sample mean
$\mu$ , the population mean

Show answer and explanation

N

, the population size

The 500 dogs form the entire group of interest, the population, so 500 is the population size and is denoted

N

. The 40 weighed dogs form the sample, so their count would be

n = 40

Question 2

Which variable below is qualitative rather than quantitative?

The number of pages in a textbook
The brand name printed on a textbook cover
The price of a textbook in dollars
The weight of a textbook in grams

Show answer and explanation

The brand name printed on a textbook cover

Brand name is a category label; averaging or adding brand names produces no meaningful result, so it is qualitative. Page count, price, and weight are all measurable amounts where arithmetic makes sense, so they are quantitative.

Question 3

A sample of 5 exam scores is CAD 70, 85, 90, 60, 95. What is the sample mean

\bar{x}

78.000000
80.000000
82.500000
75.000000

Show answer and explanation

80.000000

The sum is

70+85+90+60+95 = 400

, and dividing by the sample size

n=5

gives

\bar{x} = 400/5 = 80.000000

Question 4

Why is a computed sample statistic, such as

\bar{x}

, generally not exactly equal to the true population parameter,

\mu

Because statistics always use rounding errors that parameters never use
Because a statistic is based on only part of the population, so it is an estimate rather than a complete measurement
Because Greek letters are always larger in value than Latin letters
Because parameters are only used for qualitative variables

Show answer and explanation

Because a statistic is based on only part of the population, so it is an estimate rather than a complete measurement

A statistic comes from sample data, which covers only some individuals from the population. Since it is not built from every individual, it serves as an estimate of the parameter and will typically differ from it somewhat.

Key terms

Population: The complete collection of all individuals or objects that a study wants to describe or draw conclusions about.
Sample: A subset of the population that is actually observed or measured and used to learn about the population.
Individual: A single person, object, or item that belongs to the population or sample and about which data are recorded.
Variable: A characteristic of an individual that can take different values from one individual to another.
Parameter: A fixed numerical summary that describes an entire population, usually denoted with Greek letters such as $\mu$ or $\sigma$ .
Statistic: A numerical summary computed from sample data, usually denoted with Latin letters such as $\bar{x}$ or $s$ , used to estimate a parameter.
Quantitative variable: A variable whose values are numbers that represent a measurable amount, so arithmetic on them is meaningful.
Qualitative variable: A variable whose values are category labels or names rather than measurable amounts, sometimes called a categorical variable.

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Published by DoAssignment. This AI-assisted lesson follows Athabasca University MATH 215: Introduction to Statistics, study topic 1.1. It is a study resource, not an official curriculum publication.

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