Source: Lecture notes / study guide
Difficulty: Introductory | Prerequisites: None. This is typically the first unit in an introductory statistics course.
This unit lays the groundwork for everything else in your statistics course. Before you can run a hypothesis test or build a regression model, you need to understand what data is, how it is collected, and what kind of variable you are working with. The concepts here (population vs. sample, levels of measurement, sampling methods) will come back repeatedly when you study confidence intervals, hypothesis testing, and experimental design. If you are joining the course late, start here: nothing that follows makes sense without this vocabulary.
Statistics splits into two camps: descriptive (organising and summarising data) and inferential (drawing conclusions about a population from a sample). Variables come in quantitative and qualitative flavours, measured on four scales (nominal, ordinal, interval, ratio). Data is gathered through surveys, records, or observation, and samples are drawn using random, systematic, stratified, or cluster techniques.
Statistics
The science of conducting studies to collect, organise, summarise, analyse, and interpret data in order to draw conclusions. Think of it as the toolkit for turning raw numbers into decisions.
Variable
A characteristic or attribute under study that can take on different values. In simple terms, it is whatever you are measuring or categorising about each subject.
Random variable
A variable whose values are determined by chance. Think of it as the outcome you cannot predict before the experiment runs.
Data
The values (observations or measurements) that a variable can assume. In simple terms, data is the raw material you collect.
Data set
A collection of observations (data values) on one or more variables.
Population
The complete set of all subjects (people, objects, events) being studied. In simple terms, it is the entire group you care about.
Sample
A subset of subjects selected from a population. Think of it as a smaller, manageable slice of the population you actually collect data from.
Descriptive statistics
The branch of statistics concerned with collecting, organising, summarising, and presenting data. Tables, charts, and summary values like the mean belong here.
Inferential statistics
The branch of statistics that uses sample data to make generalisations about a population, including estimation, hypothesis testing, and prediction.
Hypothesis testing
A decision-making process for evaluating claims about a population, based on sample information. Think of it as putting a claim on trial using data as evidence.
Quantitative variable
A variable that is numerical and can be ordered or ranked (e.g. age, height, weight). It splits into discrete (countable) and continuous (measured on a scale with infinite possible values).
Qualitative variable (categorical variable)
A variable that describes a characteristic or attribute and falls into distinct categories (e.g. gender, religion, eye colour). You cannot do arithmetic on it.
Discrete variable
A quantitative variable whose possible values can be counted. Think of it as values you can list: 0, 1, 2, 3.
Continuous variable
A quantitative variable that can take infinitely many values between any two points, obtained by measuring. Think of it as values on a number line with no gaps.
Nominal level of measurement
Classifies data into mutually exclusive, exhaustive categories with no natural order. Examples: gender, postcode, eye colour, nationality.
Ordinal level of measurement
Classifies data into categories that can be ranked, but the precise differences between ranks are unknown. Examples: rating scales (poor, good, excellent), class rank.
Interval level of measurement
Ranks data with equal, meaningful distances between units, but there is no true zero point. Examples: IQ scores, temperature in Celsius.
Ratio level of measurement
Possesses all the properties of interval measurement plus a true zero, allowing meaningful ratios. Examples: height, weight, time, salary.
Random sampling
A sampling technique in which every possible sample of a given size has an equal chance of being selected. Think of it as the gold standard for avoiding selection bias.
Systematic random sampling
Selecting every kth element from a list after a random starting point. Think of it as picking every 10th name on a roster.
Stratified random sampling
Dividing the population into subgroups (strata) and then randomly sampling within each. Ensures all important subgroups are represented.
Cluster sampling
Selecting an entire intact group (cluster) that is representative of the population. Convenient and cost-effective, but the cluster may be too homogeneous.
Statistics is the science of collecting, organising, summarising, analysing, interpreting, and drawing conclusions from data. It is used across fields: survey analysis, scientific research, quality control, operations research, estimation, and prediction.
The discipline splits into two broad areas:
Descriptive statistics organises and summarises data using tables, charts, graphs, and summary measures (e.g. the mean). The goal is to present data clearly, not to draw conclusions beyond the data you have.
Inferential statistics generalises from a sample to a population. This includes hypothesis testing, estimation, determining relationships among variables, and making predictions. Whenever you hear "we can infer that...", you are in inferential territory.
A statistic is a measure calculated from a sample. A parameter is the same kind of measure calculated from the entire population. Much of the course is about using statistics to estimate parameters.
Variables fall into two families, and quantitative variables subdivide further:
Quantitative (numerical) variables can be ordered or ranked and arithmetic makes sense on them.
Discrete: values that can be counted (number of children, number of defects).
Continuous: values obtained by measuring, with infinitely many possible values between any two points (height, temperature, time).
Qualitative (categorical) variables place subjects into distinct categories based on some attribute. You cannot meaningfully add or average them (gender, religion, geographic location).
Knowing which type you have determines which summary measures, graphs, and inferential tests are appropriate later in the course.
The mnemonic NOIR (Nominal, Ordinal, Interval, Ratio) is a handy way to remember the four scales, which build on one another:
Nominal: categories with no natural order. Examples: eye colour, nationality, postcode. You can count frequencies but cannot rank or compute distances.
Ordinal: categories that can be ranked, but the distances between ranks are not equal or known. Examples: letter grades (A, B, C), satisfaction ratings (poor, fair, excellent).
Interval: ranked data with equal, meaningful distances between units, but no true zero. A reading of 0 does not mean "none." Examples: temperature in Fahrenheit or Celsius, IQ scores.
Ratio: all the properties of interval plus a true zero point, so ratios are meaningful ("twice as heavy" makes sense). Examples: height, weight, time, salary, age.
As you move from nominal to ratio, the amount of information and the range of permissible calculations increase. Ratio data supports every operation; nominal data supports only counting and mode.
There are three broad methods for gathering data:
Surveys
Surveys are the most common collection method and come in several forms:
Telephone surveys: less costly, respondents may be more candid because they are not face to face. Drawback: some people have no phone, will not answer, or have unlisted numbers.
Mailed questionnaires: inexpensive, respondents can remain anonymous. Drawbacks: low response rates, people may misunderstand or give inappropriate answers.
Personal interviews: yield the most in-depth responses. Drawbacks: interviewers must be trained, interviewer bias can creep in, and they are costly.
Surveying records
Existing records (hospital files, census data, financial reports) are reviewed and data is extracted.
Direct observation
The researcher observes situations directly and records what happens. Useful when subjects may not report behaviour accurately.
We use samples rather than censuses because they save time and money, can yield more detailed information, and sometimes a census is simply impractical.
Four basic techniques:
Random sampling: every possible sample of size n has an equal chance of being selected. Methods include the fishbowl draw (put numbered cards in a bowl, mix, draw) and random number tables or generators. Limitation: numbering a very large population is time-consuming.
Systematic random sampling: pick a random starting point between 1 and k, then select every kth element from the list. Advantage: easy to execute. Risk: if the list has a hidden periodic pattern that aligns with k, the sample can be biased.
Stratified random sampling: divide the population into subgroups (strata) that share a key characteristic, then randomly sample within each stratum. Ensures every important subgroup is represented. Drawback: identifying and dividing into strata can require a great deal of effort when there are many variables of interest.
Cluster sampling: select an entire pre-existing group (cluster) that is believed to be representative of the population. Cost-effective and convenient. Drawback: if the cluster is too homogeneous, it may not represent the full population well.
No major formulas in this unit. The key relationships to internalise are conceptual:
Population → Parameter; Sample → Statistic
NOIR hierarchy: Nominal < Ordinal < Interval < Ratio (each level adds capability)
Sampling fraction k in systematic sampling: k = population size / desired sample size
Pharmaceutical companies use stratified random sampling to ensure clinical trials include sufficient numbers of participants from different age groups, sexes, and ethnic backgrounds. Political polling relies on random sampling to estimate voter preferences from a small fraction of the electorate. Quality-control engineers in manufacturing use systematic sampling to inspect every kth item on a production line, catching defects without halting the entire line.
Students often confuse ordinal and interval scales. Just because you can assign numbers (1st, 2nd, 3rd) does not mean the gaps between them are equal. Class rank is ordinal, not interval.
Students sometimes treat temperature as ratio data. Celsius and Fahrenheit have no true zero (0°C is not "no temperature"), so they are interval. Kelvin, which does have a true zero, is ratio.
A large sample does not fix a bad sampling method. A huge convenience sample is still biased; a small random sample can be more representative.
"Random" does not mean haphazard. Random sampling follows a precise procedure that gives every member of the population a known chance of selection.
⚠️ Expect questions that give you a variable and ask you to identify its type (quantitative vs. qualitative) and level of measurement (NOIR). Practise with examples you have not seen before.
⚠️ Know the advantages and disadvantages of each sampling technique. A common exam format: "A researcher does X. What sampling method is this? Name one advantage and one disadvantage."
⚠️ Be able to distinguish between a population and a sample, and between a parameter and a statistic. This distinction underpins every inference topic that follows.
⚠️ Data collection method questions often present a scenario and ask you to identify which method (telephone, mailed questionnaire, interview, observation, records) is being used and to name a limitation.
True or False: A sample is a subset of a population. (True)
True or False: Ordinal data has equal intervals between ranks. (False, the intervals are unknown)
Fill in the blank: The two main branches of statistics are ______ and ______. (Descriptive, Inferential)
True or False: In systematic sampling, the first element is chosen at random. (True)
Fill in the blank: A measure computed from an entire population is called a ______. (Parameter)
Q: A researcher records the eye colour of 200 participants. What type of variable is eye colour, and at what level of measurement?
A: Eye colour is a qualitative variable measured at the nominal level (categories with no natural order).
Q: A hospital administrator wants to study patient satisfaction and selects every 15th patient from the daily check-in list, starting from a randomly chosen number between 1 and 15. What sampling method is this?
A: Systematic random sampling.
Q: Explain why temperature in Celsius is interval rather than ratio.
A: Celsius has no true zero point. 0°C does not represent the absence of temperature; it is simply the freezing point of water. Because there is no true zero, you cannot form meaningful ratios (40°C is not "twice as hot" as 20°C in any absolute sense).
Q: A pollster divides voters into age groups (18 to 29, 30 to 44, 45 to 64, 65+) and randomly selects 100 voters from each group. Name the sampling method and one advantage.
A: Stratified random sampling. One advantage is that it ensures representation of all age subgroups in the sample.
Q: What is the difference between a statistic and a parameter?
A: A statistic is a numerical measure calculated from a sample. A parameter is the corresponding measure calculated from the entire population.
This connects to frequency distributions and graphs because the type of variable and level of measurement determine which graph (histogram, bar chart, pie chart) is appropriate. It also connects to measures of central tendency: you cannot compute a mean for nominal data, only a mode. Later, hypothesis testing and confidence intervals rely entirely on the population/sample distinction and on understanding what a parameter is versus a statistic.
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