Statistics › Section A · Sampling and data › Topic A.1
Statistics, probability, and key terms
Before any formula, statistics is a way of thinking: a language for describing a whole group by studying part of it, and a way of saying how confident you can be in what you found. This topic sets up the vocabulary every later topic uses.
Why statistics?
News feeds, weather apps, sports commentary and election polls all present numbers, and the numbers never quite repeat themselves: no two days have identical temperatures, and no two polls report exactly the same figure. Statistics is the science of collecting, analysing, interpreting and presenting data, and its purpose is to learn from data in the presence of this variability. Averages, charts and margins of error are its tools. Because almost every field - medicine, business, psychology, engineering, law - produces data, statistical thinking has become a basic life skill: it helps you judge whether a claim, a study or an advertisement deserves your trust.
Descriptive and inferential statistics
Statistical work comes in two stages. Organising and summarising the data you actually have is called descriptive statistics: making a table, drawing a graph, or computing an average. Drawing formal conclusions that reach beyond the data, from a sample to the larger group it came from, is called inferential statistics. Inference is where probability enters: statistical inference uses probability to say how confident we can be that a conclusion is correct. This course builds both halves in order - first the descriptive tools for summarising data, then the probability that powers inference.
Statistical models: deterministic and probabilistic
Like the rest of mathematics, statistics describes the world with models. A deterministic model predicts one exact outcome: given a circle's radius, its area follows exactly; given a fixed price and quantity, the cost is fixed. A probabilistic model instead assigns chances to outcomes: the number of customers arriving at a café in the next ten minutes could take several values, each with its own probability. Much of statistics is building and checking probabilistic models, because real data - measurements, counts, survey answers - vary even when conditions look identical. A model that captures how the variation behaves lets you predict patterns without promising any single outcome.
Probability: the pattern inside chance
Probability measures how likely an outcome is, on a scale from 0 (impossible) to 1 (certain). Individual chance events are unpredictable, but repetition reveals a stable pattern: the relative frequency of an outcome - its count divided by the number of trials - settles toward a fixed value as the number of trials grows. Imagine tossing a fair coin 10,000 times and getting 5,040 heads: the relative frequency is 5,040 ÷ 10,000 = 0.504, close to the probability 0.5. Tossing only 100 times, 60 heads would give 0.60, further away. Probability describes long-run behaviour, not the result of any single toss.
Population and sample
A population is the entire group of people, things or objects a study is about: all voters in a country, all bottles produced by a factory line, all students at a school. A sample is the part of the population that is actually selected and measured. Examining a whole population is usually too expensive or slow, so sampling exists to gain information about the population at a fraction of the cost: an opinion poll of roughly a thousand voters is meant to stand in for millions. Data are the values the sampling produces, and they are the raw material everything else builds on.
Parameter, statistic, variable, and data
Four more terms complete the core vocabulary. A variable is the characteristic being measured, one value per individual: we write X = the sleep time of one student. Data are the actual values of the variable that the sample yields (a single value is called a datum, so "the data are" is plural). A parameter is a number that describes a population, such as its mean; a statistic is the corresponding number computed from a sample. Notation keeps them apart: the population mean \(\mu\) and population proportion \(p\) are parameters, while the sample mean \(\bar{x}\) and sample proportion \(\hat{p}\) are statistics. The statistic is what estimates the parameter.
Worked example: a study of student sleep
A school wants to know the mean number of hours its 1,250 students sleep. It randomly surveys six students, who report 7, 6, 8, 7.5, 5 and 9 hours. Identify the key terms. 1. Population: all 1,250 students. 2. Sample: the six surveyed students. 3. Variable: X = hours one student sleeps. 4. Data: 7, 6, 8, 7.5, 5, 9. 5. Parameter: the mean sleep time of all 1,250 students, unknown without a census. 6. Statistic: the sample mean. Add: \(7+6+8+7.5+5+9 = 42.5\); divide by 6: \(42.5 \div 6 \approx 7.08\). Check: the six values total 42.5, and 42.5 ÷ 6 = 7.083..., so 7.08 hours is right to two decimals.
Worked example: a proportion study
A streaming service wants the proportion of its about 1.2 million subscribers who finish a new series within a month of release. It selects 500 subscribers at random and finds that 215 finished it. Population: all subscribers. Sample: the 500 selected. Variable: X = whether one subscriber finished the series, yes or no - a categorical variable. Data: the 500 yes-or-no answers. Parameter: the proportion for all subscribers. Statistic: the sample proportion \(\hat{p} = \frac{215}{500} = 0.43\). Check: 215 ÷ 500 = 0.43 exactly, and 43% of 500 is 215, so the statistic is consistent.
Kinds of variables
Variables come in kinds, and the kind decides which summaries make sense later. A categorical (qualitative) variable sorts individuals into groups: blood type, favourite app, whether a subscriber finished a series. A quantitative variable records numbers on which arithmetic makes sense: hours of sleep, minutes to fill an order. Quantitative variables split further into discrete (countable values, like the number of siblings) and continuous (measurements, like time or weight). Digits alone do not make a variable quantitative: a student ID number is categorical, because adding two ID numbers is meaningless. Most of this course handles one kind at a time, so classify the variable first.
Common mistakes to avoid
First, swapping parameter and statistic: the parameter belongs to the population, the statistic to the sample, so "the sample's parameter" is a contradiction in terms. Second, calling one value "data": data are plural, and a single value is a datum. Third, assuming digits mean quantitative: ID numbers, zip codes and phone numbers are categorical even though they are written with digits. Fourth, expecting probability to predict a single event: a coin's 0.5 says nothing about your next toss, only about long-run proportions. Fifth, confusing the variable with the data: hours of sleep is the variable; the value 7 is one datum.
Key takeaways
Statistics collects, analyses, interprets and presents data to learn from variation. Descriptive statistics organise and summarise what you have; inferential statistics use probability to draw conclusions about a population from a sample. Models are deterministic (one exact outcome) or probabilistic (outcomes with chances). Probability, on a 0-to-1 scale, describes long-run relative frequency, not single events. Every study has a population, a sample drawn from it, a variable X, and data; a parameter (a population number such as \(\mu\) or \(p\)) is estimated by a statistic (the matching sample number such as \(\bar{x}\) or \(\hat{p}\)). Variables are categorical or quantitative, and quantitative ones are discrete or continuous.
Flashcards
Flashcard
The science of collecting, analysing, interpreting and presenting data
Test your understanding
Question 1 of 5
A researcher charts the daily high temperatures of one city for a year and computes their average. Summarising data you already have like this is called:
Question 2 of 5
An airline wants to know what proportion of all its passengers would choose a window seat. It surveys 250 passengers picked at random. What is the sample?
Question 3 of 5
The mean number of hours slept by all students at one school is an example of a:
Question 4 of 5
Four randomly selected orders took 20, 25, 30 and 25 minutes to fill. What is the sample mean?
Question 5 of 5
Which of these is a quantitative variable?