Statistics is the practice of learning from data while accounting for variation. It helps us decide what to measure, how to collect observations, which patterns matter and how far a conclusion can reasonably extend. Calculating an average is one part of that work, not the whole subject.
Begin by defining the question and deciding what evidence could answer it. Statistics Canada’s introductory resource connects statistical information with data quality, collection, exploration and communication. This article follows that sequence through a small, deliberately synthetic example.
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Start with a question that allows variation
“How long did one learner’s journey take today?” asks for a single observation. “How do learners’ journey times vary across this school?” asks a statistical question. Different learners can have different times, and the same learner’s time may change from day to day.
Specify the population and the measurement before collecting numbers. A workable question might be: “What were the one-way journey times, in minutes, for learners attending this school on a specified Tuesday?” That is different from usual weekly travel time, distance from home or time spent waiting for a bus. These are related variables, but they do not answer the same question.
The observational unit is the person, object or event described by one record. The variable is the characteristic recorded. If a learner supplies a time for five days, there are five measurements but only one learner. Repeated records should not silently become five independent people.
Follow the statistical investigation
A statistical investigation links a question to a collection plan, then to analysis and an interpretation. An unexpected pattern may send you back to the records or the original question. This is not a requirement to keep collecting until a preferred answer appears. It is a reason to document what was planned and what changed.
A statistical investigation connects five decisions
Diagram explanation and exact data
The question specifies the population and variable. The design determines who can enter the sample. Collection produces records; analysis describes variation; conclusions remain limited by the design and evidence.
| Stage | Concrete decision |
|---|---|
| Question | Who, what, when? |
| Design | Who can be selected? |
| Collect | Record consistently |
| Analyse | Summarise variation |
| Conclude | State the limits |
A school might sample learners from a complete enrolment list rather than asking only people standing beside the bicycle racks. It would use one question and one definition of journey time, record missing answers, and check unusually large or small values against their original records. These decisions can matter more than an extra decimal place in the final average.
Describe the observations you actually have
Consider these ten synthetic journey times, in minutes:
Their total is 180 minutes. The mean divides this total equally among the ten records:
The two middle observations are 14 and 16, so the median is 15 minutes. The range is minutes. Each summary describes something different: the mean balances all recorded values, the median locates the middle, and the range compares the extremes.
Saying only “the average journey was 18 minutes” conceals the unusual 48-minute observation and does not say which average was used. A stronger descriptive statement is: “In this illustrative set of ten observations, the mean was 18 minutes and the median was 15 minutes; recorded times ranged from 8 to 48 minutes.” A plot would show how those observations are distributed.
These calculations describe the supplied records. They do not establish the usual experience of a real school, because the numbers were created for teaching.
Separate description from inference
Descriptive statistics summarise observed data. Statistical inference uses data and a sampling or probability model to estimate or assess a feature beyond those observations. The population quantity is a parameter; a quantity calculated from a sample is a statistic.
If the ten times were a defensible random sample from a larger school, their mean could estimate the school’s population mean for the defined day. Another random sample would usually produce a different mean. That sampling variation is why an estimate needs an account of uncertainty.
A sample mean can be calculated from a convenient group too. What becomes questionable is its interpretation as an estimate for everyone. A narrow numerical interval cannot repair a collection method that systematically excludes particular learners. Populations and Samples and Sampling Bias develop that distinction.
Observation is not the same as an experiment
An observational study records what happens without assigning the condition under investigation. If learners who attend a homework club have different completion rates from learners who do not, that association may also reflect prior habits, workload or other differences.
An experiment deliberately assigns a condition. Random assignment can make comparison groups similar on average before treatment, supporting a causal analysis when the design, implementation and follow-up are sound. It does not automatically make the participants representative of a wider population.
Random sampling concerns who enters the study. Random assignment concerns which condition participants receive. A study can have one, both or neither. State these separately rather than using the word “random” as a general assurance of quality.
Choose summaries that suit the question
For a category such as travel mode, counts and proportions are useful. Adding category codes and reporting their mean would usually have no meaningful interpretation. For numerical journey times, a histogram, median and interquartile range can reveal features that a category chart cannot.
For two numerical variables measured on the same observational units, a scatter plot can show a relationship. A correlation or fitted line may then help describe it, but the original paired observations remain important. Later pages explain why a strong correlation does not establish causation and why predictions outside the observed range are risky.
No single statistic is a complete description. A useful report connects centre, spread, shape, sample size and the data collection method to the question being answered.
Communicate what the evidence supports
A statistical conclusion should identify the population or records, the variable and unit, the time period, the relevant summary and any limitation that changes the interpretation. Use the actual denominator for percentages, distinguish a missing response from zero, and avoid implying precision that the measurement does not support.
State the collection and measurement conditions that affect the claim. “Eighteen of 30 respondents chose cycling” is clear. “Sixty per cent of all learners prefer cycling” adds a population claim that needs evidence about how those respondents were selected.
Practice questions
- For the ten synthetic journey times above, calculate the mean and median.
- Is “the mean of these ten records is 18 minutes” a descriptive or inferential statement?
- A researcher randomly assigns volunteer learners to two study routines. Does this alone make the volunteers representative of all learners?
Check your answers
- The total is 180 minutes, so the mean is minutes. The middle values are 14 and 16, so the median is minutes.
- It is descriptive. The statement summarises the observed records without extending the result to a larger population.
- No. Random assignment addresses the comparison between conditions. Representativeness depends on how participants entered the study and which population the researcher intends to describe.
Related topics
Start with Types of Data to identify suitable measurements, then use Populations and Samples and Sampling Methods to plan collection. Mean, Median and Mode begins the numerical-summary sequence.