Why Statistics?

Imagine you want to know how much time students at a university spend studying each week. There is a perfectly straightforward way to find out: ask every single student, verify every answer, and calculate the result.

There are two problems. First, this would be expensive. Second, the answers might not be entirely accurate. What counts as studying? Does watching a lecture count? What about having the textbook open while watching television?

Welcome to statistics. The mathematics is often the easy part.

We rarely see the whole picture

The population is the complete group about which we want to know something. If our question concerns all students currently enrolled at a particular university, they constitute the population.

A sample is the subset we actually observe. We might survey 200 students and estimate something about the university’s 20,000 students.

A number summarizing the population—such as its true average study time—is a parameter. A number calculated from our sample is a statistic. Parameters are generally unknown. Statistics are what we have in our hands.

A statistic can be useful without being an exact representation of the parameter. The question is how much uncertainty accompanies it.

Two different jobs

Descriptive statistics summarize observed information. A histogram of the 200 students’ reported study hours describes that sample. So do its mean and median.

Inferential statistics use observed data to make claims beyond the observations. If we use the 200 students to estimate the average for all 20,000, we are doing inference.

That second job is harder than it looks. Which students were asked? Did they volunteer? Did everyone respond? Could the questionnaire have nudged respondents toward particular answers?

A beautifully precise average from a hopelessly biased sample is still a hopelessly biased answer.

Variation is not an inconvenience

Suppose the five students we asked report 2, 4, 6, 8, and 10 hours. Their mean is 6 hours. That does not mean anyone must study exactly 6 hours, or that the next student will.

Statistics begins by accepting that observations vary. Rather than pretending variation does not exist, we describe it and ask whether our results are stable in the presence of it.

This becomes especially important when comparing groups. A difference of two points on a test could be meaningful, irrelevant, or entirely unsurprising given the variability in scores. The difference itself cannot tell us which.

What this course will do

We will start by learning how to describe data. Then we will study randomness and sampling. Eventually we will build toward confidence intervals, hypothesis tests, regression models, and the rather remarkable problem of distinguishing patterns from coincidence.

The goal is not to memorize a table of formulas. The goal is to know which question each formula answers, what assumptions it requires, and where it can lead us astray.

Check your understanding

1. A city surveys 400 residents about transit. Identify the population, the sample, a possible parameter, and a possible statistic.
2. A news report says, “The survey found that 61% support the proposal.” Does that number necessarily describe the population?
3. Give one way that a large sample could still produce a misleading estimate.

Suggested responses

1. Population: the city’s residents (if that is the target). Sample: the 400 surveyed. Parameter: the true citywide support rate. Statistic: the observed support rate among 400 respondents.

2. No. It describes the respondents; generalizing requires information about how they were selected and how uncertainty was assessed.

3. The sample might systematically exclude residents without internet access, or respondents might self-select.

A design problem, before any formulas

Here is a challenge you will return to in the Week 1 investigation. A university wants to know whether a statistics-preparation workshop causes students to perform better. Someone surveys the students who attended, compares their marks with those who did not, and claims that the difference is the benefit of attending.

What did the researcher actually learn?

For the people measured, the researcher may have described an association between attendance and scores. But students who attend voluntarily may differ from those who decline in prior knowledge, time, motivation, or access. We do not know what their scores would have been without attending. And even a carefully randomized experiment involving willing volunteers would not necessarily represent all university students.

These are three separate limitations:

• Description: do we correctly summarize the observations we collected?
• Causal inference: is the observed difference a credible estimate of the effect of an intervention?
• Generalization: which population could this result describe?

The point of the first week is not to learn a formula for each problem. It is to stop one kind of claim from quietly turning into another. Go to the interactive study-design workshop and predict what happens when participation and treatment selection change.

Specify the claim before collecting its denominator

A research question should identify the target population, measurement, time window, and quantity to estimate. “Do students study enough?” is underspecified. “What fraction of currently enrolled undergraduates report at least ten hours of independent study during the previous seven days?” names a measurable proportion, although the quality of self-report still needs investigation.

Suppose a university has 20,000 undergraduates and selects 200 by simple random sampling. Of 160 respondents, 96 report at least ten hours. The observed respondent proportion is $96/160=0.60$. It is not $96/200=0.48$ unless nonresponse has explicitly been coded as nonsupport for a justified target, which would answer a different question. The response rate is $160/200=0.80$. Distinguish the denominator selected, the denominator responding, and the population to which an estimate is intended to apply.

An inferential report connects question, design, evidence, estimate, uncertainty, and limitations. Each link can fail independently. More elaborate computation cannot repair an undefined outcome, a missing population, or a self-selected comparison silently described as random assignment.

Further investigation

1. Write a fully specified question about travel time to campus, including its population and time window.
2. In the example above, how many selected students did not respond, and what is the response rate?
3. Give one limitation that would remain even if every selected student responded.

Worked further responses

1. One defensible example is the mean door-to-door travel time of undergraduates attending in-person classes during a specified teaching week. Define how multiple journeys are combined and whether nontravellers are in the target.
2. Forty did not respond; the response rate is 80%.
3. Misremembered study hours, ambiguous wording, frame omissions, or the original sampling design could still limit interpretation. Complete response addresses one mechanism, not every error source.