Parameters and Variables | Introduction to Statistics

Opening

A single statistic from a sample of 1,000–2,000 people is used to represent the views of an entire country in presidential elections. How can such a small sample say anything about millions of people?

The idea

A parameter describes a whole population, a statistic describes a sample, and variables can be numerical or categorical.

From sample to population

Because it takes time and money to examine an entire population, sampling is practical. From sample data we calculate a statistic, which estimates a population parameter. A key concern is how accurately a statistic estimates a parameter, which depends on how well the sample represents the population.

Vocabulary· 2 terms

Easily confused

Looks like

A statistic and a parameter sound like the same number, just computed on different groups.

Actually

A statistic is computed from the sample and estimates the parameter, which describes the entire population.

The distinction matters because the statistic is what we can measure, but the parameter is what we want to know.

Vocabulary· 2 terms

Worked example

A study wants to know the average amount first-year students spend on school supplies (excluding books) at ABC College. A random sample of 100 first-year students is surveyed; three spent $150, $200, and $225.

1
Identify the population: all first-year students attending ABC College this term.
The population is the entire group the study wants to describe.

5 more steps

Vocabulary· 2 terms

Another example: crash test dummies

In the crash test study, cars with dummies in the front seats were crashed into a wall at 35 miles per hour. The population is all cars containing dummies in the front seat; the sample is the 75 cars selected by simple random sample. The parameter is the proportion of driver dummies who would have suffered head injuries in the population; the statistic is that proportion in the sample. The variable X is whether a dummy would have suffered a head injury, and the data are yes or no.

Vocabulary· 2 terms

Numerical vs. categorical variables

Variables may be numerical or categorical. Numerical variables take values with equal units, like weight in pounds or time in hours. Categorical variables place the person or thing into a category, like party affiliation. You can do math with numerical values (like averaging points), but it makes no sense to do math with categorical values (averaging party affiliations is meaningless).

Now try classifying a variable.

Mean and proportion

Two words that come up often are mean and proportion. If you take three exams and score 86, 75, and 92, your mean score is (86+75+92)/3 = 84.3 to one decimal place. If a class has 40 students, 22 men and 18 women, the proportion of men is 22/40 and the proportion of women is 18/40. Mean and proportion are discussed in more detail in later chapters.

Computing a mean

Recap

Say each point to yourself, then tap it to check.