In statistics, a population is the entire group you want to learn about, while a sample is the smaller group you actually measure. A parameter is a number that describes a population, such as the true mean height of all students in a school. A statistic is a number computed from a sample, such as the mean height of 50 selected students.
This distinction matters because most real populations are too large, costly, or impossible to measure completely.
We use sampling to collect data from part of the population and then use sample statistics to estimate population parameters. For example, the sample mean x̄ can estimate the population mean μ, and the sample proportion p̂ can estimate the population proportion p. Different samples usually give different statistics, so estimates have uncertainty.
Good sampling methods reduce bias and make statistics more reliable for drawing conclusions about the population.
Understanding Statistics: Parameters vs Statistics
A parameter is fixed for one clearly defined population, even when nobody knows its value. It does not change because a different group of people is measured. A statistic can change each time a new sample is chosen.
This is called sampling variability. Imagine taking several random groups of fifty students from the same school. Their average sleep time will not be identical.
Some groups will contain more students who sleep longer, while others will contain more students who sleep less. The changing sample averages form a pattern around the population average. Larger random samples usually produce statistics that stay closer to the parameter.
Random sampling is important because it protects against systematic bias. Bias occurs when the method of collecting data tends to favor some outcomes over others. A school survey sent only to members of a sports club cannot fairly estimate exercise habits for every student.
Students who do not reply can create bias too, especially if their views differ from those who reply. The list used to choose people matters as well. If the list leaves out a group, that group has no chance of being selected.
A large sample does not automatically fix a biased method. A smaller well chosen random sample can be more useful than a huge unrepresentative one.
Students meet these ideas in opinion polls, product testing, health studies, and school decisions. A poll may report that a certain share of voters supports a candidate. That reported share is a statistic from respondents, not a direct reading of every voter.
A factory may inspect a sample of light bulbs to estimate the rate of defects in a day’s production. Researchers may study a sample of patients to estimate how a treatment works in a wider group. In every case, the population must be stated carefully.
The target might be all voters in a country, all bulbs made during one shift, or patients with a particular condition. Changing the target population changes the parameter being estimated.
When reading a result, look beyond the single reported number. Check who was sampled, how they were selected, how many observations were included, and when the data were collected. A sample result should be reported with a measure of uncertainty, often called a margin of error or confidence interval.
This range reflects the fact that random samples differ. It does not remove problems caused by bias, inaccurate measurements, or misleading questions. Keep the roles separate while solving problems.
The parameter belongs to the full target group and is usually unknown. The statistic comes from observed sample data and is used as evidence about that unknown population value.
Key Facts
- Parameter: a numerical value that describes a population, such as μ, σ, or p.
- Statistic: a numerical value computed from a sample, such as x̄, s, or p̂.
- Population mean: μ = sum of all population values / N.
- Sample mean: x̄ = sum of sample values / n.
- Sample proportion: p̂ = number of successes in sample / n.
- Statistics estimate parameters, so x̄ estimates μ and p̂ estimates p.
Vocabulary
- Population
- The complete group of individuals, objects, or measurements that a study is about.
- Sample
- A smaller subset selected from a population to collect data from.
- Parameter
- A fixed numerical value that describes a characteristic of an entire population.
- Statistic
- A numerical value calculated from sample data.
- Sampling variability
- The natural change in a statistic from one random sample to another.
Common Mistakes to Avoid
- Calling x̄ a parameter is wrong because x̄ is calculated from a sample, so it is a statistic.
- Calling μ a statistic is wrong because μ describes the entire population, even if its value is unknown.
- Assuming one sample statistic equals the exact parameter is wrong because samples vary and estimates usually contain sampling error.
- Using a biased sample is wrong because a statistic from an unrepresentative sample may give a poor estimate of the population parameter.
Practice Questions
- 1 A school has 1,200 students, and a random sample of 60 students has an average height of 167 cm. Identify the population, sample, parameter being estimated, and statistic calculated.
- 2 In a sample of 200 voters, 118 support a proposal. Calculate the sample proportion p̂ and state which population parameter it estimates.
- 3 A researcher surveys only people leaving a gym to estimate the average weekly exercise time for an entire city. Explain why the sample may be biased and how that affects the statistic as an estimate of the parameter.