which of the following statements best describes the relationship between a parameter and a statistic?
Which of the following statements best describes the relationship between a parameter and a statistic?
Answer: A parameter and a statistic are two different concepts in the field of statistics. To understand their relationship, let’s break down each term and then look at how they interact:
1. Parameter
- Definition: A parameter is a numerical value that summarizes a characteristic of a population. The population includes all members of a specified group, making the parameter a true fixed value.
- Example: If you’re interested in the average height of adults in a country, the mean height calculated from the entire population is the parameter.
- Role: Parameters are often unknown because it is usually impractical, or impossible, to collect data from every member of a population.
2. Statistic
- Definition: A statistic is a numerical value that represents a characteristic of a sample — a subset of the population.
- Example: Using the same example, if you measure the average height of 1,000 randomly selected adults in the country, that average is a statistic.
- Role: Statistics are used to estimate parameters. They provide an approximate value from a sample that represents the larger population.
3. Relationship Between Parameter and Statistic
- Estimation: A statistic is calculated to estimate the parameter. For instance, the sample mean (a statistic) serves as an estimate for the population mean (a parameter).
- Approximation: While a parameter is a fixed value, a statistic can vary depending on the sample. Different samples may yield different statistics.
- Inference: Statistical methods, including confidence intervals and hypothesis testing, use statistics to make inferences about parameters. For example, you might calculate a confidence interval to estimate the range in which the true population mean (parameter) is likely to lie based on a sample mean (statistic).
4. Examples of Parameters and Statistics
| Parameter | Statistic |
|---|---|
| Population mean (\mu) | Sample mean (\bar{x}) |
| Population standard deviation (\sigma) | Sample standard deviation (s) |
| Population proportion (p) | Sample proportion (\hat{p}) |
5. Importance in Research and Surveys
- Accuracy: Understanding the difference helps in designing surveys and experiments, ensuring data collection methods accurately reflect the population.
- Bias and Variability: While parameters are devoid of bias or variability as they reflect an entire population, statistics may have bias or variability due to sampling errors.
6. Practical Implications
Practically, this relationship emphasizes the necessity of good sampling methodologies. Random sampling, for example, seeks to minimize bias, ensuring that the statistic (e.g., the sample mean) is a good estimator for the parameter (e.g., the population mean).
Reiterating the core idea, a parameter is what statisticians ultimately wish to know about a population, while a statistic based on a sample helps them to make informed guesses about the parameter.
In summary, the relationship between a parameter and a statistic is one of estimator and estimated. Parameters are fixed characteristics of a population, while statistics are variable measurements derived from samples used to infer those parameters.