Statistic: Descriptive Measure Calculated from Data Sampled from a Population

August 25, 2024 3 min read Mathematics Statistics Data Population Sample Descriptive Measure Statistic

Definition and detailed explanation of a statistic, including types, examples, and related terms.

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A statistic is a numerical value that describes a characteristic of a sample, which is a subset of the population. It is used to estimate, infer, or make decisions about the characteristics of the population from which the sample was drawn.

Types of Statistics

Descriptive Statistics

Descriptive statistics summarize or describe the characteristics of a data set. Common examples include the mean, median, mode, variance, and standard deviation.

Inferential Statistics

Inferential statistics use sample data to make generalizations about a larger population. Examples include hypothesis testing, confidence intervals, and regression analysis.

Key Formulas in Statistics

Mean

The arithmetic mean of a dataset is given by:

\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

Variance

The variance of a dataset is:

\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{N}

where $\mu$ is the population mean.

Standard Deviation

The standard deviation is the square root of the variance:

\sigma = \sqrt{\sigma^2}

Special Considerations

When calculating statistics, it’s vital to consider the quality and size of the sample as these factors can significantly affect the accuracy and reliability of the statistics derived.

Examples

Mean Salary: Suppose you survey the annual salaries of 50 employees in a company and calculate the average (mean) salary to be $50,000. This mean salary is a statistic.
Test Scores: The variance and standard deviation of test scores from a sample of 30 students can provide insights into the performance and consistency of scores within the sample.

Historical Context

Statistics as a field has roots tracing back to the 18th century, with significant contributions from mathematicians like Carl Friedrich Gauss. Over time, particularly in the 20th century, the field expanded to include methods for handling larger and more complex datasets—a response to advancements in computing and data collection technologies.

Applications

Statistics are ubiquitous in various fields:

Economics: For understanding economic indicators like GDP, inflation, employment rates.
Medicine: In clinical trials to evaluate the effectiveness of new treatments.
Engineering: For quality control and reliability testing.

Comparisons

Statistic vs. Parameter

Statistic: A measure derived from a sample.
Parameter: A measure that describes an entire population.

Statistic vs. Test Statistic

Statistic: General measure from a sample.
Test Statistic: A specific statistic calculated during hypothesis testing to make inferences about the population.

Population: The complete set of items that data can be drawn from.
Sample: A subset of the population used to estimate characteristics of the whole population.
Parameter: A descriptive measure calculated from the population data.

FAQs

What is the difference between a statistic and a parameter?

A statistic is calculated from a sample, while a parameter is calculated from the entire population.

Why are statistics important?

Statistics allow us to make inferences about populations, test hypotheses, and make informed decisions based on data.

How are statistics used in research?

Statistics are used to describe data, test hypotheses, and determine the reliability and validity of research findings.

References

Rice, J. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W.W. Norton & Company.
Moore, D. S., McCabe, G. P., & Craig, B. (2013). Introduction to the Practice of Statistics. W.H. Freeman.

Summary

A statistic is a valuable tool for summarizing and understanding data drawn from a sample. By employing descriptive and inferential statistics, researchers and analysts can make informed decisions, draw conclusions, and predict future trends effectively. As data continues to grow in volume and complexity, the role of statistics in various industries remains crucial.