Statistic, Statistical Hypothesis Test(ing), Test Statistic, t-test and p-value

Yao Yao on September 20, 2014

1. Statistic

1.1 Definition

A statistic, is a single measure of some attribute of a sample (e.g. sample mean). It is calculated by applying a function to the values of the sample.

More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample’s distribution; that is, the function can be stated before realization of the data. The term statistic is used both for the function and for the value of the function on a given sample.

A statistic is distinct from a statistical parameter, which is not computable because often the population is much too large to examine and measure all its items.

• A statistic is an observable random variable, computed on a sample.
• A parameter is a generally unobservable quantity describing a property of a statistical population, which can only be computed exactly if the entire population can be observed without error.

However, a statistic, when used to estimate a population parameter, is called an estimator. For instance, the sample mean is a statistic that estimates the population mean, which is a parameter.

1.2 Types

When a statistic (a function) is being used for a specific purpose, it may be referred to by a name indicating its purpose:

• in descriptive statistics, a descriptive statistic is used to describe the data;
• in estimation theory, an estimator is used to estimate a parameter of the distribution (population);
• in statistical hypothesis testing, a test statistic is used to test a hypothesis, e.g.
• t statistics
• chi-squared statistics
• f statistics

1.3 Statistical Properties

Important potential properties of statistics include

• completeness
• consistency
• sufficiency
• unbiasedness
• minimum mean square error
• low variance
• robustness
• computational convenience

2. Statistical Hypothesis Test(ing)

A statistical hypothesis test is a method of statistical inference. In statistics, a result is called statistically significant if it has been predicted as unlikely to have occurred by chance alone, according to a pre-determined threshold probability, the significance level.

Statistical hypothesis testing is sometimes called confirmatory data analysis, in contrast to EDA, which may not have pre-specified hypotheses.

1. 提出 $H_0$, $H_a$
2. 建立 test statistic
3. 计算是否应该 reject hypothesis

3. Test Statistic

A test statistic is a statistic used in statistical hypothesis testing.

4. t-test

A t-test is a statistical hypothesis test in which the test statistic follows a Student’s t distribution if the null hypothesis is supported.