Comparison tests, p-value, z-score

Comparison tests are statistical methods used to determine if there are significant differences between groups, they are crucial tools in inferential statistics, allowing researchers to draw conclusions about the populations from which their samples are drawn, and to make informed decisions based on statistical evidence.

Student's t-test

The Student's t-test is used to compare the means of two groups and is most effective when the data is normally distributed and the variances are equal.

Non-parametric tests

Non-parametric tests, such as the Mann-Whitney U test or Kruskal-Wallis test, are alternatives to t-tests that do not assume a normal distribution and are used when data does not meet parametric test assumptions.

ANOVA

ANOVA (Analysis of Variance) is used to compare the means of three or more groups. It assesses the overall variance to determine if there is at least one significant difference between group means.

P-values

P-values are a measure of the strength of evidence against the null hypothesis in statistical tests. A p-value indicates the probability of obtaining test results at least as extreme as the results observed, assuming that the null hypothesis is true. In hypothesis testing, a low p-value (typically ≤ 0.05) suggests that the null hypothesis can be rejected, indicating that there is a statistically significant difference between groups. Conversely, a high p-value suggests that there is insufficient evidence to reject the null hypothesis, implying that any observed differences may be due to random chance.

Z-score

A z-score, also known as a standard score, is a statistical measurement that describes a data point's relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean.

The z-score of a data point xxx is calculated using the formula:

$z = \frac{x - \mu}{\sigma}$

where $x$ is the value of the data point, $\mu$ is the mean of the dataset, and $\sigma$ is the standard deviation of the dataset.

A positive z-score indicates that the data point is above the mean, while a negative z-score indicates that the data point is below the mean. The magnitude of the z-score reflects the number of standard deviations the data point is from the mean. A larger absolute value indicates the data point is further from the mean.

Z-scores are used for standardization, making different datasets comparable by converting data into a common scale. They also help identify outliers, as data points with z-scores beyond a certain threshold (commonly ±2 or ±3) are considered unusual. Additionally, in a normal distribution, z-scores correspond to probabilities, helping determine the likelihood of a data point occurring within a certain range.

For example, suppose we have a dataset with a mean (μ) of 50 and a standard deviation (σ) of 10. For a data point x = 70, the z-score is calculated as:

$z = \frac{70 - 50}{10} = \frac{20}{10} = 2$

This z-score of 2 means that the data point is 2 standard deviations above the mean.