Descriptive statistics: definitions and types

• Descriptive statistics involves methods for summarizing and describing the main features of a dataset.

Definitions

• A population is the entire group of individuals or items of interest in a study.
• A sample is a subset of the population that is selected for analysis.
• A variable is a characteristic that is measured or observed for each individual in the population or sample.
• A parameter is a numerical value that describes a characteristic of the population.
• A statistic is a numerical value that describes a characteristic of the sample.

Types of Variables

• A discrete quantitative variable is obtained by counting and can only take a limited number of values.
• Examples of discrete quantitative variables include the number of children or cars.
• A continuous quantitative variable is obtained by measuring and can take any value within a given range.
• Examples of continuous quantitative variables include height, weight, and temperature.
• A nominal qualitative variable categorizes data into mutually exclusive, unordered categories.
• Examples of nominal qualitative variables include eye color, sex, and nationality.
• An ordinal qualitative variable categorizes data into ordered categories.
• Examples of ordinal qualitative variables include satisfaction level and education level.

Measures of Central Tendency

• The arithmetic mean is the sum of all values divided by the number of values, denoted as $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$.
• The median is the value that separates the dataset into two equal halves.
• The mode is the value that appears most frequently in the dataset.

Measures of Dispersion

• The range is the difference between the maximum and minimum values in a dataset.
• Variance measures the average of the squared differences from the mean, calculated as $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$.
• Standard deviation is the square root of the variance, denoted as $s = \sqrt{s^2}$.
• The coefficient of variation is the standard deviation divided by the mean, expressed as $CV = \frac{s}{\bar{x}}$.

Measures of Position

• Quantiles divide the data into equal parts.
• Quartiles divide the data into four equal parts.
• Deciles divide the data into ten equal parts.
• Percentiles divide the data into one hundred equal parts.

Plackett-Burman Screening Design

• Plackett-Burman designs are used for identifying significant factors affecting a response variable.

Objective

• Designed to quickly identify the factors that have significant effects on the outcome.

Application Scope

• Used in the early stages of a study to assess many independent factors with minimal trials.
• Assumes that the factors are independent.

Assumptions

• Assumes no interactions between factors.
• Assumes the response is approximately linear within the studied range.

Experimental Design

• The number of trials is a multiple of 4, such as 8, 12, 16, etc.
• The number of factors studied is the number of trials minus 1, corresponding to 7, 11, 15, etc.
• The number of center point repetitions ranges from 1 to 3 to evaluate curvature.
• Variables are coded in with the following formula:
• $\qquad X_i = \frac{x_i - \frac{x_{i,max} + x_{i,min}}{2}}{\frac{x_{i,max} - x_{i,min}}{2}}$

Experimental Matrix

• Example matrix for 7 factors with 8 trials, showing '+' and '-' levels for each factor:

Results Analysis

• Main Effects:
  • $E_i = \frac{\sum Y_{+i} - \sum Y_{-i}}{n/2}$
• $E_i$ represents the estimated effect of factor i.
• $\sum Y_{+i}$ is the sum of experimental results when factor i is at its high level (+).
• $\sum Y_{-i}$ is the sum of experimental results when factor i is at its low level (-).
• $n$ is the number of trials.
• Student's t-test:
  • $t_i = \frac{E_i}{S_{Ei}}$
• $t_i$ is the t-value for factor i.
• $E_i$ is the estimated effect of factor i.
• $S_{Ei}$ is the standard error of the estimated effect of factor i.
• Confidence Interval:
  • $IC_i = E_i \pm t_{\alpha/2, v} \cdot S_{Ei}$
    • $IC_i$ is the confidence interval for the effect of factor i.
    • $E_i$ is the estimated effect of factor i.
    • $t_{\alpha/2, v}$ is the t-value for a significance level $\alpha$ and $v$ degrees of freedom.
    • $S_{Ei}$ is the standard error of the estimated effect of factor i.
• Analysis of Variance (ANOVA) determines if the studied factors have a significant effect on the response.
• Pareto Chart visualizes the relative importance of the factor effects.
• JMP, Minitab, and Design-Expert are software commonly used for this type of analysis.