Statistical Concepts: Sampling, Variables, and Data Visualization

Fundamentals of Statistical Analysis

Population and Sampling

All statistical study refers to a set of elements called the population. When the population is very large, and the goal is to focus on only a part of it, a sample is taken. This sample can represent the entire population or be a partial survey.

Analyzing Representative Samples

A representative sample can be analyzed using two main methods based on strata (subgroups) of the population:

• Equal Allocation: The sample is taken such that the number of elements selected from each stratum of the population is the same.
• Proportional Allocation: The representative sample is taken in proportion to the number of elements in each stratum.

Characteristics and Variables

The characteristics (or qualities) possessed by the elements of the population are called variables. These variables can be:

• Qualitative: These characteristics have different forms or situations that can occur. They cannot be measured numerically and are described by words (e.g., color, gender).
• Quantitative: These characteristics are described by measurable numbers. The assigned numerical value is called the variable, which can be:
  • Discrete: If the variable takes only certain specific values (e.g., number of children).
  • Continuous: If the variable takes any value between two consecutive points (e.g., height, weight).

Classification of Statistics

Statistics are generally classified into two types:

• Descriptive Statistics: Involves the collection, organization, and reduction of data.
• Inductive (Inferential) Statistics: Involves gaining knowledge about certain data sets from observations made on a sample, allowing for generalizations about the population.

Graphical Representation of Data

Common Statistical Charts

Bar Chart

Uses Cartesian axes where the horizontal axis represents the variable. Bars are raised to a height equal to the absolute frequency. If the upper ends of each bar are joined, a polygonal line called the frequency polygon is obtained. Note: When representing a discrete quantitative variable, a standard bar is used, but when representing a qualitative attribute, a wider bar may be used for visual distinction.

Pictogram

A representation using drawings or icons related to the theme, where the size of the drawing is proportional to the absolute frequency for each attribute.

Histogram

A drawing of adjacent rectangles where the area of each rectangle is equal or proportional to the frequency of its corresponding interval. Histograms are typically used for continuous quantitative variables.

Pie Chart (Circular Chart)

A circle is divided into circular sectors. The angle of each sector is determined using a rule of three, matching the 360° central angle of the circle to the total number of elements. The angle for each data point is proportional to its absolute frequency.

Data Collection Tables (Frequency Distribution)

For collecting and organizing data, frequency distribution tables are used, typically structured with the following columns:

1. Variable (X): The specific value or category being measured.
2. Absolute Frequency (f): The number of times a variable is repeated. The sum of all absolute frequencies equals the total number of elements (N).
3. Relative Frequency (fr): The ratio between the absolute frequency and the total number of variables (f/N).
4. Percentage (%): The relative frequency multiplied by 100 (fr * 100).
5. Cumulative Absolute Frequency (F): The running total of the absolute frequencies.
6. Cumulative Relative Frequency (Fr): The running total of the relative frequencies.
7. Sector Angle (for Pie Charts): The angle required for graphical representation in a pie chart (Relative Frequency * 360°).