Discrete vs. Continuous - Algebra-1

1. Fundamental Concepts

Discrete Data: Data obtained through counting, which can only take specific values. There are clear gaps between its values, and it cannot take any value within these gaps.
Continuous Data: Data obtained through measurement, which can take any value within a certain interval. Its values change continuously and can theoretically be infinitely subdivided.

Characteristics of Discrete Data:
- Generated by counting, and the results are usually integers (or specific enumerable values), such as "the number of students in a class" or "the number of apples in a box".
- There are indivisible gaps between values. For example, there are no other possible counting results between "3 people" and "4 people".
- Can only be represented by specific points and cannot be infinitely subdivided.
Characteristics of Continuous Data:
- Generated by measurement, and the results can be decimals or fractions, such as "a person's height" or "the weight of an object".
- Can take any value within a certain range. For example, height can be 175cm, 175.5cm, 175.55cm, etc., and can be infinitely refined.
- Depends on the precision of the measuring tool. The higher the precision, the more accurate the data (e.g., measuring the same object with rulers of different precisions may yield different results).
Relationship Between the Two: Discrete data and continuous data are two subdivided types of quantitative data, both presented in numerical form. However, they differ in value-taking methods and properties, requiring different processing methods in statistical analysis.

1:
- Discrete Data: The number of children in a family (1, 2, 3, etc., cannot be 1.5).
- Continuous Data: The volume of a glass of water (500mL, 500.2mL, 500.25mL, etc., which can be infinitely subdivided).
2:
- Discrete Data: The number of mobile phones sold in a shopping mall in one day (0, 1, 2... There cannot be 2.3 phones).
- Continuous Data: The commuting time from home to school (15 minutes, 15.5 minutes, 15.53 minutes, etc., which can be accurate to seconds or even milliseconds).
3:
- Discrete Data: The number of traffic accidents occurring in a city within one year (100, 101, etc., the number must be an integer).
- Continuous Data: The annual precipitation in a region (800mm, 800.1mm, 800.12mm, etc., which can be measured to more detailed values with precision instruments).

Distinguishing Between Discrete and Continuous Data:
- Determine the source of the data: Data obtained through "counting" is discrete (e.g., "how many"), while data obtained through "measuring" is continuous (e.g., "how long, how heavy").
- Check if it can be subdivided: Discrete data cannot take intermediate values between two adjacent values (e.g., there is no 5.5 pens between "5 pens" and "6 pens"); continuous data can be infinitely subdivided (e.g., there are 2.5kg, 2.55kg, etc., between "2kg" and "3kg").
Data Processing and Representation:
- Discrete Data: Often represented by bar charts, with the horizontal axis as specific values and the vertical axis as frequency (e.g., "distribution of the number of students in different classes").
- Continuous Data: Often represented by histograms or line charts, with the horizontal axis as data intervals and the vertical axis as frequency (e.g., "height distribution of a group").
Notes on Application Scenarios:
- Discrete data is suitable for describing "countable independent individuals or events", such as the quantity of products, the number of votes, etc.
- Continuous data is suitable for describing "measurable physical quantities or processes", such as temperature, time, speed, etc. When analyzing, the impact of measurement precision on results should be considered.