Positive and Negative Correlation - Algebra-1

1. Fundamental Concepts

Positive correlation and negative correlation are important concepts in scatter plots that describe the relationship between two variables:
- Positive Correlation: When the value of one variable increases, the value of the other variable tends to increase as well. The data points show a trend from the bottom-left to the top-right.
- Negative Correlation: When the value of one variable increases, the value of the other variable tends to decrease. The data points show a trend from the top-left to the bottom-right.
- The strength of correlation can be measured by the correlation coefficient r, where : The closer r is to 1, the stronger the positive correlation; the closer r is to -1, the stronger the negative correlation; if r is close to 0, the correlation is weak or there is no obvious association.

2. Key Concepts

Trend Lines: A straight line used to approximate the relationship between variables, with the equation (the slope m determines the direction of correlation: indicates positive correlation, indicates negative correlation).
Difference between Correlation and Causation: Correlation only indicates an association between variables, not that one variable directly causes changes in the other (e.g., there is a positive correlation between swimsuit sales and ice bag sales, but it is not a causal relationship).
Characteristics of Data Distribution: Both positive and negative correlations describe the "trend" of variables, not a strict one-to-one correspondence. Data points may be scattered around the trend line to some extent.

3. Examples

Easy Level

Question: The scatter plot below shows data for students in a class: weekly reading time (hours) and Chinese test scores. The data points are (2, 65), (3, 70), (5, 78), (7, 85), (9, 92). The overall trend of the points is from the bottom-left to the top-right, indicating a ______ correlation between the two variables.

Explanation: When one variable (reading time) increases, the other variable (test score) also increases, and the data points show a trend from the bottom-left to the top-right, which fits the definition of a positive correlation.
Answer: positive

Medium Level

Question: A city recorded data for monthly average temperature (°C) and heating usage (hours): (5, 180), (10, 120), (15, 80), (20, 40), (25, 10). The scatter plot shows that as temperature rises, heating usage decreases, with points roughly distributed from the top-left to the bottom-right. This indicates a ______ correlation.

Explanation: When one variable (temperature) increases, the other variable (heating usage) decreases, and the data points show a trend from the top-left to the bottom-right, which is characteristic of a negative correlation.
Answer: negative

Hard Level

Question: Researchers collected data on annual medical expenses (yuan) and age (years) for residents in a region: (20, 1,200), (30, 1,500), (40, 2,800), (50, 3,200), (60, 4,500), (45, 1,800) (an outlier). The scatter plot shows that as age increases, medical expenses generally rise. This overall trend is still a ______ correlation.

Explanation: Despite some outliers, the overall trend is that as one variable (age) increases, the other variable (medical expenses) also increases, which aligns with the core feature of a positive correlation.
Answer: positive

4. Problem-Solving Techniques

Draw a Scatter Plot: Plot the corresponding data of two variables as coordinate points and observe the overall trend.
Determine the Direction of Correlation:
- If the points gather from the lower left to the upper right, it is a positive correlation;
- If the points gather from the upper left to the lower right, it is a negative correlation.
Fit a Trend Line: Approximately represent the relationship between variables by the straight line . The sign of the slope m directly reflects the direction of correlation.
Distinguish between Correlation and Causation: Even if there is a strong correlation, it is necessary to verify whether there is a causal relationship through logic or experiments to avoid misjudgment (for example, "ice cream sales and drowning accidents" are positively correlated, but both are actually affected by temperature).