Simple Linear Regression

Revision Notes – Simple Linear Regression

Simple linear regression is a quantitative method used to analyze how a change in one variable (the independent variable) affects another (the dependent variable).
It provides a mathematical model in the form of an equation:

Y = a + bX

Where:

• Y = dependent variable (e.g., sales revenue)

• X = independent variable (e.g., advertising expenditure)

• a = intercept (value of Y when X = 0)

• b = slope (rate at which Y changes with X)

In business, this model helps estimate and forecast future outcomes—such as predicting sales based on advertising costs or production output based on labor hours.

Regression analysis supports decision-making by quantifying relationships between variables, helping managers justify investments, and evaluating performance efficiency.

Scatter Diagrams

A scatter diagram visually displays the relationship between two variables. Each point represents a data pair, plotted along two axes—the independent variable on the X-axis and the dependent variable on the Y-axis.

Scatter diagrams are used to:

• Identify relationships between factors such as sales and promotion expenditure.

• Detect patterns, trends, and outliers in datasets.

• Decide whether applying a regression model is appropriate.

Types of Correlation in Scatter Diagrams:

• Strong Positive Correlation: Both variables increase together in a consistent pattern (e.g., smartphone sales and phone case demand).

• Weak Positive Correlation: Both variables increase, but not consistently (e.g., meal size and dessert consumption).

• Strong Negative Correlation: One variable increases while the other decreases closely (e.g., Coca-Cola price and Pepsi demand).

• Weak Negative Correlation: The inverse relationship exists but is not strong (e.g., fruit consumption and illness frequency).

• No Correlation: No identifiable relationship between the variables (e.g., employee parental leave and commuting method).

A clear scatter pattern often indicates a suitable case for further regression analysis.

Line of Best Fit

The line of best fit represents the general trend in a scatter diagram. It passes through the middle of the plotted points, minimizing the distance between itself and all the points.

• It can be drawn manually by estimating the center of the data points or calculated statistically using the least squares method.

• The slope of this line indicates the rate of change between variables.

• When extended, the line can help predict values outside the observed range (known as extrapolation).

In business, this allows managers to anticipate future sales, costs, or performance levels based on historical data trends.

Correlation and Extrapolation

Correlation
Correlation measures how strongly two variables are related and in what direction.

• A positive correlation means both variables move in the same direction.

• A negative correlation means they move in opposite directions.

• A zero correlation means no relationship exists between the variables.

Correlation vs Causation
While correlation shows the degree of association, it does not necessarily mean that one variable causes the other to change. For example, ice cream sales and sunburn cases are positively correlated but one does not cause the other—they are both influenced by temperature.

Extrapolation
Extrapolation is the process of extending the line of best fit beyond the observed data points to predict future outcomes.
Managers often use extrapolation to forecast sales, market growth, or seasonal demand patterns.
However, extrapolation carries risks because future trends might not follow historical patterns. Economic shifts, new competitors, or technological changes can make extrapolations unreliable if not evaluated carefully.

Evaluation of Simple Linear Regression

Benefits

1. Quantitative justification for decisions: It helps managers make evidence-based decisions using measurable relationships.

2. Forecasting and planning: Allows businesses to predict sales, production, and resource requirements based on trends.

3. Risk reduction: Provides early insight into potential business shifts, supporting proactive strategies.

4. Data simplification: Condenses large datasets into understandable trends and relationships.

Limitations

1. Dependent on data accuracy: Unreliable or incomplete data can lead to incorrect predictions.

2. Influence of outliers: Extreme values can distort regression results and weaken reliability.

3. Assumes linearity: Real-world relationships may be nonlinear, reducing accuracy.

4. Seasonal and external factors: Changes in economic conditions or consumer behavior may alter trends unexpectedly.

5. Correlation is not causation: Even if variables are correlated, other underlying factors may be influencing both.

Therefore, managers should use regression analysis alongside qualitative judgment, market research, and contextual understanding.