## **Learning Outcomes**

After reading this article, you will be able to explain the concepts of correlation and regression in the context of cost estimation. You will understand how to draw and interpret scatter diagrams, calculate and interpret the correlation coefficient and coefficient of determination, and construct and use lines of best fit for forecasting costs or revenues. You will be able to apply regression analysis for budgeting and comment on the reliability and limitations of such models.

## **ACCA Management Accounting (MA) Syllabus**

For ACCA Management Accounting (MA), you are required to understand statistical and analytical techniques used in forecasting costs. This article focuses on the following key areas that may be examined:

- The purpose and construction of scatter diagrams (lines of best fit)
- The distinction between dependent and independent variables in cost estimation
- Calculation and interpretation of the correlation coefficient and coefficient of determination
- Application of regression analysis to estimate cost functions
- Use of linear regression outputs for budgeting and forecasting
- Understanding the limitations of regression analysis and the importance of correlation in interpreting results

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What does a correlation coefficient close to 0 indicate about the relationship between two variables?
2. Which variable is typically considered the independent variable when analysing how production volume affects total cost?
3. Why is it important to look at the coefficient of determination, $ r^2 $, after calculating a regression equation?
4. How is a line of best fit used in forecasting future costs for budgeting purposes?

## **Introduction**

Reliable forecasts of costs are essential for budgeting and operational decision making. Relationships between variables such as costs and activity levels often need to be quantified. Management accountants use statistical methods—particularly correlation, regression analysis, and scatter diagrams—to identify patterns and predict future costs.

A clear understanding of these analytical techniques helps you explain fluctuations in cost data, select appropriate forecasting models, and appreciate the reliability of generated estimates.

## **SCATTER DIAGRAMS AND LINES OF BEST FIT**

Scatter diagrams are often the first step in exploring potential relationships between two variables.

A scatter diagram plots data points representing pairs of values for two variables, with the horizontal axis showing the independent variable and the vertical axis the dependent variable. This graphical approach provides a visual indication of any connection between the variables and whether the relationship is likely to be linear.

> **Key Term: Scatter diagram**
> A graph that displays values for two numerical variables as dots on a two-dimensional plane, used to visually assess potential relationships.

When the plotted data points suggest a pattern—that is, as one variable increases, the other also tends to increase (or decrease)—you may draw a line of best fit. This line visually represents the general direction or trend of the data and can be drawn by eye for a quick approximation. The primary use of such a line is to forecast new values of the dependent variable based on known independent variable inputs.

### **Worked Example 1.1**

A company records maintenance costs (in $) against machine hours for six months as follows:

| Machine hours (x) | Maintenance cost ($) (y) |
| ----------------- | ------------------------ |
| 100               | 2,000                    |
| 120               | 2,200                    |
| 110               | 2,100                    |
| 130               | 2,300                    |
| 140               | 2,450                    |
| 150               | 2,600                    |

**Question:** Briefly describe how you would use a scatter diagram to analyse this data.

> **Answer:** Plot each (x, y) value as a point. Examine whether points roughly align in a straight line. If so, draw a line of best fit by eye. Use this line to estimate the maintenance cost for a given machine-hours value.

## **CORRELATION**

Correlation measures the strength and direction of a linear relationship between two variables. It answers: as the independent variable changes, does the dependent variable tend to increase, decrease, or stay the same?

> **Key Term: Correlation coefficient**
> A numerical value, typically denoted as $ r $, that quantifies the strength and direction of a linear relationship between two variables, ranging from -1 to +1.

- $ r = 1 $: perfect positive relationship—variables increase together
- $ r = -1 $: perfect negative relationship—one variable increases as the other decreases
- $ r = 0 $: no linear relationship

Correlation can be positive, negative, or nonexistent. However, a strong correlation does not guarantee causation.

### **Worked Example 1.2**

Given the maintenance cost data above, suppose the calculated $ r $ is 0.99.

**Question:** What does this tell you about the data?

> **Answer:** There is a very strong positive linear relationship. As machine hours increase, maintenance cost increases nearly perfectly in step.

### **Coefficient of determination ($ r^2 $)**

The coefficient of determination, or $ r^2 $, represents the proportion of variation in the dependent variable that can be explained by changes in the independent variable using the regression line.

> **Key Term: Coefficient of determination**
> The square of the correlation coefficient ($ r^2 $), indicating the proportion of variation in the dependent variable accounted for by the regression model.

An $ r^2 $ of 0.85 implies that 85% of the variation in cost is explained by the changes in activity level; the remaining 15% is attributed to other factors.

## **REGRESSION ANALYSIS**

Regression analysis quantifies the linear relationship between variables by finding the equation of the line of best fit. This allows you to make numerical forecasts.

The general form of a linear regression equation is:

$$ y = a + bx $$

Where:

- $ y $: Dependent variable (e.g., cost)
- $ x $: Independent variable (e.g., activity level)
- $ a $: Intercept (estimated fixed cost)
- $ b $: Slope (variable cost per unit)

> **Key Term: Regression analysis**
> A statistical technique used to estimate the relationship between a dependent variable and one or more independent variables, typically resulting in an equation to predict future values.

### Calculating regression coefficients

Given data for $ n $ observations ($ x_i $, $ y_i $), the slope $ b $ and intercept $ a $ can be calculated:

$$ b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} $$

$ a = \overline{y} - b\overline{x} $, where $ \overline{x} $ and $ \overline{y} $ are the sample means.

### **Worked Example 1.3**

A cost accountant examines the following data for the past six months:

| Production (x) | Semi-variable cost (y) |
| -------------- | ---------------------- |
| 600            | \$4,800                |
| 650            | \$5,050                |
| 700            | \$5,250                |
| 725            | \$5,400                |
| 775            | \$5,650                |
| 800            | \$5,800                |

Compute the regression line $ y = a + bx $ (rounded to the nearest $).

**Given sums:**  
$ \sum x = 4,250 $, $ \sum y = 31,950 $, $ \sum x^2 = 3,040,625 $, $ \sum xy = 22,745,000 $, n = 6.

> **Answer:**  
> $$ b = [6(22,745,000) - 4,250 \times 31,950] / [6(3,040,625) - (4,250)^2] $$
> $$ b = (136,470,000 - 135,787,500)/(18,243,750 - 18,062,500) = 682,500/181,250 = 3.77 $$
> Round to $ b = 4 $.  
> $$ a = (31,950/6) - 4 \times (4,250/6) = 5,325 - 2,833 = 2,492 $$
> Regression equation: $ y = 2,492 + 4x $

If next month’s production is expected to be 850 units, forecast cost is:  
$$ y = 2,492 + 4 \times 850 = 2,492 + 3,400 = 5,892 $$

## **INTERPRETING AND APPLYING REGRESSION RESULTS**

Equations obtained via regression analysis can be used to predict costs and to prepare flexible budgets for different activity levels. The slope $ b $ gives the variable cost per unit of activity, and the intercept $ a $ estimates the fixed cost.

Be aware that the model is most reliable within the historical range of data (interpolation). Predictions made outside this range (extrapolation) are less dependable unless a strong logical basis exists for continuing the identified trend.

### **Exam Warning**

> Regression analysis assumes a linear relationship between the variables. Always check the scatter diagram. If the data does not align close to a straight line, the regression and correlation calculations may give misleading results.

## **LIMITATIONS OF REGRESSION ANALYSIS**

While regression is a powerful forecasting tool, its practical use carries some limitations:

- Only two variables are considered at one time in simple regression analysis, while actual costs may depend on several factors.
- Historical data is assumed to be relevant for future predictions, which may not be correct if processes or markets change.
- Regression assumes a linear relationship; non-linear or step changes may distort the outputs.
- Outliers (unusual data points) can distort both the slope and intercept, reducing the accuracy of your forecasts.

### **Revision Tip**

> Always inspect your raw data visually before calculation and look up the correlation coefficient. If $ |r| $ is low (for example, below 0.5), the regression equation may not be suitable for reliable forecasting.

## **Summary**

Cost functions can often be described mathematically using the techniques of correlation and regression. Scatter diagrams provide a graphical view of the relationship, while regression analysis offers a formula for forecasting costs based on activity levels. The correlation coefficient and the coefficient of determination help you assess the strength and reliability of these relationships. Remember, high correlation supports the use of regression for budgeting, but always consider model limitations and data quality before making forward-looking decisions.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- The use of scatter diagrams and lines of best fit in visualising relationships between variables
- Meaning and interpretation of the correlation coefficient and strength of association
- The purpose of the coefficient of determination ($ r^2 $)
- Calculation and interpretation of linear regression equations for cost estimation
- How to use regression equations to forecast costs within the relevant range
- The limitations and cautions in applying regression analysis for budgeting

## **Key Terms and Concepts**

- Scatter diagram
- Correlation coefficient
- Coefficient of determination
- Regression analysis

Cost estimation and forecasting - Correlation, regression, and lines of best fit