Overview

• What you’ll learn: The high-low method for estimating cost functions, simple regression analysis, interpreting R-squared, choosing between estimation methods, and the data requirements for reliable cost estimation.
• Prerequisites: Lesson 6 — Determining How Costs Behave
• Estimated reading time: 20 minutes

Introduction

The Grand Historian records: In Lesson 6, we surveyed the landscape of cost estimation — the industrial engineering method, account analysis, and the scatter plot. Each has its merits, but each also has a fatal flaw: subjectivity. Two analysts examining the same scatter plot may draw different lines and arrive at different cost functions. The management accountant who presents subjective estimates to the CFO had better have an excellent explanation — or an updated resume.

This lesson introduces two quantitative methods that replace human judgment with mathematical precision. The high-low method offers quick-and-dirty estimation using only two data points. Regression analysis offers rigorous, statistically defensible estimation using all available data points. Together, they form the quantitative backbone of cost estimation. The high-low method is the cavalry scout — fast but imprecise. Regression is the siege artillery — slower to deploy but devastatingly accurate.

The High-Low Method

The high-low method estimates the variable and fixed components of a mixed cost using only the highest and lowest activity observations in the dataset. It draws a straight line between the two extreme points.

Step-by-Step Process

Given the following monthly maintenance cost data:

Step 1: Identify the highest and lowest activity levels:

• Highest: June — 4,000 machine hours, $19,800
• Lowest: January — 1,500 machine hours, $10,200

Step 2: Calculate the variable cost per unit of activity (the slope):

b = (Y_high – Y_low) / (X_high – X_low) = ($19,800 – $10,200) / (4,000 – 1,500) = $9,600 / 2,500 = $3.84 per machine hour

Step 3: Calculate the fixed cost component (the y-intercept):

a = Y_high – b(X_high) = $19,800 – $3.84(4,000) = $19,800 – $15,360 = $4,440

Step 4: Write the cost function:

Y = $4,440 + $3.84X

Advantages of the High-Low Method

• Simple, fast, requires no special software — a calculator suffices
• Easy to understand and explain to non-accountants
• Useful for quick preliminary estimates

Limitations of the High-Low Method

• Uses only two data points: All other observations are ignored. If the two extreme points happen to be outliers, the estimate will be severely distorted.
• Sensitive to outliers: A single unusual month at the high or low end can skew the entire cost function.
• No measure of goodness of fit: Unlike regression, there is no R-squared or statistical test to evaluate accuracy.
• Ignores the pattern: The relationship suggested by the middle observations is completely ignored.

The high-low method is acceptable for quick estimates and exam problems, but in professional practice, regression analysis is almost always preferred.

Regression Analysis

Regression analysis is a statistical method that fits a line to the data by minimizing the sum of squared differences between the actual and predicted values. It uses all data points, not just two, and provides statistical measures of the quality of the fit.

Simple Linear Regression

Simple linear regression estimates the equation Y = a + bX where:

• Y = dependent variable (total cost)
• a = intercept (estimated fixed cost)
• b = slope (estimated variable cost per unit of activity)
• X = independent variable (cost driver / activity level)

The regression algorithm finds the values of a and b that minimize the sum of squared residuals — the vertical distances between each data point and the regression line. This is called the least-squares method, and it produces the mathematically “best” line through the data.

Interpreting Regression Output

Most spreadsheet programs (Excel, Google Sheets) and statistical software produce regression output that includes:

R-Squared: The Coefficient of Determination

R-squared (R²) is the single most important statistic for evaluating a cost function. It measures the percentage of variation in the dependent variable (cost) that is explained by the independent variable (cost driver):

• R² = 0.90 means 90% of the variation in cost is explained by the cost driver — excellent fit.
• R² = 0.50 means only 50% is explained — moderate fit, consider other drivers.
• R² = 0.10 means only 10% is explained — poor fit, the wrong driver was chosen.

An R-squared close to 1.0 is ideal, but context matters. In some industries, R² = 0.70 may be the best achievable. The key question is always: “Is this the best cost driver available?”

Comparing High-Low and Regression

Using our maintenance cost data, the regression output might show:

• Intercept (a) = $4,120 (vs. $4,440 from high-low)
• Slope (b) = $3.92/hour (vs. $3.84 from high-low)
• R² = 0.993 (excellent fit)

The regression estimates differ from the high-low estimates because regression uses all six data points, not just two. In this case, the differences are modest, but with noisier data or outliers, the gap can be enormous.

Choosing the Right Method

In professional practice, regression analysis is the gold standard. It uses all available data, provides statistical measures of fit (R², t-statistics, p-values), and can be extended to multiple drivers (multiple regression, covered in Lesson 8).

Data Requirements for Reliable Estimation

Even regression analysis cannot rescue bad data. For reliable results:

• Sufficient observations: At least 15–20 data points for simple regression; more for multiple regression. Fewer observations reduce statistical reliability.
• Adequate variation: The cost driver must vary meaningfully across observations. If machine hours are nearly the same every month, the data cannot reveal the cost-activity relationship.
• Consistent environment: All observations should come from the same operating environment. If the company switched to a new production process midway through the data period, use only post-switch data.
• Matched periods: Costs and activity must be measured in the same time period. If maintenance is billed with a one-month lag, adjust the data so costs align with the activity that caused them.
• Outlier investigation: Do not mechanically exclude outliers. Investigate each one — it may be a data error (exclude it), an unusual but real event (consider excluding), or a signal that the cost structure has changed (segment the data).

Key Takeaways

• The high-low method estimates variable and fixed costs using only the highest and lowest activity observations — fast but ignores all other data and has no measure of fit.
• Regression analysis uses all data points and the least-squares method to find the best-fit line, providing R-squared, t-statistics, and p-values for evaluation.
• R-squared measures the proportion of cost variation explained by the cost driver — closer to 1.0 is better.
• Regression is the professional standard for cost estimation; high-low is a quick preliminary tool.
• Reliable estimation requires sufficient observations, adequate variation, a consistent environment, matched periods, and careful outlier investigation.

What’s Next

In Lesson 8, we venture into advanced territory: multiple regression (using two or more cost drivers simultaneously), nonlinear cost functions, learning curves, and the criteria for choosing cost drivers in complex environments. This is where cost estimation becomes both an art and a science.