2. Explain how a scattergraph is used to separate a mixed cost into its fi xed and variable components.
G U I D E D U N I T P R E P A R A T I O N
A large part of analyzing a business decision is predicting the level of cost that will be incurred. Once you know how a particular cost behaves, estimating the total cost is relatively simple. In this unit, we will learn how to use several tech- niques for making these estimates.
Total cost is a combination of fi xed and variable costs.1 It can be predicted using the standard algebraic equation
3. Explain how the high-low method is used to separate a mixed cost into its fi xed and variable components for cost estimation.
4. Given a choice between the high-low method, a scattergraph, or regression analysis, which method would you prefer for separating a mixed cost into its fi xed and variable components? Why?
5. Explain the concept of the relevant range. How does a company’s relevant range differ from the steps found in a step cost?
mx 1 b 5 y
$0.06(100) 1 $10 5 $16 where:
m 5 the variable cost per unit;
x 5 the level of activity (such as number of units);
b 5 total fi xed cost; and y 5 total cost.
Think back to the cost of natural gas, which we examined in Unit 2.1. This mixed cost included a fi xed charge of $10 per month and a variable charge of
$0.06 per cubic foot. The total cost of service at any level of usage can be esti- mated using the following equation:
$0.06 (number of cubic feet used) 1 $10 5 Total gas cost
T T T T
m x b y
So for 100 cubic feet of gas, the estimated total cost would be
While using this equation to predict the total cost is a simple task, we don’t always know the total fi xed cost and variable cost per unit. So we need to deter- mine those costs before we can predict future costs. Let’s look at three methods of estimating costs: scattergraphs, the high-low method, and regression analysis.
Scattergraphs
Scattergraphs are the simplest method for estimating the fi xed and variable com- ponents of a mixed cost. A scattergraph is simply a graph that shows total costs in relation to volume or activity level. The data needed to create a scattergraph can be gathered from weekly or monthly reports. Once you have plotted the individual points, draw a line through them to estimate the cost relationship.2
Unit 2.2 Cost Estimation 55 The following table shows the delivery costs that Universal Sports Exchange
incurred last year with its outside delivery service. Exhibit 2-5 shows the same information in the form of a scattergraph. Notice that the level of activity—
number of deliveries in this case—is plotted on the horizontal axis and the total delivery cost is plotted on the vertical axis. This is the customary format for a scattergraph. Based on a visual inspection of the scattergraph, a linear relation- ship appears to exist between the number of deliveries and delivery cost.
Month Number of Deliveries Total Delivery Cost
January 2,000 $ 3,650
February 985 1,875
March 1,500 2,600 April 2,500 4,000
May 800 1,400
June 600 1,500
July 2,800 4,800
August 1,200 2,125
September 1,350 2,200
October 725 1,600
November 1,850 3,050
December 2,200 3,400
Total 18,510 $32,200
EXHIBIT 2-5
Scattergraph of delivery costs.
Total Delivery CostTT
Number of Deliveries
$0
$500
$1,000
$1,500
$2,000
$2,500
$3,000
$3,500
$4,000
$4,500
$5,000
$5,500
0 500 1,000 1,500 2,000 2,500 3,000
To estimate the fi xed and variable cost components using a scattergraph, it is necessary to visually “fi t” a line to the plotted points. You need to draw the line so that it appears to fi t the data well, minimizing the distance between the line and the data points. Once you have drawn the line, you can calculate the fi xed and variable costs using basic algebra. Exhibit 2-6 shows one possible line fi tted to the plotted points. Notice that it crosses the y-axis (where the level of activity is 0) at $500. Thus, the estimate for fi xed delivery cost is $500.
You can calculate the variable cost of a delivery as the slope of the line using any visually identifi ed point on the line. Using the point representing 1,500 deliveries and $2,600 of total delivery cost,
EXHIBIT 2-6 Scattergraph of delivery costs
with fi tted line.
Total Delivery Cost
Number of Deliveries
$0
$500
$1,000
$1,500
$2,000
$2,500
$3,000
$3,500
$4,000
$4,500
$5,000
0 500 1,000 1,500 2,000 2,500 3,000
Variable cost per delivery= Change in total cost Change in number of deliveries x= $2,600 –$500
1,500–0 x= $2,100
1,500
x= $1.40 per delivery Thus, total delivery cost can be estimated using the equation
The estimated delivery cost for 1,000 deliveries would be
1$1.403 1,000 deliveries2+ $500 =Total delivery cost
1$1.40⫻1,000 deliveries2+ $500 =$1,900
The scattergraph method of estimating costs is subject to some limitations. Most importantly, the choice of the line used to estimate the cost components is subjective.
Using a different line will produce different estimates of the fi xed and variable costs.
High-Low Method
Another “quick and dirty” approach to estimating the fi xed and variable com- ponents of a mixed cost is the high-low method. This method is similar to the scattergraph in that you begin by examining the cost data from a number of periods. Unlike the scattergraph, the high-low method requires only two data points—the lowest point of activity and the highest point of activity.
To estimate total cost using the high-low method:
1. Identify the highest and lowest levels of activity
2. Compute the variable cost per unit (the slope of the line):
Variable cost per unit= Change in total cost Change in activity
Unit 2.2 Cost Estimation 57 3. Calculate the fi xed cost using either the high point or the low point such that:
Fixed costs 5 Total cost 2 Variable cost 4. Complete the cost equation by showing that
(Variable cost per unit 3 Units) 1 Fixed cost 5 Total cost
Let’s return to the data on delivery cost that we used in the scattergraph example.
Step 1: The highest level of activity, 2,800 deliveries, occurred in July at a total cost of $4,800. The lowest level of activity, 600 deliveries, occurred in June at a total cost of $1,500.
Step 2:
Variable cost per delivery= Change in total cost Change in number of deliveries x= $4,800 –$1,500
2,800 –600 x= $3,300
2,200
x=$1.50 per delivery Step 3: Using the high point,
1Variable cost per unit3Number of units2+Fixed cost =Total cost 1$1.5032,8002+Fixed cost =$4,800 $4,200+Fixed cost= $4,800
Fixed cost= $4,800–$4,200 Fixed cost=$600
Or, using the low point,
1Variable cost per unit3Number of units2 + Fixed cost=Total cost 1$1.50 36002+Fixed cost =$1,500 $900 +Fixed cost= $1,500
Fixed cost= $1,500–$900 Fixed cost=$600
Step 4:
1$1.503 Number of deliveries2+ $600 =Total delivery cost
We can now use our equation to estimate the delivery cost at any level of activity.
For example, at 1,000 deliveries, estimated total delivery cost would be:
1$1.50⫻1,000 deliveries2+$600 =$2,100
Like the scattergraph, the high-low method does have some limitations.
Because it is based on only two extreme points, the high and low activity levels, the cost equation may not be truly representative of the cost relationship.
Be careful not to use an obvious outlier as either the high or the low point, or it will greatly skew your cost estimate.
Always select the high and low points based on level of activity, not total cost.
WATCH OUT!
In using the high-low method to separate a mixed cost into its fi xed and variable components, students often stop after Step 2, forgetting half the solution. Remember to insert either the high point or the low point into the total cost equation to calculate the fi xed cost component of the mixed cost (Step 3).
Then write out the total cost estimation equation (Step 4).
Use only the high point or the low point to complete Step 3, since those were the only two points used to estimate the variable cost per unit.
WATCH OUT!
Regression Analysis
A more precise approach to separating a mixed cost is regression analysis, a sta- tistical technique that identifi es the line of best fi t for the points plotted in a scat- tergraph. As shown in Exhibit 2-7, spreadsheet software such as Microsoft Excel makes regression analysis easy. After entering the data points in the spreadsheet, use the INTERCEPT and SLOPE functions to determine the fi xed cost and vari- able cost per unit, respectively.
Using the same data represented in our scattergraph (Exhibit 2-5), regression analysis results in a total fi xed cost of $388.94 and a variable cost of $1.49 per delivery. Thus, the equation to estimate total delivery cost would be
1$1.493Number of deliveries2+$388.94 =Total delivery cost Using this equation, the estimated delivery cost for 1,000 deliveries would be
1$1.49⫻ 1,0002+ $388.94= $1,878.94
You have now learned three ways to estimate the fi xed and variable com- ponents of a mixed cost: the scattergraph, the high-low method, and regression analysis. Let’s compare the results of the three methods:
EXHIBIT 2-7 Regression using Microsoft Excel.
Fixed Cost Component
Variable Cost Component
Unit 2.2 Cost Estimation 59 Why go through all these estimations? Remember, we started the chapter by
asking what Universal’s income would be if the company had achieved its original sales volume target. But we can’t estimate how income will change when the activ- ity level changes until we can estimate how costs will change with the activity level.
Cost Estimation and the Relevant Range
A fi nal word of caution about cost behavior and estimation: Cost behaviors and esti- mates are valid only within the relevant range, or the normal level of operating activ- ity. Beyond the relevant range, cost relationships are likely to change, and with them, cost estimates. (For more about relevant range see the box at the top of this page.)
Consider the graph shown in Exhibit 2-8, which represents a cost relation- ship that is often encountered in business. Note that the cost relationships on either side of the shaded relevant range differ markedly from the highlighted
How wide is the relevant range of activity? When does a company leave one relevant range and enter another one with a different cost function? The answers to these questions are company-specifi c and can greatly affect the cost estimates used in decision making.
Consider a 2008 announcement by the BMW Group that it would spend $750 million to ex- pand its plant in Spartanburg, South Carolina. When the three-year expansion was complete, the plant was 50% larger and employed 1,600 more workers. Production capacity increased from 160,000 to 240,000 cars. But the company isn’t fi nished expanding. On the heels of this expansion, the company announced an additional three-year, $900 million expansion that will increase production by up to 300,000 additional cars and add 300 more jobs. Growth of this magnitude will defi nitely move BMW’s production costs into a new relevant range.
Sources: BMW Group, “BMW Group Invests US $750 Million in US Plant,” news release, March 10, 2008; “BMW Announces Plant Expansion,” news release, March 10, 2008, http://www.bmwusfactory.com/news-center/press- releases/2008/bmw-announces-plant-expansion/ (accessed March 11, 2008). Michelle Krebs, “Plant Expansion Linchpin in BMW Plan to Dominate U.S. Luxury Market,” Edmunds AutoObserver, October 14, 2010, http//www.
edmunds.com/autoobserver.com/2010/00/plant-expansion-in-bmw-plan-to-dominate-us-luxury-market.html (accessed March 8, 2011); South Carolina Department of Commerce,“BMW Announces Plant Expansion in Spartanburg County,” press release, January 12, 2012, http://sccommerce.com/news/press-releases/bmw-announces- plant-expansion-spartanburg-county (accessed August 13, 2012).
Growth of this magnitude will defi nitely move BMW’s production costs into a new
relevant range.
REALITY CHECK —Where does the relevant range end?
Scott Olson/Getty Images News/Getty Images
Why do these three estimation techniques result in different predictions for delivery cost? Which one is right?