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Graphic Analysis
Introduction to Graphic Analysis . When you only have two data points, you must generally
assume a linear relationship. When you get more data, you can examine the data to determine if
there is truly a linear relationship.
You should always graph the data before performing an algebraic analysis.
• Graphic analysis is the best way of developing an overall view of cost-volume
• Graphic analysis is useful in analyzing cost-volume relationships, particularly, when the
cost and volume numbers involved are relatively small.
• Even when actual analysis is performed algebraically you can use graphs to demonstrate
cost-volume analysis to others.
Steps of Graphic Analysis . There are four steps in using graph paper to analyze cost-volume
Step 1. Determine the scale that you will use. Volume is considered the independent variable
and will be graphed on the horizontal axis. Cost is considered the dependent variable and will be
graphed on the vertical axis. The scales on the two axes do not have to be the same. However, on
each axis one block must represent the same amount of change as every other block of the same
size on that axis. Each scale should be large enough to permit analysis , and small enough to
permit the graphing of all available data and anticipated data estimates.
Step 2. Plot the available cost-volume data. Find the volume given for one of the data points
on the horizontal axis. Draw an imaginary vertical line from that point. Find the related cost on
the vertical axis and draw an imaginary horizontal line from that point. The point where the two
lines intersect represents the cost for the given volume. (If you do not feel comfortable with
imaginary lines you may draw dotted lines to locate the intersection.) Repeat this step for each
data point.
Step 3. Fit a straight line to the data. In this section of text, all data points will fall on a straight
line. All that you have to do to fit a straight line is connect the data points. Most analysts use
regression analysis to fit a straight line when all points do not fall on the line.
Step 4. Estimate the cost for a given volume. Draw an imaginary vertical line from the given
volume to the point where it intersects the straight line that you fit to the data points. Then move
horizontally until you intersect the vertical axis. That point is the graphic estimate of the cost for
the given volume of the item.
Example of Graphic Analysis . The four steps of cost-volume-profit analysis can be used to graph
and analyze any cost-volume relationship. Assume that you have been asked to estimate the cost
of 400 units given the following data:
Units Cost
200 $100,000
500 $175,000
600 $200,000
Step 1. Determine the scale that you will use .
Step 2. Plot the available cost-volume data.
Step 3. Fit a straight line to the data.
Step 4. Estimate the cost for a given volume. From the graph, you can estimate that the total
cost of 400 units will be $150,000.
In addition you can also estimate fixed cost. The cost of making zero units, $50,000, is the fixed
cost for this set of data.
You can use any two points on the line to calculate the equation of the line. Follow the
procedures in the section on Algebraic Analysis.
2.3 Analyzing The Price-Volume Relationship
Analysis Situations . In situations where you do not have offeror cost data, you can use the
principles of the cost-volume relationship in price-volume analysis. Price-volume analysis is
based on pricing information that is typically available to the Government negotiator.
Quantity Price Discounts . Unit prices normally decline as volume increases, primarily because
fixed costs are being divided by an increasing number of units. Buyers see these price reductions
with increasing volume in the form of quantity discounts.
Quantity discounts complicate the pricing decision, because a price that is reasonable for one
volume may not be reasonable for a different volume.
Offers quote quantity discounts because their costs per unit declines as volume increases. As a
result, even though offered prices include profit, you can use the cost-volume equation to
estimate prices at different quantities.
Steps for Estimating a Quantity Price Discount . Estimating a quantity price discount is a 4-step
Step 1. Calculate the variable element.
Step 2. Calculate the fixed element using data from the available data points.
Step 3. Develop an estimating equation.
Step 4. Estimate the price for a given quantity
Example of Estimating a Quantity Price Discount . You know that the unit price for 100 units is
$3,000 and the unit price for 500 units is $2,500. You are about to purchase 250 units. How can
you use the available information to estimate the price for 250 units?
Approach this question just as though you were being asked to estimate cost given information
for two different quantities. The only difference is that you use price data instead of cost data in
the analysis.
Step 1. Calculate the variable element.
V U = (C 2 – C1 ) / (Q 2 – Q 1 )
= ($1,250,000 – $300,000) / (500 – 100)
= $950,000 / 400
= $2,375
Step 2. Calculate the fixed element using data from the available data points.
C = F + V U (Q)
$300,000 = F + $2,375 (100)
$300,000 – $2,375 (100) = F
$300,000 – $237,500 = F
$62,500 = F
Step 3. Develop an estimating equation.
C = F + V U (Q)
= $62,500 + $2,375(Q)
Step 4. Estimate the price for 250 units.
C = $62,500 + $2,375 (Q)
= $62,500 + $2,375 (250)
= $62,500 + $593,750
= $656,250 total price and a $2,625 unit price
2.4 Analyzing The Cost-Volume-Profit Relationship
Until now, we have only looked at the cost-volume or price-volume relationship. Now, we are
going to expand that relationship to consider the relationship between cost, volume, and profit .
Cost-Volume-Profit Equation . The revenue taken in by a firm is equal to cost plus profit. That
can be written:
Revenue = Total Cost + Profit
We have already seen that total cost (C) is:
C = F + V U (Q)
Using this information, we can rewrite the revenue equation as:
Revenue = F + V U (Q) + Profit
In the cost-volume-profit equation, profit can be positive, negative, or zero . If profit is
negative, we normally refer to it as a loss. If profit is zero, the firm is breaking even with no
profit or loss. If we let P stand for profit, we can write the equation:
Revenue = F + V U (Q) + P
Revenue is equal to selling price per unit (R U ) multiplied by volume.
Revenue = R U (Q)
If we assume that the firm makes all the units that it sells, and sells all the units that it makes, we
can complete the cost-volume-profit equation:
R U (Q) = F + V U (Q) + P
Application of the Cost-Volume-Profit Equation . This equation and limited knowledge of a
contractor’s cost structure can provide you with extremely valuable information on the effect
purchase decisions can have on a firm’s profitability.
Using the Cost-Volume-Profit Equation to Estimate Selling Price . Given the following product
information, a firm prepared an offer for an indefinite quantity contract with the Government for
a new product developed by the firm. There are no other customers for the product. In
developing the offer unit price estimate (R U ), the firm used its estimated costs and its best
estimate of the quantity that it would sell under the contract.
Fixed Cost = $10,000
Variable Cost per Unit = $20
Contract Minimum Quantity = 3,000 units
Contract Maximum Quantity = 6,000 units
Firm’s Best Estimate of Quantity = 5,000 units
Target Profit = $5,000
R U (Q) = F + V U (Q) + P
R U (5,000) = $10,000 + $20 (5,000) + $5,000
R U (5,000) = $10,000 + $100,000 + $5,000
R U (5,000) = $115,000
R U = $115,000 / 5,000
R U = $23.00
Using the Cost-Volume-Profit Equation to Estimate Profit . Managers of the firm wanted to
know how profits would be affected if it actually sold the maximum quantity (6000 units) at
$23.00 per unit.
R U (Q) = F + V U (Q) + P
$23 (6,000) = $10,000 + $20 (6,000) + P
$138,000 = $10,000 + $120,000 + P
$138,000 = $130,000 + P
$138,000 – $130,000 = P
$8,000 = P
If the firm sells 6,000 units at $23.00 per unit, profit will be $8,000. That is a $3,000 increase
from the original $5,000 target profit, or an increase of 60 percent. Note that the firm’s profit
would increase solely because sales were higher than estimated.
Managers were even more concerned about how profits would be affected if they only sold the
minimum quantity (4000 units) at $23.00 per unit.
R U (Q) = F + V U (Q) + P
$23(4,000) = $10,000 + $20(4,000) + P
$92,000 = $10,000 + $80,000 + P
$92,000 = $90,000 + P
$2,000 = P
If the firm sells 4,000 at $23.00 per unit, profit will be $2,000. That is $3,000 less than the
original $5,000 target profit. Note that the firm’s profit would decrease solely because sales were
lower than estimated.
Using the Cost-Volume-Profit Equation to Estimate Break-Even Sales . In a final effort to
analyze the risk to the firm under the proposed indefinite deliver contract, managers wanted to
know the level of sales that would be required for the firm to break even (zero profit).
R U (Q) = F + V U (Q) + P
$23 (Q) = $10,000 + $20 (Q) + 0
($23 – $20) (Q) = $10,000
$3 (Q) = $10,000
Q = $10,000 / $3
Q = 3,333.33 units
The calculations show that the firm would break even at 3,333.33 units. Assuming that the firm
could not sell .33 units, the firm must sell 3,334 units to assure that all costs are covered. Selling
3,333 units would still result in a $.10 loss.
Contribution Income . The difference between revenue and variable cost is contribution income
(I). The term contribution income comes from the contribution made to covering fixed costs and
profit. If contribution income is positive, increasing sales will increase profits or reduce losses. If
contribution income is negative, increasing sales will reduce profits or create greater losses.
Contribution Income = Revenue – Variable Cost
Using symbols:
I = R U (Q) – V U (Q) or
I = (R U -V U ) (Q)
Knowledge of a contractor’s cost structure and contribution income can be valuable in analysis of
proposed costs.
Contribution Income Example . In evaluating an offeror’s proposal for 500 units at $900 each,
your analysis reveals the following cost structure:
Fixed Cost = $100,000
Variable Cost per Unit = $1,000
How would this affect your analysis of contract risk?
I = (R U – V U ) (Q)
= ($900 – $1,000) (500)
= -$100 (500)
= -$50,000
The contribution income from the sale is a negative $50,000. The firm would be substantially
worse off for having made the sale. Unless the firm can offer a positive rationale for such a
pricing decision, you must consider pricing as an important factor as you analyze the risk of
contract performance.
2.5 Identifying Issues And Concerns
Questions to Consider in Analysis . As you perform price/cost analysis, consider the issues and
concerns identified in this section, whenever you use cost-volume-profit analysis concepts.
• Has the contractor’s cost structure changed substantially?
Application of cost-volume-profit analysis assumes that the period covered by the analysis is too
short to permit facilities expansion or contraction or other changes that might affect overall
pricing relationships. If the contractor has substantially changed its cost structure, your ability to
use cost-volume-profit analysis may be limited. Examples of possible changes include:

o Downsizing to reduce fixed costs; and
o Increased investment in automated equipment to reduce variable costs of labor
and material.
• Is the straight-line assumption reasonable?
The cost-volume-profit relationship is not usually a straight- line relationship. Instead, it is a
curvilinear relationship. A straight-line analysis works as long as the straight line is a good
approximation of the cost-volume-profit relationship. Most computer programs designed to fit a
straight-line to a set of data provide measures of how well the line fits the data. For example, a
regression program will usually provide the coefficient of determination (r 2 ).
• Are current volume estimates within the relevant range of available data?
If the current business volume is substantially higher or lower than the volumes used to develop
the cost-volume-profit equation, the results may be quite unreliable. The contractor should be
expected to change the way it does business and its cost structure if volume increases or
decreases substantially.
• 3.0 – Chapter Introduction
• 3.1 – Identifying Situations For Use
• 3.2 – Measuring Central Tendency
• 3.3 – Measuring Dispersion
• 3.4 – Establishing A Confidence Interval
• 3.5 – Using Stratified Sampling
• 3.6 – Identifying Issues and Concerns
3.0 – Chapter Introduction
In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and
interpret data for contract pricing.
Categories of Statistics . Statistics is a science which involves collecting, organizing,
summarizing, analyzing, and interpreting data in order to facilitate the decision-making process.
These data can be facts, measurements, or observations. For example, the inflation rate for
various commodity groups is a statistic which is very important in contract pricing. Statistics can
be classified into two broad categories:
• Descriptive Statistics. Descriptive statistics include a large variety of methods for
summarizing or describing a set of numbers. These methods may involve computational
or graphical analysis. For example, price index numbers are one example of a descriptive
statistic. The measures of central tendency and dispersion presented in this chapter are
also descriptive statistics, because they describe the nature of the data collected.
• Inferential Statistics. Inferential or inductive statistics are methods of using a sample
data taken from a statistical population to make actual decisions, predictions, and
generalizations related to a problem of interest. For example, in contract pricing, we can
use stratified sampling of a proposed bill of materials to infer the degree it is overpriced
or under-priced.
Populations and Samples . The terms population and sample are used throughout any discussion
of statistics.
• Population. A population is the set of all possible observations of a phenomenon with
which we are concerned. For example, all the line items in a bill of materials would
constitute a population. A numerical characteristic of a population is called a parameter.
• Sample. A sample is a subset of the population of interest that is selected in order to
make some inference about the whole population. For example, a group of line items
randomly selected from a bill of materials for analysis would constitute a sample. A
numerical characteristic of a sample is called a statistic.
In contract pricing, you will most likely use statistics because you do not have complete
knowledge of the population or you do not have the resources needed to examine the population
data. Because most pricing applications involve the use of sample data, this chapter will
concentrate on statistical analysis using sample data. If you are interested in learning about
descriptive or inferential analysis of numerical population data, consult a college level statistics
Measure of Reliability . Since a sample contains only a portion of observations in the population,
there is always a chance that our inference or generalization about the population will be
inaccurate. Therefore, our inference should be accompanied by a measure of reliability. For
example, let’s assume that we are 90 percent sure that the average item in a bill of materials
should cost 85 percent of what the contractor has proposed plus or minus 3 percent. The 3
percent is simply a boundary for our prediction error and it means there is a 90 percent
probability that the error of our prediction is not likely to exceed 3 percent.
3.1 – Identifying Situations For Use
Situations for Use . Statistical analysis can be invaluable to you in:
• Developing Government objectives for contract prices based on historical values.
Historical costs or prices are often used as a basis for prospective contract pricing. When
several historical data points are available, you can use statistical analysis to evaluate the
historical data in making estimates for the future. For example, you might estimate the
production equipment set-up time based on average historical set-up times.
• Developing minimum and maximum price positions for negotiations. As you prepare
your negotiation objective, you can also use statistical analysis to develop minimum and
maximum positions through analysis of risk. For example, if you develop an objective of
future production set-up time based on the average of historical experience, that average
is a point estimate calculated from many observations. If all the historical observations
are close to the point estimate, you should feel confident that actual set-up time will be
close to the estimate. As the differences between the individual historical observations
and the point estimate increase, the risk that the future value will be substantially
different than the point estimate also increases. You can use statistical analysis to assess
the cost risk involved and use that assessment in developing your minimum and
maximum negotiation positions.
• Developing an estimate of risk for consideration in contract type selection. As
described above, you can use statistical analysis to analyze contractor cost risk. In
addition to using that analysis in developing your minimum and maximum negotiation
positions you can use it in contract type selection. For example, if the risk is so large that
a firm fixed-price providing reasonable protection to the contractor could also result in a
wind-fall profit, you should consider an incentive or cost-reimbursement contract instead.
• Developing an estimate of risk for consideration in profit/fee analysis. An analysis of
cost risk is also an important element in establishing contract profit/fee objectives. The
greater the dispersion of historical cost data, the greater the risk in prospective contract
pricing. As contractor cost risk increases, contract profit/fee should also increase.
• Streamlining the evaluation of a large quantity of data without sacrificing quality.
Statistical sampling is particularly useful in the analysis of a large bill of materials. The
stratified sampling techniques presented in this chapter allow you to:
o Examine 100 percent of the items with the greatest potential for cost reduction;
o Use random sampling to assure that there is no general pattern of overpricing
smaller value items.
o The underlying assumption of random sampling is that a sample is representative
of the population from which it is drawn.
o If the sample is fairly priced, the entire stratum is assumed to be fairly priced; if
the sample is overpriced, the entire stratum is assumed to be proportionately
3.2 – Measuring Central Tendency
Measures of Central Tendency . You are about to prepare a solicitation for a product that your
office has acquired several times before. Before you begin, you want to know what your office
has historically paid for the product. You could rely exclusively on the last price paid, or you
could collect data from the last several acquisitions. An array of data from several acquisitions
will likely mean little without some statistical analysis.
To get a clearer picture of this array of data, you would likely want to calculate some measure of
central tendency. A measure of central tendency is the central value around which data
observations (e.g., historical prices) tend to cluster. It is the central value of the distribution. This
section will examine calculation of the three most common and useful measures of central
• Arithmetic mean;
• Median; and
• Mode.
Calculating the Arithmetic Mean . The arithmetic mean (or simply the mean or average) is the
measure of central tendency most commonly used in contract pricing. To calculate the mean,
sum all observations in a set of data and divide by the total number of observations involved. The
formula for this calculation is:
= Sample mean
S = Summation of all the variables that follow the symbol (e.g., S X represents the sum of all X
X = Value for an observation of the variable X
n = Total number of observations in the sample
For example: Suppose you are trying to determine the production lead time (PLT) for an
electronic component using the following sample data:
Item 1 2 3 4 5 6 7 8
9 7 9 9 11 8 11 8
Calculating the Median . The median is the middle value of a data set when the observations are
arrayed from the lowest to the highest (or from the highest to the lowest). If the data set contains
an even number of observations, the median is the arithmetic mean of the two middle
It is often used to measure central tendency when a few observations might pull the measure
from the center of the remaining data. For example, average housing value in an area is
commonly calculated using the median, because a few extremely high-priced homes could result
in a mean that presents an overvalued picture of the average home price.
For the PLT example: You could array the data from lowest to highest:
X 7 8 8 9 9 9 11 11
Since there is an even number of observations, calculate median using the arithmetic mean of the
two middle observations:
Calculating the Mode . The mode is the observed value that occurs most often in the data set
(i.e. the value with the highest frequency).
It is often used to estimate which specific value is most likely to occur in the future. However, a
data set may have more than one mode. A data set is described as:
• Unimodal if it has only one mode.
• Bimodal if it has two modes.
• Multimodal if it has more than two modes.
For the PLT example: The mode is nine, because nine occurs three times, once more than any
other value.

Sample Answer

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