Variances
Construction Engineering · FE Reference Handbook section
Learning objectives
What you must be able to do before leaving this section.
This chapter section covers Variances within Construction Engineering. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.
- Explain, in your own words, what variances describes physically and when it applies.
- State every one of the 2 relations the handbook lists here and name each symbol with its unit.
- Select the correct relation from the wording of an exam stem within 20 seconds.
- Carry a complete solution from givens to a "most nearly" answer with the correct unit.
- Recognise the distractors generated by the unit trap: crew-hours, cubic yards and dollars per unit.
Lecture
Why this section exists. Variances is the part of Construction Engineering that lets you connect an activity network or production operation to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.
How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.
How it is examined. Items from this page are written as a schedule, productivity or earned-value computation. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.
The habit that earns the points. Unit discipline. crew-hours, cubic yards and dollars per unit. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.
How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.
Photo 1. Where this shows up in practice: variances.
Capstone Studio instructional photograph
Construction Engineering — Variances: reference schematic for orienting the symbols used in this section.
Theory, developed
Read this before the equations — it is what makes them memorable.
The physical situation
Every item from this section describes an activity network or production operation. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.
The governing principle
The 2 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.
Assumptions and limits of validity
Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.
Solution procedure you should automate
1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.
Photo 2. Construction Engineering: the physical system the theory above idealises.
Capstone Studio instructional photograph
Notation used in this section
| CV | Quantity produced by "CV = BCWP - ACWP (Cost variance = Earned - Actual)" — read its definition and unit from the handbook line directly above the equation. |
|---|---|
| SV | Quantity produced by "SV = BCWP - BCWS (Schedule variance = Earned - Planned)" — read its definition and unit from the handbook line directly above the equation. |
Core formulas for this FE topic
Definitions, applicability, units, assumptions and worked examples for each relation.
Worked exam-style examples
The four ways this section is written on the real exam — thoughts first, then equations, then substitution.
A $959,000 contract is 35% complete. Planned value is $527,450 and actual cost is $302,085. Compute CV, SV, CPI and SPI.
Given
- BAC = $959,000
- % complete = 35%
- PV = $527,450
- AC = $302,085
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $335,650 − $302,085 = $33,565
Schedule variance — SV = EV − PV = $335,650 − $527,450 = $-191,800
Cost index
Schedule index
Answer: CV = $33,565, SV = $-191,800, CPI = 1.11, SPI = 0.64
Why the other options are there
- CPI = 0.90 (ratio inverted)
- CV = $225,365 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $1,057,000 contract is 30% complete. Planned value is $422,800 and actual cost is $332,955. Compute CV, SV, CPI and SPI.
Given
- BAC = $1,057,000
- % complete = 30%
- PV = $422,800
- AC = $332,955
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $317,100 − $332,955 = $-15,855
Schedule variance — SV = EV − PV = $317,100 − $422,800 = $-105,700
Cost index
Schedule index
Answer: CV = $-15,855, SV = $-105,700, CPI = 0.95, SPI = 0.75
Why the other options are there
- CPI = 1.05 (ratio inverted)
- CV = $89,845 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $1,341,000 contract is 45% complete. Planned value is $536,400 and actual cost is $693,968. Compute CV, SV, CPI and SPI.
Given
- BAC = $1,341,000
- % complete = 45%
- PV = $536,400
- AC = $693,968
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $603,450 − $693,968 = $-90,518
Schedule variance — SV = EV − PV = $603,450 − $536,400 = $67,050
Cost index
Schedule index
Answer: CV = $-90,518, SV = $67,050, CPI = 0.87, SPI = 1.13
Why the other options are there
- CPI = 1.15 (ratio inverted)
- CV = $-157,568 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $1,316,000 contract is 35% complete. Planned value is $526,400 and actual cost is $460,600. Compute CV, SV, CPI and SPI.
Given
- BAC = $1,316,000
- % complete = 35%
- PV = $526,400
- AC = $460,600
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $460,600 − $460,600 = $0
Schedule variance — SV = EV − PV = $460,600 − $526,400 = $-65,800
Cost index
Schedule index
Answer: CV = $0, SV = $-65,800, CPI = 1.00, SPI = 0.87
Why the other options are there
- CPI = 1.00 (ratio inverted)
- CV = $65,800 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $2,196,000 contract is 45% complete. Planned value is $1,317,600 and actual cost is $889,380. Compute CV, SV, CPI and SPI.
Given
- BAC = $2,196,000
- % complete = 45%
- PV = $1,317,600
- AC = $889,380
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $988,200 − $889,380 = $98,820
Schedule variance — SV = EV − PV = $988,200 − $1,317,600 = $-329,400
Cost index
Schedule index
Answer: CV = $98,820, SV = $-329,400, CPI = 1.11, SPI = 0.75
Why the other options are there
- CPI = 0.90 (ratio inverted)
- CV = $428,220 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $1,968,000 contract is 55% complete. Planned value is $984,000 and actual cost is $1,082,400. Compute CV, SV, CPI and SPI.
Given
- BAC = $1,968,000
- % complete = 55%
- PV = $984,000
- AC = $1,082,400
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $1,082,400 − $1,082,400 = $0
Schedule variance — SV = EV − PV = $1,082,400 − $984,000 = $98,400
Cost index
Schedule index
Answer: CV = $0, SV = $98,400, CPI = 1.00, SPI = 1.10
Why the other options are there
- CPI = 1.00 (ratio inverted)
- CV = $-98,400 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $1,757,000 contract is 45% complete. Planned value is $1,229,900 and actual cost is $869,715. Compute CV, SV, CPI and SPI.
Given
- BAC = $1,757,000
- % complete = 45%
- PV = $1,229,900
- AC = $869,715
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $790,650 − $869,715 = $-79,065
Schedule variance — SV = EV − PV = $790,650 − $1,229,900 = $-439,250
Cost index
Schedule index
Answer: CV = $-79,065, SV = $-439,250, CPI = 0.91, SPI = 0.64
Why the other options are there
- CPI = 1.10 (ratio inverted)
- CV = $360,185 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $878,000 contract is 45% complete. Planned value is $658,500 and actual cost is $355,590. Compute CV, SV, CPI and SPI.
Given
- BAC = $878,000
- % complete = 45%
- PV = $658,500
- AC = $355,590
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $395,100 − $355,590 = $39,510
Schedule variance — SV = EV − PV = $395,100 − $658,500 = $-263,400
Cost index
Schedule index
Answer: CV = $39,510, SV = $-263,400, CPI = 1.11, SPI = 0.60
Why the other options are there
- CPI = 0.90 (ratio inverted)
- CV = $302,910 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $2,424,000 contract is 35% complete. Planned value is $1,818,000 and actual cost is $890,820. Compute CV, SV, CPI and SPI.
Given
- BAC = $2,424,000
- % complete = 35%
- PV = $1,818,000
- AC = $890,820
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $848,400 − $890,820 = $-42,420
Schedule variance — SV = EV − PV = $848,400 − $1,818,000 = $-969,600
Cost index
Schedule index
Answer: CV = $-42,420, SV = $-969,600, CPI = 0.95, SPI = 0.47
Why the other options are there
- CPI = 1.05 (ratio inverted)
- CV = $927,180 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
A $1,921,000 contract is 60% complete. Planned value is $960,500 and actual cost is $1,094,970. Compute CV, SV, CPI and SPI.
Given
- BAC = $1,921,000
- % complete = 60%
- PV = $960,500
- AC = $1,094,970
Find
CV, SV, CPI, SPI
Start with the thinking
- Earned value is percent complete times budget.
- Cost indices compare EV with AC; schedule indices compare EV with PV.
Step-by-step solution
Earned value
Cost variance — CV = EV − AC = $1,152,600 − $1,094,970 = $57,630
Schedule variance — SV = EV − PV = $1,152,600 − $960,500 = $192,100
Cost index
Schedule index
Answer: CV = $57,630, SV = $192,100, CPI = 1.05, SPI = 1.20
Why the other options are there
- CPI = 0.95 (ratio inverted)
- CV = $-134,470 (PV used instead of EV)
Reference: FE Reference Handbook — Construction Engineering → Variances
Self-check
Answer these without notes before moving on.
- Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
- Which assumption, if violated, makes the main relation of this section invalid?
- Given an activity network or production operation, what is the first quantity you would compute, and why that one first?
- Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
- Rework Example 1 above from the givens alone, without reading the solution lines.
Chapter summary
- Variances contains 2 relations; you must be able to find this page in under 15 seconds.
- Exam style: a schedule, productivity or earned-value computation.
- Unit rule: crew-hours, cubic yards and dollars per unit.
- Work the 10 examples until the solution path, not the answer, is automatic.
Common traps in this section
- crew-hours, cubic yards and dollars per unit
- Answering the intermediate quantity instead of the quantity requested.
- Rounding intermediate values before the final step.
- Using a relation from an adjacent handbook section that shares a symbol.
- Skipping the sketch — most lost points on this page start with a misread geometry.