Approximations

Probability and Statistics · FE Reference Handbook section

Probability and Statistics

7 formulas

10 exam-style examples

~59 min

All Probability and Statistics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Approximations within Probability and Statistics. 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 approximations describes physically and when it applies.
State every one of the 7 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: probabilities are dimensionless and must land in [0, 1].

Lecture

Why this section exists. Approximations is the part of Probability and Statistics that lets you connect a sample of measurements from a construction or materials process 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 one distribution or one counting rule, then a single probability or interval. 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. probabilities are dimensionless and must land in [0, 1]. 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.

Three engineers in hard hats and safety vests reviewing drawings on a truck tailgate. — Photo 1. Where this shows up in practice: approximations.
Capstone Studio instructional photograph

Probability and Statistics — Approximations: 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 a sample of measurements from a construction or materials process. 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 7 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.

Notation used in this section

σt	Quantity produced by "σt = R d2" — read its definition and unit from the handbook line directly above the equation.
σR	Quantity produced by "σR = d3 σt" — read its definition and unit from the handbook line directly above the equation.
σS	Quantity produced by "σS = σt 1 − c 42" — read its definition and unit from the handbook line directly above the equation.
vt	Quantity produced by "vt = an estimate of σ" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

The following table and equations may be used to generate initial approximations of the items indicated.
n c4 d2 d3
2 0.7979 1.128 0.853
3 0.8862 1.693 0.888
4 0.9213 2.059 0.880
5 0.9400 2.326 0.864
6 0.9515 2.534 0.848
7 0.9594 2.704 0.833
8 0.9650 2.847 0.820
9 0.9693 2.970 0.808
10 0.9727 3.078 0.797
where

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.

Example 1

Sample size required for a target margin of error — Approximations

Concrete strength has a known standard deviation of 180.0 psi. How many cylinders must be tested so that the mean is estimated within ±27 psi at a z of 1.645?

Given

σ = 180.0 psi
E = 27 psi
z = 1.645

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.645 \times 180.0/27)^{2}$
Inside the bracket — 10.9667
Evaluate
$n = 120.3 \to 121 s p ec im e n s$

Answer: n = 121 specimens

Why the other options are there

11 (forgot to square)
61 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 2

Sample size required for a target margin of error — Approximations (2)

Concrete strength has a known standard deviation of 390.0 psi. How many cylinders must be tested so that the mean is estimated within ±46 psi at a z of 1.645?

Given

σ = 390.0 psi
E = 46 psi
z = 1.645

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.645 \times 390.0/46)^{2}$
Inside the bracket — 13.9467
Evaluate
$n = 194.5 \to 195 s p ec im e n s$

Answer: n = 195 specimens

Why the other options are there

14 (forgot to square)
98 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 3

Sample size required for a target margin of error — Approximations (3)

Concrete strength has a known standard deviation of 155.0 psi. How many cylinders must be tested so that the mean is estimated within ±30 psi at a z of 2.576?

Given

σ = 155.0 psi
E = 30 psi
z = 2.576

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (2.576 \times 155.0/30)^{2}$
Inside the bracket — 13.3093
Evaluate
$n = 177.1 \to 178 s p ec im e n s$

Answer: n = 178 specimens

Why the other options are there

14 (forgot to square)
89 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 4

Sample size required for a target margin of error — Approximations (4)

Concrete strength has a known standard deviation of 305.0 psi. How many cylinders must be tested so that the mean is estimated within ±38 psi at a z of 1.96?

Given

σ = 305.0 psi
E = 38 psi
z = 1.96

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.96 \times 305.0/38)^{2}$
Inside the bracket — 15.7316
Evaluate
$n = 247.5 \to 248 s p ec im e n s$

Answer: n = 248 specimens

Why the other options are there

16 (forgot to square)
124 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 5

Sample size required for a target margin of error — Approximations (5)

Concrete strength has a known standard deviation of 175.0 psi. How many cylinders must be tested so that the mean is estimated within ±30 psi at a z of 2.576?

Given

σ = 175.0 psi
E = 30 psi
z = 2.576

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (2.576 \times 175.0/30)^{2}$
Inside the bracket — 15.0267
Evaluate
$n = 225.8 \to 226 s p ec im e n s$

Answer: n = 226 specimens

Why the other options are there

16 (forgot to square)
113 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 6

Sample size required for a target margin of error — Approximations (6)

Concrete strength has a known standard deviation of 185.0 psi. How many cylinders must be tested so that the mean is estimated within ±20 psi at a z of 1.96?

Given

σ = 185.0 psi
E = 20 psi
z = 1.96

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.96 \times 185.0/20)^{2}$
Inside the bracket — 18.1300
Evaluate
$n = 328.7 \to 329 s p ec im e n s$

Answer: n = 329 specimens

Why the other options are there

19 (forgot to square)
165 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 7

Sample size required for a target margin of error — Approximations (7)

Concrete strength has a known standard deviation of 395.0 psi. How many cylinders must be tested so that the mean is estimated within ±12 psi at a z of 1.96?

Given

σ = 395.0 psi
E = 12 psi
z = 1.96

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.96 \times 395.0/12)^{2}$
Inside the bracket — 64.5167
Evaluate
$n = 4, 162 \to 4163 s p ec im e n s$

Answer: n = 4163 specimens

Why the other options are there

65 (forgot to square)
2082 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 8

Sample size required for a target margin of error — Approximations (8)

Concrete strength has a known standard deviation of 395.0 psi. How many cylinders must be tested so that the mean is estimated within ±44 psi at a z of 1.645?

Given

σ = 395.0 psi
E = 44 psi
z = 1.645

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.645 \times 395.0/44)^{2}$
Inside the bracket — 14.7676
Evaluate
$n = 218.1 \to 219 s p ec im e n s$

Answer: n = 219 specimens

Why the other options are there

15 (forgot to square)
110 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 9

Sample size required for a target margin of error — Approximations (9)

Concrete strength has a known standard deviation of 205.0 psi. How many cylinders must be tested so that the mean is estimated within ±30 psi at a z of 2.576?

Given

σ = 205.0 psi
E = 30 psi
z = 2.576

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (2.576 \times 205.0/30)^{2}$
Inside the bracket — 17.6027
Evaluate
$n = 309.9 \to 310 s p ec im e n s$

Answer: n = 310 specimens

Why the other options are there

18 (forgot to square)
155 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

Example 10

Sample size required for a target margin of error — Approximations (10)

Concrete strength has a known standard deviation of 300.0 psi. How many cylinders must be tested so that the mean is estimated within ±36 psi at a z of 1.96?

Given

σ = 300.0 psi
E = 36 psi
z = 1.96

Find

Required sample size n

Start with the thinking

Sample size scales with the square of the ratio σ/E — halving the margin quadruples the testing.
Always round the computed sample size up to the next whole specimen.

Step-by-step solution

Formula
$n = (z σ / E)^{2}$
Substituting
$n = (1.96 \times 300.0/36)^{2}$
Inside the bracket — 16.3333
Evaluate
$n = 266.8 \to 267 s p ec im e n s$

Answer: n = 267 specimens

Why the other options are there

17 (forgot to square)
134 (halved the requirement)

Reference: FE Reference Handbook — Probability and Statistics → Approximations

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 a sample of measurements from a construction or materials process, 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

Approximations contains 7 relations; you must be able to find this page in under 15 seconds.
Exam style: one distribution or one counting rule, then a single probability or interval.
Unit rule: probabilities are dimensionless and must land in [0, 1].
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

probabilities are dimensionless and must land in [0, 1]
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.

Approximations

Learning objectives

Lecture

Theory, developed

Notation used in this section

Handbook notes for this section

Core formulas for this FE topic

σt = R d2Needs technical review

σt = S c4Needs technical review

σR = d3 σtNeeds technical review

σS = σt 1 − c 42Needs technical review

vt=anestimateofσ

σR = an estimate of the standard deviation of the ranges of the samplesNeeds technical review

σS = an estimate of the standard deviation of the standard deviations of the samplesNeeds technical review

Worked exam-style examples

Self-check

Chapter summary

Common traps in this section