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10 FE practice problems: statistics: standard deviation and probability, with solutions

By Dr. Mohamed Elansary, PhD Environmental EngineeringUpdated

These ten original problems practice statistics: standard deviation and probability, a topic from the FE exam specification, using the Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values part of the FE Reference Handbook 10.6. Each problem gives the situation and the values with units. Work it with the handbook PDF open and commit to an answer before you open the solution. Every solution shows the handbook page, the equation, the substitution with units, a size check, and the mistake behind each wrong option, so a wrong pick tells you exactly what to fix.

How to use these problems

Give each problem an honest attempt before you open the solution: write the given values with units, find the equation in the FE Reference Handbook PDF, and commit to an answer. Then compare line by line. If you picked a wrong option, read the note for that option; each one names the mistake that produces it.

The problems

Problem 1 · FE, Statistics: standard deviation and probability

A lab reports the tensile strength of rebar samples: 528, 492, 527, 440, 576, 522, 509 MPa. The sample standard deviation is most nearly:

  • A 41.3 MPa
  • B 68.0 MPa
  • C 1,710 MPa
  • D 28.4 MPa

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
A (41.3 MPa)
Given
data = 528, 492, 527, 440, 576, 522, 509 MPa
Find
sample standard deviation s (MPa)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 3,594/7 = 513.4 MPa
  2. Σ(x - x̄)² = 10,260 MPa²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(10,260/6) = 41.3 MPa
Result
41.3 MPa, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 38.3 MPa, as it must be.
Why the others are wrong
  • B: is half the range, not a standard deviation
  • C: is the sample variance (no square root)
  • D: is the mean absolute deviation, not the standard deviation

Problem 2 · FE, Statistics: standard deviation and probability

A lab reports the influent BOD5 of composite samples: 301, 278, 237, 235, 199 mg/L. The sample standard deviation is most nearly:

  • A 35.7 mg/L
  • B 1,600 mg/L
  • C 51.0 mg/L
  • D 39.9 mg/L

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
D (39.9 mg/L)
Given
data = 301, 278, 237, 235, 199 mg/L
Find
sample standard deviation s (mg/L)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 1,250/5 = 250.0 mg/L
  2. Σ(x - x̄)² = 6,380 mg/L²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(6,380/4) = 39.9 mg/L
Result
39.9 mg/L, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 35.7 mg/L, as it must be.
Why the others are wrong
  • A: divided by n (population) instead of n - 1 for a sample
  • B: is the sample variance (no square root)
  • C: is half the range, not a standard deviation

Problem 3 · FE, Statistics: standard deviation and probability

A lab reports the tensile strength of rebar samples: 548, 562, 508, 589, 447, 575, 590 MPa. The sample standard deviation is most nearly:

  • A 38.9 MPa
  • B 71.5 MPa
  • C 51.8 MPa
  • D 48.0 MPa

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
C (51.8 MPa)
Given
data = 548, 562, 508, 589, 447, 575, 590 MPa
Find
sample standard deviation s (MPa)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 3,819/7 = 545.6 MPa
  2. Σ(x - x̄)² = 16,130 MPa²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(16,130/6) = 51.8 MPa
Result
51.8 MPa, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 48.0 MPa, as it must be.
Why the others are wrong
  • A: is the mean absolute deviation, not the standard deviation
  • B: is half the range, not a standard deviation
  • D: divided by n (population) instead of n - 1 for a sample

Problem 4 · FE, Statistics: standard deviation and probability

A lab reports the tensile strength of rebar samples: 538, 481, 585, 596, 524, 467 MPa. The sample standard deviation is most nearly:

  • A 52.6 MPa
  • B 64.5 MPa
  • C 48.0 MPa
  • D 41.2 MPa

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
A (52.6 MPa)
Given
data = 538, 481, 585, 596, 524, 467 MPa
Find
sample standard deviation s (MPa)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 3,191/6 = 531.8 MPa
  2. Σ(x - x̄)² = 13,830 MPa²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(13,830/5) = 52.6 MPa
Result
52.6 MPa, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 48.0 MPa, as it must be.
Why the others are wrong
  • B: is half the range, not a standard deviation
  • C: divided by n (population) instead of n - 1 for a sample
  • D: is the mean absolute deviation, not the standard deviation

Problem 5 · FE, Statistics: standard deviation and probability

Each water sample from a distribution system has a probability of 0.15 of failing a coliform test, independently. For 11 samples, the probability that exactly 1 fail is most nearly:

  • A 0.492
  • B 0.833
  • C 0.325
  • D 1.65

Handbook: Engineering Probability and Statistics, Binomial Distribution, FE Reference Handbook 10.6

Show the worked solution
Answer
C (0.325)
Given
n = 11, k = 1, p = 0.15
Find
P(X = 1)
Handbook
Engineering Probability and Statistics, Binomial Distribution, page 67
Equation
P(x) = C(n, x) pˣ qⁿ⁻ˣ, with q = 1 - p
Substitute
  1. C(11, 1) = 11!/[1!(10)!] = 11
  2. P(1) = C(n, x) pˣ qⁿ⁻ˣ = 11 × (0.15)^1 × (0.85)^10 = 0.325
Result
0.325, 3 significant figures
Check
P(1) is less than the cumulative P(X ≤ 1) = 0.492, and every probability is between 0 and 1.
Why the others are wrong
  • A: is P(X ≤ 1), the cumulative probability
  • B: is P(X ≥ 1), not exactly 1
  • D: left out the (1 - p)^(n - x) term

Problem 6 · FE, Statistics: standard deviation and probability

A lab reports the 28-day compressive strength of concrete cylinders: 5,270, 3,370, 4,350, 5,900, 4,920 psi. The sample standard deviation is most nearly:

  • A 1,270 psi
  • B 722 psi
  • C 859 psi
  • D 960 psi

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
D (960 psi)
Given
data = 5,270, 3,370, 4,350, 5,900, 4,920 psi, age = 28-day
Find
sample standard deviation s (psi)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 23,810/5 = 4,762 psi
  2. Σ(x - x̄)² = 3,685,000 psi²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(3,685,000/4) = 960 psi
Result
960 psi, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 859 psi, as it must be.
Why the others are wrong
  • A: is half the range, not a standard deviation
  • B: is the mean absolute deviation, not the standard deviation
  • C: divided by n (population) instead of n - 1 for a sample

Problem 7 · FE, Statistics: standard deviation and probability

The tensile strength of rebar samples follows a normal distribution with mean 483 MPa and standard deviation 47 MPa. The standard normal value z for a result of 586 MPa is most nearly:

  • A 0.456
  • B 12.5
  • C 15.0
  • D 2.19

Handbook: Engineering Probability and Statistics, Normal Distribution, FE Reference Handbook 10.6

Show the worked solution
Answer
D (2.19)
Given
mu = 483 MPa, sigma = 47 MPa, x = 586 MPa
Find
standard normal value z
Handbook
Engineering Probability and Statistics, Normal Distribution (Gaussian Distribution), page 68
Equation
z = (x - μ)/σ
Substitute
  1. z = (x - μ)/σ = (586 - 483)/47 = 2.19
Result
2.19, 3 significant figures
Check
the result is 2.19 standard deviations above the mean; z has no units.
Why the others are wrong
  • A: inverted the ratio
  • B: did not subtract the mean
  • C: divided by the square root of σ

Problem 8 · FE, Statistics: standard deviation and probability

A lab reports the 28-day compressive strength of concrete cylinders: 3,940, 3,750, 5,750, 4,900, 5,670, 5,630 psi. The sample standard deviation is most nearly:

  • A 743 psi
  • B 1,000 psi
  • C 904 psi
  • D 825 psi

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
C (904 psi)
Given
data = 3,940, 3,750, 5,750, 4,900, 5,670, 5,630 psi, age = 28-day
Find
sample standard deviation s (psi)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 29,640/6 = 4,940 psi
  2. Σ(x - x̄)² = 4,083,000 psi²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(4,083,000/5) = 904 psi
Result
904 psi, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 825 psi, as it must be.
Why the others are wrong
  • A: is the mean absolute deviation, not the standard deviation
  • B: is half the range, not a standard deviation
  • D: divided by n (population) instead of n - 1 for a sample

Problem 9 · FE, Statistics: standard deviation and probability

A lab reports the influent BOD5 of composite samples: 271, 230, 176, 221, 220, 228 mg/L. The sample standard deviation is most nearly:

  • A 47.5 mg/L
  • B 18.7 mg/L
  • C 918 mg/L
  • D 30.3 mg/L

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
D (30.3 mg/L)
Given
data = 271, 230, 176, 221, 220, 228 mg/L
Find
sample standard deviation s (mg/L)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 1,346/6 = 224.3 mg/L
  2. Σ(x - x̄)² = 4,589 mg/L²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(4,589/5) = 30.3 mg/L
Result
30.3 mg/L, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 27.7 mg/L, as it must be.
Why the others are wrong
  • A: is half the range, not a standard deviation
  • B: is the mean absolute deviation, not the standard deviation
  • C: is the sample variance (no square root)

Problem 10 · FE, Statistics: standard deviation and probability

A lab reports the influent BOD5 of composite samples: 255, 289, 231, 203, 260, 235, 264 mg/L. The sample standard deviation is most nearly:

  • A 27.7 mg/L
  • B 769 mg/L
  • C 21.6 mg/L
  • D 43.0 mg/L

Handbook: Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, FE Reference Handbook 10.6

Show the worked solution
Answer
A (27.7 mg/L)
Given
data = 255, 289, 231, 203, 260, 235, 264 mg/L
Find
sample standard deviation s (mg/L)
Handbook
Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values, page 64
Equation
s = √[Σ(xi - x̄)²/(n - 1)]
Substitute
  1. Mean: x̄ = Σx/n = 1,737/7 = 248.1 mg/L
  2. Σ(x - x̄)² = 4,613 mg/L²
  3. s = √[Σ(x - x̄)²/(n - 1)] = √(4,613/6) = 27.7 mg/L
Result
27.7 mg/L, 3 significant figures
Check
s is a little larger than the population value √(Σ(x - x̄)²/n) = 25.7 mg/L, as it must be.
Why the others are wrong
  • B: is the sample variance (no square root)
  • C: is the mean absolute deviation, not the standard deviation
  • D: is half the range, not a standard deviation

A new problem posts every day inside the lab, with the full worked solution the same evening.

Frequently asked questions

Where is statistics: standard deviation and probability in the FE Reference Handbook?

Look in the Engineering Probability and Statistics, Dispersion, Mean, Median, and Mode Values part of FE Reference Handbook 10.6. Each solution gives the exact page, so you can practice finding it in the PDF the way you will on exam day.

Are these real FE exam questions?

No. They are original problems generated from the handbook formulas and checked by code. Real exam questions are confidential, and sharing them breaks the NCEES agreement every examinee accepts.

How do I check my answer before opening the solution?

Check the units of your result and whether its size makes sense for the situation. Each solution ends with the same kind of check, so you can compare your habit with ours.

Sources

  1. NCEES FE Civil CBT exam specifications (PDF). Retrieved October 3, 2026.
  2. NCEES FE Environmental CBT exam specifications (PDF). Retrieved October 3, 2026.
  3. NCEES FE Mechanical CBT exam specifications (PDF). Retrieved October 3, 2026.
  4. NCEES FE Reference Handbook 10.6 (free PDF in MyNCEES). Retrieved October 3, 2026.