Probability and Statistics is 4-6 questions on the FE Mechanical exam. It is one of the smaller areas, but it is also one of the most mechanical: nearly every item reduces to picking the right distribution and reading one handbook table.
It connects to two other parts of the paper. Measurement uncertainty is a statistics question wearing an instrumentation label, and reliability in Mechanical Design is a probability question wearing a machine-element label. Time spent here pays out three times.
Exam weight: NCEES lists Probability and Statistics at 4-6 questions (4-5%) of the 110-question FE Mechanical exam. Work each problem below on paper first, then reveal the worked solution — reading a solution you have not attempted builds recognition, not recall.
What NCEES Tests in Probability and Statistics
The specification covers measures of central tendency and dispersion, probability distributions, expected value, confidence intervals, hypothesis testing, and linear regression. The distributions that actually appear are normal, binomial and exponential; the exam does not ask you to derive them, only to apply them.
Regression items usually give a small data set or a fitted line and ask for a slope, an intercept, or a predicted value. The coefficient of determination shows up as a concept question about how well a fit explains the data, not as a hand calculation.
5 Free Probability and Statistics Practice Problems
Each problem below comes from the PECivilClick FE Mechanical question bank and matches the style, difficulty and format of the real exam. Attempt each one under a three-minute limit — roughly the pace the exam demands.
Problem 1 — B. Measures of central tendencies and dispersions
A quality control engineer measures the diameter of 6 steel rods (in mm): 25.2, 25.0, 24.8, 25.1, 25.3, 24.6. What is the sample mean?
A) 24.9 mm
B) 25.0 mm
C) 25.1 mm
D) 25.2 mm
Answer: B) 25.0 mm
The sample mean is calculated as:
\(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\)
\(\bar{X} = \frac{25.2 + 25.0 + 24.8 + 25.1 + 25.3 + 24.6}{6} = \frac{150.0}{6}\)
\(\boxed{\bar{X} = 25.0 \text{ mm}}\)
Problem 2 — B. Measures of central tendencies and dispersions
A sample of 5 measurements has values: 12, 18, 15, 20, 10. What is the sample standard deviation?
A) 4.0
B) 3.58
C) 16.0
D) 12.8
Answer: A) 4.0
Step 1: Calculate the mean
\(\bar{X} = \frac{12 + 18 + 15 + 20 + 10}{5} = 15\)
Step 2: Sum of squared deviations = 68
Step 3: Sample variance
\(s^2 = \frac{\sum(X_i - \bar{X})^2}{n-1} = \frac{68}{4} = 17\)
Step 4: Sample standard deviation
\(\boxed{s = \sqrt{17} \approx 4.0}\)
Problem 3 — A. Probability distributions - Binomial
A quality inspector tests 8 components. Each component has a 70% probability of passing inspection. What is the variance of the number of components that will pass?
A) 1.30
B) 1.68
C) 2.40
D) 5.60
Answer: B) 1.68
For a binomial distribution, the variance is:
\(\sigma^2 = npq\)
where: \(n = 8\), \(p = 0.70\), \(q = 0.30\)
\(\sigma^2 = 8 \times 0.70 \times 0.30 = 8 \times 0.21\)
\(\boxed{\sigma^2 = 1.68}\)
Problem 4 — B. Measures of central tendencies and dispersions
For a data set with values 3, 5, 7, 9, 11, 13, 15, what is the median?
A) 9
B) 8
C) 7
D) 11
Answer: A) 9
For a data set arranged in increasing order with \(n\) values:
When \(n\) is odd, the median is the \(\left(\frac{n+1}{2}\right)^{th}\) item.
Here \(n = 7\) (odd)
\(\text{Median position} = \frac{7+1}{2} = 4^{th} \text{ item}\)
Data in order: 3, 5, 7, 9, 11, 13, 15
\(\boxed{\text{Median} = 9}\)
Problem 5 — B. Measures of central tendencies and dispersions
The following data set represents test scores: 78, 82, 85, 85, 88, 90, 92, 85. What is the mode?
A) 85
B) 82
C) 86
D) 88
Answer: A) 85
The mode is the value that occurs with the greatest frequency.
Counting occurrences:
- 78: 1 time
- 82: 1 time
- 85: 3 times
- 88: 1 time
- 90: 1 time
- 92: 1 time
The value 85 appears 3 times, more than any other value.
\(\boxed{\text{Mode} = 85}\)
Using the FE Reference Handbook for Probability and Statistics
The unit normal distribution table is the single most-used page in this section, and it is the one candidates fumble because the handbook tabulates one specific area convention. Read the column heading once during practice and write down what it gives you, because the difference between the area to the left of z and the area between the mean and z is a wrong answer that is always among the options.
Four Mistakes That Cost Points
- Confusing population and sample standard deviation. Dividing by n instead of n-1 on a small sample changes the answer enough to hit a distractor. The exam signals which one it wants by using the word sample, and the handbook gives both formulas.
- Using a z statistic when the sample is small. With a small sample and unknown population variance, the t distribution applies. Options are built so the z answer is present and wrong.
- Reading the normal table on the wrong side. A question asking for the probability that a part exceeds a dimension needs the upper tail. Most tables give the lower area, so the answer is one minus the table value.
- Treating dependent events as independent. Sampling without replacement changes the second probability. Multiplying two identical probabilities is fast, wrong, and always an option.
Frequently Asked Questions
How many probability and statistics questions are on the FE Mechanical exam?
NCEES specifies 4-6 questions out of 110, roughly 4-5 percent. Mathematics is counted separately at 6-9 questions.
Which distributions do I actually need?
Normal, binomial and exponential cover almost everything asked. The normal distribution dominates because it supports tolerance, quality and measurement-uncertainty questions elsewhere on the exam. Know how to standardise a value to z and how to read the handbook table in both directions.
Do I need to memorise statistical formulas?
No. Mean, variance, standard deviation, confidence interval and regression formulas are all in the handbook. What is not in the handbook is the judgement of which one applies, so practise reading the question for the words sample, population, with replacement and confidence level.
How is this different from the statistics on the FE Civil exam?
FE Civil folds statistics into a single Mathematics and Statistics area. FE Mechanical splits them, and leans slightly more toward reliability and measurement uncertainty because of the Mechanical Design and Controls areas that depend on them.
Keep Going
These topics feed into each other on the exam:
- FE Mechanical Mathematics practice problems — 6-9 questions on the exam
- FE Mechanical Measurements, Instrumentation, and Controls practice problems — 5-8 questions on the exam
- FE Mechanical Mechanical Design and Analysis practice problems — 10-15 questions on the exam
Browse every knowledge area from the free FE Mechanical practice problem hub, see what the full bank covers on the FE Mechanical exam prep page, or plan your schedule with the FE study timeline.