Statistics for the FE Civil Exam

NCEES FE Civil spec: Mathematics and Statistics, D. Statistics (e.g., distributions, mean, mode, standard deviation, confidence interval, regression and curve fitting). FE Reference Handbook: Engineering Probability and Statistics chapter.

Statistics questions on the FE Civil exam use the kind of data civil engineers handle every day: concrete cylinder strengths, traffic counts, rainfall depths and test results. You need to summarize a sample, use the normal distribution, build a confidence interval and fit a straight line to data.

What the NCEES spec covers

Mathematics section D, "Statistics", lists distributions, mean, mode, standard deviation, confidence intervals, and regression and curve fitting. Basic probability (combinations, binomial and conditional probability) also shows up in this area of the exam.

Key concepts

Central tendency. The mean is the arithmetic average, the median is the middle value, and the mode is the most frequent value.
Sample vs population. The sample standard deviation divides by $n - 1$ ; the population version divides by $n$ . Most exam data are samples.
Normal distribution. Convert a value to a z-score and use the standard normal table. About 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations.
Confidence intervals. Use z when the population standard deviation is known; use Student's t with $n - 1$ degrees of freedom when you only have the sample standard deviation.
Least-squares regression finds the line that minimizes the sum of squared vertical errors. The correlation coefficient r measures how well it fits, from −1 to +1.

Core formulas

\overset{x}{ˉ} = \frac{1}{n} \sum x_{i} s = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1} z = \frac{x - μ}{σ}

\overset{x}{ˉ} \pm z_{α /2} \frac{σ}{n} \overset{x}{ˉ} \pm t_{α /2, n - 1} \frac{s}{n}

Least-squares line $y = a + b x$ :

b = \frac{n \sum x _{i} y _{i} - \sum x _{i} \sum y _{i}}{n \sum x _{i}^{2} - ( \sum x _{i} ) ^{2}} a = \overset{y}{ˉ} - b \overset{x}{ˉ}

Where it is in the FE Reference Handbook

The Engineering Probability and Statistics chapter has the dispersion and central tendency formulas, probability laws, the binomial and normal distributions, the unit normal distribution table, the Student's t table, confidence interval formulas and least-squares curve fitting. Practice reading the normal table, since it lists several areas (one tail, two tails, center) side by side.

Worked example

Five concrete cylinders from one pour break at 32, 35, 30, 34 and 29 MPa. Find the sample mean, the sample standard deviation and a 95% confidence interval for the mean strength.

Step 1: mean.

\overset{x}{ˉ} = \frac{32 + 35 + 30 + 34 + 29}{5} = \frac{160}{5} = 32.0 MPa

Step 2: standard deviation. Deviations are 0, 3, −2, 2 and −3, so the squares sum to 26.

s = \frac{26}{5 - 1} = 6.5 = 2.55 MPa

Step 3: confidence interval. The population σ is unknown and n is small, so use t with 4 degrees of freedom: $t_{0.025, 4} = 2.776$ .

32.0 \pm 2.776 (\frac{2.55}{5}) = 32.0 \pm 3.17 \Rightarrow 28.8 to 35.2 MPa

Using z = 1.96 instead of t would give a narrower interval (±2.23 MPa) and overstate how precisely five cylinders define the mean.

How it shows up on the exam

Expect a mix of descriptive statistics (mean, median, mode, standard deviation of a short list), normal distribution probabilities from a z-score, confidence intervals, and the slope or intercept of a least-squares line. Probability questions on binomial outcomes and conditional events also appear. The data sets are short so that the calculation fits in a few minutes, which makes your calculator's statistics mode the fastest route.

Civil engineers use these tools for acceptance testing of concrete and asphalt, quality control of aggregate gradation, traffic speed studies and hydrologic frequency analysis. Reading the question carefully matters more than the arithmetic: note whether it asks for a sample or population value, a one-sided or two-sided probability, and which confidence level. Then pick the matching formula and table column before you start computing.

Common mistakes

Dividing by n instead of n − 1 for a sample standard deviation.
Using z instead of t for a small sample with an unknown population standard deviation.
Reading the wrong column of the normal table (one-tail area versus area between −z and +z).
Forgetting the square root of n when computing the standard error of the mean.
Mixing up variance and standard deviation.

Your calculator's one-variable statistics mode will give the mean and both standard deviations directly; the TI-36X Pro guide shows how. More examples: the statistics example and the confidence interval example.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

What is the sample standard deviation of 2, 4, and 6? See the worked solution or practice the Math & Statistics Refresher deck
A distribution has mean μ = 70 and standard deviation σ = 10. What is the z-score for a value of 85? See the worked solution or practice the Math & Statistics Refresher deck
A least-squares line is fitted to the points (1, 2), (2, 5), (3, 7) and (4, 10). The slope of the best-fit line is most nearly: See the worked solution or practice the Math & Statistics Refresher deck
A quantity is normally distributed with mean 500 and standard deviation 50. The probability that a single observation exceeds 600 is most nearly: See the worked solution or practice the Math & Statistics Refresher deck
Concrete cylinder strengths are normally distributed with mean 50 MPa and standard deviation 5 MPa. The probability that a cylinder exceeds 60 MPa is most nearly: See the worked solution or practice the Math & Statistics Refresher deck
A process produces defective items independently 20% of the time. In a sample of 5 items, the probability that exactly 2 are defective is most nearly: See the worked solution or practice the Math & Statistics Refresher deck

Related Calculators and Examples

Frequently Asked Questions

Should I use n or n - 1 for standard deviation on the FE exam?

Use n - 1 for a sample, which is the usual case when you are given test results. Use n only when the data are the entire population. Read the question wording carefully.

When do I use the t-distribution instead of z?

Use t when the population standard deviation is unknown and you are using the sample standard deviation, especially for small samples. The degrees of freedom are n - 1. Use z when the population standard deviation is known.

Are statistics tables included in the FE Reference Handbook?

Yes. The Engineering Probability and Statistics chapter includes the unit normal distribution table and Student's t table, along with formulas for confidence intervals and least-squares regression.

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