Mathematics and Statistics

8% of exam

Analytic geometry, single-variable calculus, vector operations, and statistics including distributions, central tendency, dispersion, confidence intervals, and regression.

3 concepts

Exam areas

A. Analytic geometry

Analytic Geometry and Single-Variable Calculus

Lines, distance, and conics joined to the derivative and integral — slope, optimization, the fundamental theorem, and area, the spine of the FE math problems.

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C. Vector operations

Vector Operations

Components and unit vectors, addition, the dot product and projection, the cross product and moment, and direction cosines — the algebra behind statics and dynamics.

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D. Statistics

Statistics: Distributions, Dispersion, and Confidence Intervals

Center and spread, the normal and standard-normal z-transform, confidence intervals for the mean, and least-squares regression with correlation — the FE statistics core.

Statistics on the FE Civil is where data becomes a decision: a batch of concrete cylinders becomes a strength you can specify, a column of survey residuals becomes an uncertainty, a scatter of test points becomes a design line. The problems are not hard, but they are easy to fumble — the population-versus-sample divisor, the standard-normal transform, and the choice between a

z

and a

t

interval each swing the answer. The FE Reference Handbook collects every formula you need in the Engineering Probability and Statistics chapter (dispersion and mean values, the normal distribution and its unit-normal table, confidence intervals, and least-squares regression), so the work is choosing the right tool and reading the table correctly. This concept walks the full chain from a raw data set to a fitted line.

Measures of center

Three numbers summarize where data sit. The arithmetic mean

\bar{X}

is the balance point — the sum divided by the count — and it feeds every later formula, but a single outlier drags it. The median is the middle value of the sorted data (for an even count, the average of the two middle values) and shrugs off outliers. The mode is the most frequent value. For a symmetric distribution all three coincide; for a right-skewed set (a few large values) the mean exceeds the median, a fact the exam uses to test whether you understand skew rather than just compute it.

\bar{X}=\dfrac{1}{n}\sum_{i=1}^{n}X_i

Variance and standard deviation — and the (n−1) trap

Dispersion is the average squared distance from the mean. If your numbers are the entire population you divide by

N

and call it

\sigma^2

; if they are a sample drawn to estimate a larger population — the usual engineering case — you divide by

n-1

and call it

s^2

. The

n-1

(Bessel's correction) compensates for having estimated the mean from the same data, and it makes

s^2

an unbiased estimate of

\sigma^2

. The standard deviation is the square root, restoring the original units. Deciding population versus sample is the single biggest point-swing on this topic; the word "sample" or any mention of estimating a population means

n-1

s^2=\dfrac{1}{n-1}\sum_{i=1}^{n}\left(X_i-\bar{X}\right)^2, \qquad s=\sqrt{s^2}

The normal distribution

Most engineering variability — material strength, measurement error, the mean of many readings — clusters in the bell-shaped normal distribution, symmetric about its mean

\mu

with spread set by

\sigma

. The Central Limit Theorem is why it appears everywhere: averages of many independent observations tend toward normal regardless of the parent shape. Roughly

68\%

of the area lies within

\pm 1\sigma

95\%

within

\pm 2\sigma

, and

99.7\%

within

\pm 3\sigma

— the quick mental check before you ever open the table.

f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

The standard-normal z-transform

You never integrate the bell curve; you standardize and read a table. The transform

z=(x-\mu)/\sigma

rescales any normal variable to the unit normal (mean

0

, standard deviation

1

), and a

z

value is simply the number of standard deviations from the mean. The handbook's unit-normal table gives

F(z)

, the area to the left, and

R(z)

, the area to the right; symmetry gives

F(-z)=1-F(z)=R(z)

. Probabilities between two values are differences of table areas. This single transform answers "what fraction of cylinders fall below spec" or "what is the chance a measurement exceeds a tolerance."

z=\dfrac{x-\mu}{\sigma}

Confidence interval for the mean

A confidence interval puts error bars on an estimated mean: it is the range that, with stated confidence (commonly

95\%

), brackets the true population mean

\mu

. When the population standard deviation

\sigma

is known (or

n

is large), use the

z

form; the half-width is

z_{\alpha/2}\,\sigma/\sqrt{n}

, and the

\sqrt{n}

in the denominator is why quadrupling the sample only halves the interval. When

\sigma

is unknown and estimated by

s

from a small sample, swap in Student's

t_{\alpha/2,\,n-1}

, which is wider to pay for the extra uncertainty. The handbook tabulates the common

z_{\alpha/2}

1.645

for

90\%

1.960

for

95\%

2.576

for

99\%

\bar{X}-z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\;\le\;\mu\;\le\;\bar{X}+z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}

Least-squares regression and correlation

Least squares fits the straight line

\hat{y}=\hat{a}+\hat{b}x

that minimizes the sum of squared vertical residuals. The slope is

\hat{b}=S_{xy}/S_{xx}

and the intercept follows from forcing the line through the centroid

(\bar{x},\bar{y})

, so

\hat{a}=\bar{y}-\hat{b}\bar{x}

. The sample correlation coefficient

R=S_{xy}/\sqrt{S_{xx}S_{yy}}

runs from

-1

+1

and measures how tightly the points hug a line; its square

R^2

, the coefficient of determination, is the fraction of the variation in

y

explained by the fit. An

R^2

near

1

means a strong linear relationship — but never confuse correlation with causation.

\hat{b}=\dfrac{S_{xy}}{S_{xx}}, \qquad \hat{a}=\bar{y}-\hat{b}\,\bar{x}, \qquad R=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

Exam strategy

Read for population versus sample before touching the variance key, and use your calculator's statistics registers — enter the data once and read

\bar{X}

\sigma_x

(divisor

n

), and

s_x

(divisor

n-1

) directly. For any normal question, write

z=(x-\mu)/\sigma

first, sketch the bell and shade the region you want, then translate to table areas (

F

left,

R

right) — a sketch prevents the classic complement error. Choose

z

when

\sigma

is known or

n

is large and

t

when

\sigma

is estimated from a small sample. For regression, accumulate

S_{xx}

S_{yy}

S_{xy}

with the shortcut sums, get the slope first, and remember the line passes through

(\bar{x},\bar{y})

as a built-in check.

Key equations

Arithmetic mean

\bar{X}=\dfrac{1}{n}\sum_{i=1}^{n}X_i

Balance point of the data; same units as the data. Basis for variance and regression.

Median (sorted data)

\tilde{X}=\begin{cases}X_{(n+1)/2}, & n\text{ odd}\\[2pt]\tfrac{1}{2}\!\left(X_{n/2}+X_{n/2+1}\right), & n\text{ even}\end{cases}

Middle value of ascending-sorted data; resistant to outliers. The mode is the most frequent value.

Population variance

\sigma^2=\dfrac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2

Spread of a full population of size $N$ about true mean $\mu$ . Units are (data units) $^2$ .

Sample variance / standard deviation

s^2=\dfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2,\quad s=\sqrt{s^2}

Unbiased estimate from a sample; the $n-1$ divisor accounts for estimating $\mu$ by $\bar{X}$ .

Standard-normal transform

z=\dfrac{x-\mu}{\sigma}

Number of standard deviations from the mean; converts any normal variable to the unit normal for table lookup.

Unit-normal table areas

F(-z)=1-F(z)=R(z)

$F(z)$ = area left of $z$ , $R(z)$ = area right. Use symmetry for negative $z$ ; probabilities between values are area differences.

Confidence interval for the mean (σ known)

\bar{X}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}

Brackets $\mu$ at confidence $1-\alpha$ . $z_{\alpha/2}$ : 1.645 (90%), 1.960 (95%), 2.576 (99%).

Confidence interval for the mean (σ unknown)

\bar{X}\pm t_{\alpha/2,\,n-1}\dfrac{s}{\sqrt{n}}

Use Student's $t$ with $n-1$ degrees of freedom when $\sigma$ is estimated by $s$ from a small sample; wider than the $z$ interval.

Least-squares slope and intercept

\hat{b}=\dfrac{S_{xy}}{S_{xx}}, \qquad \hat{a}=\bar{y}-\hat{b}\,\bar{x}

Best-fit line $\hat{y}=\hat{a}+\hat{b}x$ through the centroid $(\bar{x},\bar{y})$ ; minimizes squared vertical residuals.

Sums of squares

S_{xx}=\textstyle\sum x_i^2-\tfrac{1}{n}\bigl(\sum x_i\bigr)^2,\;\; S_{xy}=\textstyle\sum x_iy_i-\tfrac{1}{n}\bigl(\sum x_i\bigr)\bigl(\sum y_i\bigr)

Shortcut accumulators for regression; $S_{yy}$ has the analogous form in $y$ .

Correlation and determination

R=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}, \qquad R^2=\text{fraction of }y\text{ variation explained}

$R\in[-1,1]$ measures linear association; $R^2\in[0,1]$ is the goodness of fit. Correlation is not causation.

Worked examples

Center and spread of a strength sample

Problem. Seven concrete cylinders give 28-day strengths (psi): $42, 45, 45, 48, 50, 55, 61$ (in hundreds of psi, i.e., $4200$ – $6100$ psi). Treating these as a sample, find the mean, median, mode, and sample standard deviation (work in the hundreds-of-psi units).

Solution. Mean:

\bar{X}=(42+45+45+48+50+55+61)/7=346/7=49.43

. Sorted with

n=7

(odd), the median is the 4th value:

\tilde{X}=48

. The mode is

45

(appears twice). Sample variance:

\sum(X_i-49.43)^2=261.7

, so

s^2=261.7/(7-1)=43.62

and

s=\sqrt{43.62}=6.60

. Sanity check: the mean (

49.4

) exceeds the median (

48

), consistent with the right-skew from the

61

value; and the sample

s=6.60

exceeds the population

\sigma=\sqrt{261.7/7}=6.11

, as the

n-1

divisor requires. Final:

\bar{X}=49.4

\tilde{X}=48

, mode

=45

s=6.60

(hundreds of psi, i.e.,

\bar{X}=4{,}940

psi,

s=660

psi).

s=\sqrt{\dfrac{261.7}{7-1}}=\sqrt{43.62}=6.60

Normal distribution: fraction below spec

Problem. A concrete's compressive strength is normally distributed with mean $\mu=4{,}000\,\text{psi}$ and standard deviation $\sigma=400\,\text{psi}$ . (a) What fraction of cylinders fall below $3{,}400\,\text{psi}$ ? (b) What fraction lie between $3{,}600$ and $4{,}400\,\text{psi}$ ?

Solution. (a) Standardize:

z=(3400-4000)/400=-1.5

. The area below is

F(-1.5)=R(1.5)

. From the unit-normal table

R(1.5)=0.0668

. So about

6.68\%

of cylinders fall below

3{,}400\,\text{psi}

. (b) The bounds are

\mu\pm 1\sigma

(

z=\pm 1

). The area within

\pm 1\sigma

W(1)=1-2R(1)=1-2(0.1587)=0.6827

. About

68.3\%

lie between

3{,}600

and

4{,}400\,\text{psi}

. Sanity check:

3{,}400

1.5\sigma

below the mean, in the lower tail, so a small fraction (

\approx 7\%

) is expected; and the

\pm 1\sigma

band giving

\approx 68\%

is the classic empirical-rule value. Final: (a)

6.68\%

, (b)

68.3\%

z=\dfrac{3400-4000}{400}=-1.5,\quad P(X<3400)=R(1.5)=0.0668

95% confidence interval for the mean

Problem. A contractor measures the strength of $n=36$ cores; the sample mean is $\bar{X}=4{,}120\,\text{psi}$ . The population standard deviation is known from long experience to be $\sigma=350\,\text{psi}$ . Construct the $95\%$ confidence interval for the true mean strength.

Solution. With

\sigma

known, use the

z

interval. For

95\%

z_{\alpha/2}=1.960

. Standard error:

\sigma/\sqrt{n}=350/\sqrt{36}=350/6=58.33\,\text{psi}

. Half-width:

z_{\alpha/2}\,\sigma/\sqrt{n}=1.960\times 58.33=114.3\,\text{psi}

. Interval:

4{,}120\pm 114\Rightarrow (4{,}006,\;4{,}234)\,\text{psi}

. Sanity check: the interval is centered on

\bar{X}

and symmetric; widening to

99\%

(

z=2.576

) would enlarge it, and a larger

n

would shrink it via

\sqrt{n}

. Final:

4{,}006\,\text{psi}\le\mu\le 4{,}234\,\text{psi}

95\%

confidence.

4120\pm 1.960\dfrac{350}{\sqrt{36}}=4120\pm 114\;\text{psi}

Least-squares line and correlation

Problem. Five load-deflection readings give $x=\{1,2,3,4,5\}$ (kips) and $y=\{2.1,3.9,6.1,7.8,10.2\}$ (mm). Fit $\hat{y}=\hat{a}+\hat{b}x$ by least squares and report the correlation coefficient $R$ .

Solution. Means:

\bar{x}=3

\bar{y}=30.1/5=6.02

. Sums of squares:

S_{xx}=\sum x^2-(\sum x)^2/n=55-225/5=10

;

S_{xy}=\sum xy-(\sum x)(\sum y)/n=110.4-(15)(30.1)/5=110.4-90.3=20.1

;

S_{yy}=\sum y^2-(\sum y)^2/n=221.71-181.20=40.51

. Slope:

\hat{b}=S_{xy}/S_{xx}=20.1/10=2.01\,\text{mm/kip}

. Intercept:

\hat{a}=\bar{y}-\hat{b}\bar{x}=6.02-2.01(3)=-0.010\,\text{mm}

. Correlation:

R=S_{xy}/\sqrt{S_{xx}S_{yy}}=20.1/\sqrt{10\times 40.51}=20.1/20.13=0.999

, so

R^2=0.997

. Sanity check: a near-zero intercept and slope

\approx 2

match the data (each unit of

x

adds about

2

y

), and

R^2=0.997

confirms a near-perfect linear fit. Final:

\hat{y}=-0.010+2.01x

R=0.999

\hat{b}=\dfrac{S_{xy}}{S_{xx}}=\dfrac{20.1}{10}=2.01,\qquad R=\dfrac{20.1}{\sqrt{(10)(40.51)}}=0.999

Common pitfalls

•Dividing sample variance by $n$ instead of $n-1$ . The handbook's sample formula uses $n-1$ ; using $n$ underestimates spread. Only a full population (size $N$ ) divides by $N$ .
•Mixing up the calculator keys: $\sigma_x$ is the population divisor ( $n$ ), $s_x$ the sample divisor ( $n-1$ ). For small $n$ they differ noticeably — press the right one.
•Reading the wrong tail of the unit-normal table. $F(z)$ is area to the LEFT, $R(z)$ to the right; for negative $z$ use $F(-z)=R(z)$ . Always sketch and shade first.
•Using a $z$ interval when $\sigma$ is unknown and $n$ is small. With only $s$ estimated from a small sample, use $t_{\alpha/2,n-1}$ — the $z$ interval is too narrow and overstates precision.
•Forgetting the $\sqrt{n}$ in the standard error. The interval half-width is $z\,\sigma/\sqrt{n}$ , not $z\,\sigma$ ; it shrinks with the square root of sample size, not linearly.
•Reporting variance where the problem wants standard deviation (or vice versa). Variance carries units $^2$ ; take the square root to match the data's units.
•Reading correlation as causation, or confusing $R$ with $R^2$ . $R$ ranges $-1$ to $1$ and keeps the slope's sign; $R^2$ (always $0$ to $1$ ) is the fraction of variance explained.

References

NCEES FE Reference Handbook — Engineering Probability and Statistics: Dispersion, Mean, Median, and Mode Values — Source of the mean, population/sample variance, and standard-deviation formulas.
NCEES FE Reference Handbook — Engineering Probability and Statistics: Normal Distribution, Unit Normal Distribution table, Confidence Intervals — Standard-normal transform, table areas $F$ / $R$ / $W$ , and the $z_{\alpha/2}$ confidence-interval values.
NCEES FE Reference Handbook — Engineering Probability and Statistics: Linear Regression and Goodness of Fit (Least Squares) — Slope/intercept, sums of squares, and the sample correlation coefficient $R$ and $R^2$ .
Montgomery & Runger, Applied Statistics and Probability for Engineers — Degrees of freedom, the t-versus-z choice, and regression diagnostics.