Mathematics and Statistics · Study · FE Civil · FE → PE Prep
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
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 Xˉ 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.
Xˉ=n1i=1∑nXi
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 σ2; if they are a sample drawn to estimate a larger population — the usual engineering case — you divide by n−1 and call it s2
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 μ with spread set by σ. 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 ±1σ, 95% within ±2σ
The standard-normal z-transform
You never integrate the bell curve; you standardize and read a table. The transform z=(x−μ)/σ 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)
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 μ. When the population standard deviation σ is known (or n is large), use the z form; the half-width is zα/2σ/n
Least-squares regression and correlation
Least squares fits the straight line y^=a^+b^x that minimizes the sum of squared vertical residuals. The slope is b^=Sxy/Sxx
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 Xˉ, σx (divisor n), and sx
Key equations
Arithmetic meanXˉ=n1∑i=1nXi
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).
•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
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.
. The
n−1
(Bessel's correction) compensates for having estimated the mean from the same data, and it makes
s2
an unbiased estimate of
σ2
. 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
.
s2=n−11i=1∑n(Xi−Xˉ)2,s=s2
, and
99.7%
within
±3σ
— the quick mental check before you ever open the table.
f(x)=σ2π1e−21(σx−μ)2
, 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=σx−μ
, and the
n
in the denominator is why quadrupling the sample only halves the interval. When
σ
is unknown and estimated by
s
from a small sample, swap in Student's
tα/2,n−1
, which is wider to pay for the extra uncertainty. The handbook tabulates the common
zα/2
:
1.645
for
90%
,
1.960
for
95%
,
2.576
for
99%
.
Xˉ−zα/2nσ≤μ≤Xˉ+zα/2nσ
and the intercept follows from forcing the line through the centroid
(xˉ,yˉ)
, so
a^=yˉ−b^xˉ
. The sample correlation coefficient
R=Sxy/SxxSyy
runs from
−1
to
+1
and measures how tightly the points hug a line; its square
R2
, the coefficient of determination, is the fraction of the variation in
y
explained by the fit. An
R2
near
1
means a strong linear relationship — but never confuse correlation with causation.
b^=SxxSxy,a^=yˉ−b^xˉ,R=SxxSyySxy
(divisor
n−1
) directly. For any normal question, write
z=(x−μ)/σ
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
σ
is known or
n
is large and
t
when
σ
is estimated from a small sample. For regression, accumulate
Sxx
,
Syy
,
Sxy
with the shortcut sums, get the slope first, and remember the line passes through
(xˉ,yˉ)
as a built-in check.
Balance point of the data; same units as the data. Basis for variance and regression.
Median (sorted data)X~={X(n+1)/2,21(Xn/2+Xn/2+1),n oddn even
Middle value of ascending-sorted data; resistant to outliers. The mode is the most frequent value.
Population varianceσ2=N1∑i=1N(Xi−μ)2
Spread of a full population of size N about true mean μ. Units are (data units)2.
Sample variance / standard deviations2=n−11∑i=1n(Xi−Xˉ)2,s=s2
Unbiased estimate from a sample; the n−1 divisor accounts for estimating μ by Xˉ.
Standard-normal transformz=σx−μ
Number of standard deviations from the mean; converts any normal variable to the unit normal for table lookup.
Unit-normal table areasF(−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)Xˉ±zα/2nσ
Confidence interval for the mean (σ unknown)Xˉ±tα/2,n−1ns
Use Student's t with n−1 degrees of freedom when σ is estimated by s from a small sample; wider than the z interval.
Least-squares slope and interceptb^=SxxSxy,a^=yˉ−b^xˉ
Best-fit line y^=a^+b^x through the centroid (xˉ,yˉ)
Sums of squaresSxx=∑xi2−n1(∑xi)2,Sxy=∑xiyi−n1(∑xi)(∑yi)
Shortcut accumulators for regression; Syy has the analogous form in y.
Correlation and determinationR=SxxSyySxy,R2=fraction of y variation explained
R∈[−1,1] measures linear association; R2∈[0,1] is the goodness of fit. Correlation is not causation.
.
Sorted with
n=7
(odd), the median is the 4th value:
X~=48
. The mode is
45
(appears twice).
Sample variance:
∑(Xi−49.43)2=261.7
, so
s2=261.7/(7−1)=43.62
and
s=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
σ=261.7/7=6.11
, as the
n−1
divisor requires. Final:
Xˉ=49.4
,
X~=48
, mode
=45
,
s=6.60
(hundreds of psi, i.e.,
Xˉ=4,940
psi,
s=660
psi).
s=7−1261.7=43.62=6.60
Normal distribution: fraction below spec
Problem. A concrete's compressive strength is normally distributed with mean μ=4,000psi and standard deviation σ=400psi. (a) What fraction of cylinders fall below 3,400psi? (b) What fraction lie between 3,600 and 4,400psi?
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
z=4003400−4000=−1.5,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 Xˉ=4,120psi. The population standard deviation is known from long experience to be σ=350psi. Construct the 95% confidence interval for the true mean strength.
Solution. With σ known, use the z interval. For 95%, zα/2=1.960.
Standard error:
4120±1.96036350=4120±114psi
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 y^=a^+b^x by least squares and report the correlation coefficient R.
Solution. Means: xˉ=3, yˉ=30.1/5=6.02.
Sums of squares: Sxx=∑x2−(∑x)2/n=55−225/5=10
•Mixing up the calculator keys: σx is the population divisor (n), sx 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 σ is unknown and n is small. With only s estimated from a small sample, use tα/2,n−1 — the z interval is too narrow and overstates precision.
•Forgetting the n in the standard error. The interval half-width is zσ/n, not zσ; it shrinks with the square root of sample size, not linearly.
•Reporting variance where the problem wants standard deviation (or vice versa). Variance carries units2; take the square root to match the data's units.
•Reading correlation as causation, or confusing R with R2. R ranges −1 to 1 and keeps the slope's sign; R2 (always 0 to 1) is the fraction of variance explained.
; minimizes squared vertical residuals.
.
So about
6.68%
of cylinders fall below
3,400psi
.
(b) The bounds are
μ±1σ
(
z=±1
). The area within
±1σ
is
W(1)=1−2R(1)=1−2(0.1587)=0.6827
.
About
68.3%
lie between
3,600
and
4,400psi
.
Sanity check:
3,400
is
1.5σ
below the mean, in the lower tail, so a small fraction (