Probability and Statistics

A. Probability distributions

Probability and Distributions

The laws of probability, discrete versus continuous distributions, the binomial and Poisson models, the normal, and the expected value and variance of a random variable.

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B. Measures of central tendencies and dispersions

Descriptive Statistics, the Normal Distribution and Confidence Intervals

Mean, median, mode, variance and standard deviation, the standard-normal z transform, and the confidence interval that turns a sample into a statement about the true mean.

Almost every quantity an engineer measures scatters: a batch of bolt diameters, the yield strength of ten coupons, the flow through a valve on twenty trials. Descriptive statistics compress that scatter into a center and a spread, and the normal distribution lets you turn the spread into probabilities. On the FE Mechanical exam this area is pure points if you keep three things straight: which average you are computing, whether you are looking at a population or a sample, and whether the deviation you have is for one item or for the average of

n

items. The NCEES FE Reference Handbook (Engineering Probability and Statistics) supplies every formula and a unit-normal table; the skill being tested is choosing and reading them quickly, not memorizing them.

Mean, median, and mode

The arithmetic mean

\bar{x}

is the sum divided by the count and is the balance point of the data. The median is the middle value once the data are sorted (the average of the two middle values when

n

is even), and it ignores how far the extremes lie, so it resists outliers that drag the mean. The mode is simply the most frequent value. For a symmetric distribution the three coincide; when a few large values pull the mean above the median, the data are right-skewed. A subtle but tested point: when values carry unequal importance, use the weighted mean

\bar{x}=\sum w_i x_i/\sum w_i

, not the plain average.

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \qquad \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Variance and standard deviation

Spread is measured by averaging the squared deviations from the mean. Squaring is deliberate: it keeps positive and negative deviations from canceling and weights large excursions heavily. The standard deviation

\sigma

(or

s

) is the square root, returning to the units of the data, so a deflection scatter is reported in millimeters, not millimeters-squared. The single most common FE trap here is the divisor. A population variance divides the sum of squared deviations by

N

; a sample variance divides by

n-1

(Bessel's correction) because using the sample mean removes one degree of freedom. When the problem says 'a sample of ten parts,' use

n-1

.

s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2, \qquad \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2

The normal distribution

The bell-shaped normal (Gaussian) distribution describes the limiting behavior of sums and averages of independent effects, which is why machined dimensions, measurement errors, and material strengths so often look normal. It is fully specified by its mean

\mu

(the peak and the axis of symmetry) and standard deviation

\sigma

(the distance from the mean to each inflection point). The famous empirical rule follows directly from its shape: about

68\%

of the probability lies within

\pm 1\sigma

,

95\%

within

\pm 2\sigma

, and

99.7\%

within

\pm 3\sigma

of the mean.

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right]

The standard-normal z transform

Because no two normal curves share the same axis, the exam works in a single reference curve: the standard normal, with

\mu=0

and

\sigma=1

. Any value

x

is mapped to its standard score

z

, which counts how many standard deviations

x

sits from the mean. You then read the handbook's unit-normal table, where

F(z)

is the area to the left (cumulative probability) and

R(z)=1-F(z)

is the right-tail area. Symmetry gives

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

, so a single column of positive

z

values handles every case. A value at

z=2

has

R=0.0228

, meaning only

2.28\%

of the population exceeds it.

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

From scatter to a statement: the confidence interval

A sample mean is itself uncertain; the spread of the sample mean is the standard error

\sigma/\sqrt{n}

, which shrinks as the sample grows. A confidence interval brackets the unknown population mean

\mu

with a stated probability. When

\sigma

is known you use the normal

z

; when

\sigma

is unknown and estimated by

s

from a small sample you must use Student's

t

with

n-1

degrees of freedom, whose fatter tails widen the interval to pay for the extra uncertainty. A

95\%

interval does not mean a

95\%

chance the mean is inside this one interval; it means

95\%

of intervals built this way would capture

\mu

.

\bar{x} - t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}} \;\le\; \mu \;\le\; \bar{x} + t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}

Choosing z versus t

The decision rule is mechanical. If the population standard deviation

\sigma

is given, or the sample is large (commonly

n\ge 30

so

s\approx\sigma

), use

z

from the unit-normal table:

\bar{x}\pm z_{\alpha/2}\,\sigma/\sqrt{n}

. If

\sigma

is unknown and

n

is small, use

t_{\alpha/2,n-1}

. The common two-sided critical values are worth recognizing on sight:

z_{0.025}=1.960

for

95\%

and

z_{0.005}=2.576

for

99\%

. For small samples the

t

value is always larger than the matching

z

(e.g.,

t_{0.025,9}=2.262

versus

1.960

), and it converges to

z

as the degrees of freedom grow.

Exam strategy

First label the problem: population (

N

,

\sigma

, divide by

N

) or sample (

n

,

s

, divide by

n-1

). For any probability question on a normal variable, immediately compute

z=(x-\mu)/\sigma

, sketch the bell with the shaded region, then decide whether you need the left area

F(z)

, the right area

R(z)

, or a difference

F(z_2)-F(z_1)

. For a confidence interval, pick

z

or

t

by whether

\sigma

is known, divide

s

(or

\sigma

) by

\sqrt{n}

for the standard error, and never by

n

. Carry units through and report to three significant figures.

Key equations

Arithmetic mean

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

Center of the data; $x_i$ are observations in their physical units, $n$ is the count.

Weighted mean

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Average when observations carry unequal weights $w_i$ (counts, importances).

Sample variance and standard deviation

s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2, \quad s=\sqrt{s^2}

Spread of a sample; divide by $n-1$ . $s$ returns to the units of $x$ .

Population variance

\sigma^2 = \frac{1}{N}\sum (x_i-\mu)^2

Use when the full population of size $N$ and its mean $\mu$ are known; divide by $N$ .

Coefficient of variation

CV = \frac{s}{\bar{x}}

Dimensionless relative scatter; lets you compare variability across different units or scales.

Normal density

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

Bell curve fixed by mean $\mu$ and standard deviation $\sigma$ ; area under it is probability.

Standard-normal transform

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

Number of standard deviations $x$ is from the mean; the key into the unit-normal table.

Tail areas

R(z) = 1 - F(z), \quad F(-z) = 1 - F(z)

$F(z)$ = area left of $z$ ; $R(z)$ = right tail. Symmetry handles negative $z$ .

Standard error of the mean

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

Scatter of the sample mean; use $s$ for $\sigma$ when the population value is unknown.

Confidence interval (

\sigma

known)

\bar{x} \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}

Two-sided CI for $\mu$ with known $\sigma$ ; $z_{0.025}=1.960$ for 95%.

Confidence interval (

\sigma

unknown)

\bar{x} \pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}

Small-sample CI; use Student's $t$ with $n-1$ degrees of freedom.

Worked examples

Mean, median, and sample standard deviation

Problem. Ten machined shafts have measured diameters (mm): $12.1,\ 12.4,\ 11.8,\ 12.6,\ 12.0,\ 12.3,\ 11.9,\ 12.5,\ 12.2,\ 12.2$ . Find the mean, median, and sample standard deviation.

Solution. Sum

=122.0\,\text{mm}

, so

\bar{x}=122.0/10 = 12.20\,\text{mm}

. Sorted:

11.8,\,11.9,\,12.0,\,12.1,\,\mathbf{12.2,\,12.2},\,12.3,\,12.4,\,12.5,\,12.6

. With

n=10

even, the median is the average of the 5th and 6th values:

(12.2+12.2)/2 = 12.2\,\text{mm}

. Sum of squared deviations:

\sum(x_i-\bar{x})^2 = 0.600\,\text{mm}^2

. Sample variance

s^2 = 0.600/(10-1)=0.0667\,\text{mm}^2

.

s=\sqrt{0.0667} = 0.258\,\text{mm}

. Sanity check:

s

is about

2\%

of the mean, consistent with a tight

\pm 0.4\,\text{mm}

data spread; units are mm as expected.

Normal probability via the z-score

Problem. A bolt diameter is normally distributed with $\mu = 10.00\,\text{mm}$ and $\sigma = 0.05\,\text{mm}$ . The upper spec limit is $10.10\,\text{mm}$ . What fraction of bolts exceed the limit?

Solution. Standardize the limit:

z = (10.10-10.00)/0.05 = 2.00

. The fraction exceeding is the right-tail area

R(2.00)

. From the unit-normal table,

F(2.00)=0.9772

, so

R(2.00)=1-0.9772 = 0.0228

. Fraction exceeding

= 0.0228

, i.e.

2.28\%

. Sanity check: the limit is exactly

2\sigma

above the mean; the empirical rule puts

\approx 2.3\%

in one tail beyond

2\sigma

, matching.

95% confidence interval for the mean (small sample)

Problem. Using the same ten shafts ( $\bar{x}=12.20\,\text{mm}$ , $s=0.258\,\text{mm}$ , $n=10$ ), build a $95\%$ confidence interval for the true mean diameter. $\sigma$ is unknown.

Solution.

\sigma

unknown and

n

small, so use

t

. With

n-1=9

degrees of freedom and

\alpha/2=0.025

,

t_{0.025,9}=2.262

. Standard error:

s/\sqrt{n} = 0.258/\sqrt{10} = 0.0817\,\text{mm}

. Margin:

2.262 \times 0.0817 = 0.185\,\text{mm}

. Interval:

12.20 \pm 0.185 \Rightarrow (12.0,\ 12.4)\,\text{mm}

— more precisely

(12.015,\ 12.385)\,\text{mm}

. Sanity check: the half-width is about

0.7s

, reasonable for

n=10

; had we wrongly used

z=1.96

the interval would be too narrow.

Common pitfalls

•Dividing the sample variance by $n$ instead of $n-1$ . A 'sample of $n$ ' always uses $n-1$ in the denominator; only a full population divides by $N$ .
•Using $z$ when $\sigma$ is unknown and the sample is small. With estimated $s$ and small $n$ , the correct critical value is $t_{\alpha/2,n-1}$ , which is larger than $z$ .
•Confusing the left area $F(z)$ with the right tail $R(z)$ . 'Fraction exceeding' is $R(z)=1-F(z)$ ; always sketch the shaded region before reading the table.
•Dividing by $n$ instead of $\sqrt{n}$ in the standard error. The spread of the sample mean is $\sigma/\sqrt{n}$ , not $\sigma/n$ .
•Forgetting the median rule for even $n$ : it is the average of the two middle values, not a single middle item.
•Reading the table for negative $z$ directly. Use $F(-z)=1-F(z)$ instead of looking up a nonexistent negative entry.
•Stating a 95% CI as a probability about this single interval. The 95% refers to the long-run capture rate of the procedure, not to one realized interval.

References

NCEES FE Reference Handbook — Engineering Probability and Statistics
Montgomery & Runger, Applied Statistics and Probability for Engineers — descriptive statistics, normal distribution, and confidence intervals

C. Expected value (weighted average) in decision making

Expected Value in Decision Making

The probability-weighted average, expected monetary value, decision trees, and how to choose rationally among uncertain alternatives.

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D. Regression (linear, multiple), curve fitting, and goodness of fit

Linear Regression, Curve Fitting and Goodness of Fit

Least-squares slope and intercept, the correlation coefficient and R-squared, residuals, linearizing nonlinear data, and the standard error of the estimate.

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