Engineering Data Analysis (Probability and Statistics) - Cheat Sheet
Last-minute revision companion for the PRC Civil Engineer Licensure Examination. Covers descriptive statistics, probability rules, counting methods, and critical distributions. Board-exam ready with step-by-step templates and common pitfalls.
Descriptive Statistics
Formulas
Mean:
Meaning
x = data values; n = count of data points; = arithmetic average
Watch out
Not the same as median (middle value) or mode (most frequent). Use median if outliers are present; use mode for categorical data.
When to use
To find the central location of a dataset; always the first step in data analysis.
Population Variance:
Meaning
σ² = variance; x = each value; = mean; n = population size
Watch out
Use n in denominator for population, NOT (n−1). Variance is squared units — must take square root for standard deviation.
When to use
When analyzing the entire population; emphasis on actual squared deviations.
Sample Variance:
Meaning
s² = sample variance; (n−1) = degrees of freedom correction (Bessel's correction)
Watch out
CRITICAL: Use (n−1) for samples, n for populations. This is the #1 mistake on exams. Even one wrong denominator fails the problem.
When to use
When working with a sample from a larger population; (n−1) reduces bias in estimation.
Population Std Dev:
Meaning
σ = standard deviation (square root of variance); all other symbols as before
Watch out
Always a positive number. Must take square root after calculating variance. Common error: forgetting the square root step.
When to use
To measure spread in the same units as original data; easier to interpret than variance.
Sample Std Dev:
Meaning
s = sample standard deviation using (n−1) denominator
Watch out
Remember: sample uses (n−1); population uses n. If problem says 'sample,' the denominator is (n−1).
When to use
When reporting spread from a sample; use (n−1) denominator.
Coefficient of Variation:
Meaning
s = sample std dev; = mean; CV = relative variability as percentage
Watch out
Divide std dev by mean, then multiply by 100% for percentage. If mean is zero or negative, CV is undefined or misleading.
When to use
To compare spread between datasets with different means or units; dimensionless measure.
Common values
Value
z = ±1.96
Symbol
z₀.₀₂₅
Quantity
Standard normal critical values (α = 0.05, two-tailed)
Value
z = ±2.576
Symbol
z₀.₀₀₅
Quantity
Standard normal critical values (α = 0.01, two-tailed)
Value
±1σ (68%), ±2σ (95%), ±3σ (99.7%)
Symbol
Normal distribution
Quantity
68-95-99.7 rule coverage
Important facts
- Variance is always non-negative; zero only if all values are identical.
- Standard deviation has the same units as the original data; variance is in squared units.
- For a normal distribution, 68% of data within ±1σ, 95% within ±2σ, 99.7% within ±3σ (68-95-99.7 rule).
- Mean is sensitive to outliers; median is robust (resistant to extreme values).
- Increasing n (sample size) decreases standard error of the mean: SE = σ/√n.
Key definitions
Median
Example
For [2, 5, 7, 9, 15], median = 7. For [2, 5, 7, 9], median = (5+7)/2 = 6.
Middle value when data arranged in order; 50th percentile; unaffected by outliers.
Mode
Example
In [1, 2, 2, 3, 3, 3, 5], mode = 3. Bimodal if two values tie.
Most frequently occurring value in a dataset.
Range
Example
For [3, 7, 12, 20], range = 20 − 3 = 17.
Difference between maximum and minimum values: max − min.
Quartile (Q1, Q2, Q3)
Example
Interquartile range (IQR) = Q3 − Q1; outliers flagged if > Q3 + 1.5·IQR.
Q1 = 25th percentile; Q2 = median (50th); Q3 = 75th percentile; divide data into four equal parts.
Diagrams to know
- Box-and-whisker plot: shows median, Q1, Q3, whiskers, and outliers.
- Histogram: bars show frequency distribution; shape indicates skewness.
- Normal distribution curve: symmetric bell curve centered at mean μ.
Probability Fundamentals
Formulas
Classical Probability:
Meaning
P(A) = probability of event A; range: 0 ≤ P ≤ 1
Watch out
Numerator must be ≤ denominator. Probability is always between 0 and 1, inclusive. Count outcomes carefully to avoid double-counting.
When to use
When all outcomes equally likely (e.g., fair die, deck of cards, coin flip).
Complement Rule:
Meaning
A^c = complement of A (opposite event); P(A) + P(A^c) = 1
Watch out
P(not A) is often simpler than P(A) directly. Always check if complement rule saves time.
When to use
When it's easier to find probability of the opposite event (e.g., 'at least one' = 1 − 'none').
Addition Rule (Mutually Exclusive): if A and B cannot both occur
Meaning
A ∪ B = A or B (union); mutually exclusive = disjoint events
Watch out
Only use this form if events are mutually exclusive. Otherwise, must subtract P(A ∩ B).
When to use
When events cannot happen simultaneously (e.g., a card cannot be both king and queen).
Addition Rule (General):
Meaning
P(A ∩ B) = probability both A and B occur (intersection, overlap)
Watch out
CRITICAL: Forget the subtraction term and the answer is wrong. Overlap must be removed once. If mutually exclusive, P(A ∩ B) = 0.
When to use
For any two events (inclusive or exclusive); subtract overlap to avoid double-counting.
Multiplication Rule (Independent Events):
Meaning
A and B are independent if outcome of one doesn't affect the other
Watch out
Only multiply if independent. If dependent (without replacement), use conditional probability: P(A ∩ B) = P(A) × P(B|A).
When to use
Sequential events where first result doesn't change odds of second (with replacement in sampling).
Conditional Probability:
Meaning
P(A|B) = probability of A given B already occurred; P(B) > 0
Watch out
Denominator is P(B), the event that has occurred. Numerator is intersection P(A ∩ B), not just P(A).
When to use
When one event has already happened and we update the probability of another.
Bayes' Theorem:
Meaning
P(A) = prior; P(B|A) = likelihood; P(B) = total probability; P(A|B) = posterior
Watch out
P(B) often calculated using law of total probability: P(B) = P(B|A)P(A) + P(B|A^c)P(A^c). Do not confuse P(A|B) with P(B|A).
When to use
Diagnostic tests, reliability updates, engineering fault diagnosis; reverses condition.
Common values
Value
P(sum=7) = 6/36 = 1/6 ≈ 0.167; P(sum=2 or 12) = 1/36 each
Symbol
Classical probability
Quantity
Probability of rolling sum = k on two dice
Value
P(King) = 4/52 = 1/13; P(Heart) = 13/52 = 1/4; P(King or Heart) = 16/52 = 4/13
Symbol
Addition rule (with overlap)
Quantity
Probability of drawing from standard deck
Important facts
- If P(A) = 0, event A is impossible. If P(A) = 1, event A is certain.
- Independence ≠ mutually exclusive. Two events can be both independent and non-mutually exclusive.
- P(A and B and C) = P(A) × P(B|A) × P(C|A,B) for sequential dependent events.
- Law of Total Probability: P(B) = Σ P(B|Aᵢ) × P(Aᵢ) where events Aᵢ partition the sample space.
Key definitions
Independent Events
Example
Rolling a die twice: second roll unaffected by first. Sampling with replacement.
Events A and B where outcome of one does not affect the probability of the other; P(A|B) = P(A).
Dependent (Conditional) Events
Example
Drawing cards without replacement: first card removes one from deck, affecting next draw.
Events where outcome of one changes the probability of the other; P(A|B) ≠ P(A).
Mutually Exclusive (Disjoint) Events
Example
A coin flip: cannot be both heads and tails simultaneously.
Events that cannot both occur; P(A ∩ B) = 0.
Sample Space (S)
Example
For two dice: S has 36 outcomes (1,1) through (6,6). For a coin: S = {H, T}.
Set of all possible outcomes of an experiment.
Diagrams to know
- Venn diagram: overlapping circles for events A, B; shade union, intersection, complement.
- Tree diagram: branches show sequential outcomes and conditional probabilities.
- Contingency table: rows and columns show joint, marginal, and conditional frequencies.
Reactions or equations
Note
Foundation of the complement rule; rearrange as P(A^c) = 1 − P(A) when complement easier to calculate.
Conditions
Always true for any event A in any probability space
Note
Conditional form essential for dependent events (without replacement, sequential constraints).
Conditions
General form for any two events; becomes P(A) × P(B) if A, B independent
Counting & Combinatorics
Formulas
Factorial:
Meaning
n! = product of all positive integers ≤ n; 0! = 1 by definition
Watch out
Factorial grows very fast: 10! = 3,628,800. Common error: forgetting 0! = 1. Do not confuse with repeated factorials like (n!)!.
When to use
Building block for permutations and combinations; counts arrangements of n distinct items.
Permutations (Order Matters):
Meaning
n = total items; r = items selected; order of selection is important
Watch out
P(n,r) counts sequences. If selecting 3 from 5: P(5,3) = 5!/(5−3)! = 120/2 = 60, not 5+4+3. Order MATTERS.
When to use
Arrangements, passwords, seating in specific seats, 'first place, second place, third place' rankings.
Combinations (Order Does NOT Matter):
Meaning
n = total items; r = items selected; order is irrelevant
Watch out
C(n,r) ≤ P(n,r) always. C(5,2) = 10 (pairs) but P(5,2) = 20 (ordered pairs). Common mistake: using permutations when combinations needed.
When to use
Committees, subsets, selections where arrangement doesn't matter, 'choose r from n'.
Permutations with Repetition:
Meaning
n = total objects; nᵢ = frequency of identical object i; accounts for indistinguishable items
Watch out
Denominator is product of factorials of each frequency. If all n items distinct, denominator is 1 (reduces to n!). Must identify which items are identical.
When to use
Arrangements of letters with repeats (e.g., 'MISSISSIPPI'), objects in positions where some are identical.
Multiplication Principle:
Meaning
nᵢ = number of choices at step i; total ways = product of choices per step
Watch out
Applies only if choices at each step are independent. If later choices depend on earlier ones, use conditional logic or tree diagrams.
When to use
Sequential decisions (e.g., choosing shirt AND pants AND shoes).
Addition Principle:
Meaning
nᵢ = count in category i; sum over disjoint (non-overlapping) categories
Watch out
Categories must be disjoint (no overlap). If overlap exists, subtract: total = n₁ + n₂ − (overlap).
When to use
Mutually exclusive outcomes (e.g., rolling a 5 OR rolling a 6 on one die).
Common values
Value
0!=1, 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 10!=3,628,800
Symbol
n!
Quantity
Small factorials
Value
C(n,2) = n(n−1)/2; counts unordered pairs from n items
Symbol
Handshake lemma
Quantity
Binomial coefficient C(n,2)
Important facts
- C(n,r) = C(n,n−r) (symmetry: choosing r same as leaving n−r behind).
- P(n,r) = C(n,r) × r! (permutations = combinations × ways to arrange r items).
- C(n,1) = n; C(n,0) = 1; C(n,n) = 1 (edge cases).
- Sum of all combinations: Σ C(n,r) for r=0 to n equals 2ⁿ (total subsets).
Key definitions
Permutation
Example
Passwords, race positions. P(5,3) = 60 ways to assign 1st, 2nd, 3rd place from 5 runners.
Ordered arrangement of r items from n distinct items; each arrangement is unique by position.
Combination
Example
Committee selection, choosing toppings. C(5,3) = 10 ways to choose 3 people from 5 (no rank difference).
Unordered selection of r items from n items; order irrelevant.
Identical (Indistinguishable) Objects
Example
Letters in 'BOOK': 4!/(1!×1!×2!) = 24/2 = 12 distinct arrangements (two O's identical).
Repeated elements in an arrangement reduce the count; account via division by factorial of each frequency.
Mutually Exclusive Categories
Example
A student is either BSCE or BSME, not both in the same instant.
Non-overlapping groups; an item belongs to exactly one category.
Diagrams to know
- Pascal's triangle: rows show binomial coefficients C(n,r); each entry sum of two above.
- Tree diagram: branches for sequential choices; multiply along path for total outcomes.
Reactions or equations
Note
Rearrange: C(n,r) = P(n,r) / r!. Use to verify or derive one from the other.
Conditions
Always true; connects permutations and combinations via orderings of r items
Note
Recursive relation used in Pascal's triangle; useful for mental calculation or verification.
Conditions
Pascal's identity; true for all valid n, r
Probability Distributions — Discrete
Formulas
Binomial Distribution:
Meaning
n = number of trials; x = successes (0 ≤ x ≤ n); p = probability of success per trial; (1−p) = q = failure probability
Watch out
Requires: independent trials, constant p, exactly n trials, two outcomes. Do NOT use if trials dependent (e.g., drawing without replacement unless n << population). Common error: forgetting the binomial coefficient C(n,x).
When to use
Fixed number of independent trials, each with two outcomes (pass/fail, yes/no, defective/good). Must have constant p.
Binomial Mean:
Meaning
Expected number of successes in n trials
Watch out
Mean might not be an integer even though X itself must be integer. μ = np is simple; variance = np(1−p).
When to use
Quick estimate of central value for binomial distribution.
Binomial Variance:
Meaning
Spread around mean for binomial; σ = √[np(1−p)]
Watch out
Variance is np(1−p), NOT n times individual variance. If p = 0.5, variance is maximum = n/4.
When to use
Measure dispersion; larger p away from 0.5 gives smaller variance.
Poisson Distribution:
Meaning
λ (lambda) = average rate/count over the interval; x = actual count (0, 1, 2, ...); e ≈ 2.71828
Watch out
x can be any non-negative integer (0, 1, 2, ...). Poisson has only one parameter λ; mean = variance = λ. Must calculate e^(−λ) accurately. Common error: confusing λ with x.
When to use
Rare events over time/space interval (e.g., defects per roll, accidents per year, bacteria per mL). Approximates binomial when n large, p small, np = λ moderate.
Poisson Mean & Variance:
Meaning
Poisson is unique: mean equals variance.
Watch out
If data variance >> mean, distribution over-dispersed; if variance << mean, under-dispersed. Poisson is sweet spot (mean = variance).
When to use
Quick check: if calculated mean ≠ variance in data, distribution may not be Poisson.
Geometric Distribution: (first success on trial x)
Meaning
p = success probability; x = trial number of first success (x = 1, 2, 3, ...)
Watch out
Parameterization varies: some define X as failures before first success, shifting by 1. Check definition. Mean = 1/p; variance = (1−p)/p².
When to use
Time until first success; how many flips until first heads; number of attempts to pass.
Common values
Value
e ≈ 2.71828
Symbol
e
Quantity
e (Euler's number)
Value
λ=1: e^(−1)≈0.368; λ=2: e^(−2)≈0.135; λ=3: e^(−3)≈0.050
Symbol
Poisson
Quantity
Poisson cumulative probabilities e^(−λ) for common λ
Value
P(X≤5)=0.623; P(X≤4)=0.377 (symmetric around mean=5)
Symbol
Binomial (n=10, p=0.5)
Quantity
Binomial cumulative P(X≤k) for n=10, p=0.5
Important facts
- Binomial approximates normal when n large and 0.2 < p < 0.8; rule of thumb: np ≥ 5 and n(1−p) ≥ 5.
- Poisson approximates binomial when n large, p small, np = λ moderate (typically n ≥ 20, p ≤ 0.05).
- Geometric memoryless property: P(X > n+m | X > n) = P(X > m); failures before first success independent of past.
- Probability Σ P(X = x) = 1 for any discrete distribution (all probabilities sum to 1).
Key definitions
Binomial Trial
Example
One coin flip (heads = success, p = 0.5), or testing if a concrete sample fails (p = probability of failure).
Single experiment with exactly two outcomes (success/failure); probability p of success constant across trials.
Random Variable (X)
Example
X = number of defective units in sample of 10; X = strength of steel rod (continuous).
Function assigning numerical value to each outcome in sample space; discrete (integer values) or continuous.
Probability Mass Function (PMF)
Example
For fair die: P(X = k) = 1/6 for k = 1, 2, ..., 6; sum of all probabilities = 1.
For discrete X: function giving P(X = x) for each value x; Σ P(X = x) = 1.
Expected Value (Mean) of Discrete Distribution
Example
E[fair die] = 1(1/6) + 2(1/6) + ... + 6(1/6) = 3.5.
μ = E[X] = Σ x · P(X = x); weighted average of all values.
Diagrams to know
- Binomial PMF bar chart: peaks near np; shape depends on p (skewed if p far from 0.5).
- Poisson PMF: right-skewed for small λ; more symmetric as λ increases.
- Cumulative distribution function (CDF): S-shaped curve from 0 to 1; non-decreasing.
Reactions or equations
Note
P(X < k) = P(X ≤ k−1). Use tables for binomial/Poisson to avoid repeated calculation.
Conditions
Cumulative distribution function (CDF) for discrete X; sum up to and including k
reduces to hypergeometric if sampling without replacement
Note
Binomial: with replacement (or infinite population). Hypergeometric: without replacement (finite population, significant extraction).
Conditions
Hypergeometric: P(X=x) = [C(K,x)·C(N−K,n−x)] / C(N,n); N = population size, K = successes in population, n = sample size
Probability Distributions — Continuous
Formulas
Normal Distribution:
Meaning
μ = mean; σ = standard deviation; x = any real number; probability density at x
Watch out
This is the probability density function (PDF), not probability itself. Use area under curve (integration) or tables. P(X = exact value) = 0 (infinitesimal). Always use z-standardization for tables.
When to use
Continuous data approximately bell-shaped (heights, weights, measurement errors, test scores, many engineering processes).
Standard Normal (Z-score):
Meaning
z = standardized score; shifts mean to 0 and scales by σ to make SD = 1.
Watch out
z > 0 if x > μ; z < 0 if x < μ. Numerator (x − μ) is difference from mean; denominator σ is std dev. Common error: wrong sign or dividing by variance instead of SD.
When to use
Convert any normal distribution to standard normal (μ=0, σ=1) to use standard z-tables.
Probability from Z-table:
Meaning
Φ(z) = cumulative probability from standard normal table (area left of z)
Watch out
Standard normal tables give left-tail area Φ(z) = P(Z ≤ z). For right tail: P(Z > z) = 1 − Φ(z). Table only for z ≥ 0 in many references; use symmetry for negative z.
When to use
Look up z-value in table to find cumulative probability; Φ(−z) = 1 − Φ(z) by symmetry.
Central Limit Theorem: for large n
Meaning
Distribution of sample mean approaches normal regardless of original distribution; SE = σ/√n
Watch out
The SAMPLE MEAN distribution is normal, not individual values. SE = σ/√n, not σ. As n increases, SE decreases (sampling distribution narrows).
When to use
Large sample sizes (n ≥ 30 typical), any underlying distribution. Allows use of normal distribution for inference.
Uniform Distribution: for
Meaning
a = lower bound; b = upper bound; constant probability density across interval.
Watch out
Mean μ = (a+b)/2; variance σ² = (b−a)²/12. f(x) = 0 outside [a,b]. This is NOT the same as discrete uniform (equally likely integers).
When to use
When all values equally likely over interval (e.g., arrival time in a 1-hour window, random angle 0° to 360°).
Exponential Distribution: for
Meaning
λ = rate parameter; x = time/distance; models waiting time, lifetime.
Watch out
Mean μ = 1/λ; variance σ² = 1/λ². Not symmetric (right-skewed). Cumulative: F(x) = 1 − e^(−λx).
When to use
Time between events (failures, arrivals); memoryless property: P(X > s+t | X > s) = P(X > t).
Confidence Interval (Large Sample):
Meaning
ᾱ = significance level; z_{α/2} = critical value; s = sample SD; n = sample size
Watch out
For small samples, use t-distribution instead of z (t has wider tails, accounts for uncertainty). Margin of error = z_{α/2} · SE; wider interval if lower confidence or larger variation.
When to use
Estimate population mean with confidence level 1−α (e.g., 95% CI uses z₀.₀₂₅ = 1.96).
Common values
Value
z₀.₁₀=1.28; z₀.₀₅=1.645; z₀.₀₂₅=1.96; z₀.₀₀₅=2.576
Symbol
z_{α}
Quantity
Standard normal critical values
Value
Φ(0)=0.5; Φ(1)≈0.8413; Φ(2)≈0.9772; Φ(3)≈0.9987
Symbol
Φ(z)
Quantity
Standard normal percentiles
Value
t₀.₀₂₅,₉₉=1.98 (approaches z=1.96 as df increases)
Symbol
t_{α,df}
Quantity
t-distribution critical values (df=∞ limit to normal)
Important facts
- 68-95-99.7 Rule: In normal distribution, 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ.
- Standard normal table Φ(z) = P(Z ≤ z); by symmetry: Φ(−z) = 1 − Φ(z).
- Sample mean has SD = σ/√n (standard error); smaller than individual observation SD.
- For confidence intervals: margin of error E = z_{α/2} · SE; smaller margin requires larger n or lower confidence.
Key definitions
Probability Density Function (PDF)
Example
Normal distribution curve; probability P(a ≤ X ≤ b) = ∫_a^b f(x)dx (area under curve from a to b).
For continuous X: f(x) ≥ 0 such that ∫_{−∞}^{∞} f(x)dx = 1; area under curve = probability.
Cumulative Distribution Function (CDF)
Example
Standard normal Φ(z); F(x) ranges 0 to 1, non-decreasing.
F(x) = P(X ≤ x); for continuous X: F'(x) = f(x) (derivative of CDF is PDF).
Standard Normal Distribution
Example
Z-tables give P(Z ≤ z) for standard normal; used universally for normal probability.
Normal with μ = 0 and σ = 1; denoted Z; any normal can be standardized to this.
Confidence Level
Example
95% CI: z₀.₀₂₅ = 1.96 (two-tailed); 99% CI: z₀.₀₀₅ = 2.576.
Probability (1−α) that a confidence interval contains the true parameter; e.g., 95% confidence level means α = 0.05.
Diagrams to know
- Normal distribution bell curve: symmetric about μ; inflection points at μ±σ.
- Standard normal curve: centered at 0; marked regions for ±1σ, ±2σ, ±3σ.
- Exponential decay curve: steep at x=0, long right tail; always positive.
- Uniform rectangle: constant height 1/(b−a) from a to b, zero elsewhere.
Reactions or equations
Note
Subtract lower z-score probability from upper to get probability in interval.
Conditions
For normal X with mean μ, SD σ; standardize both endpoints
Note
Equivalent to P(Z > z) where z = (k−μ)/σ.
Conditions
Right-tail probability; uses complement rule
Statistical Inference & Hypothesis Testing
Formulas
Standard Error of Mean: or
Meaning
σ or s = population or sample SD; n = sample size; SE = standard deviation of sample mean
Watch out
SE is NOT the same as population SD (σ). SE decreases as √n, so doubling n decreases SE by √2 (≈1.41). Do NOT confuse SE with sample variance.
When to use
In confidence intervals and hypothesis tests; smaller SE with larger n or less variation.
Test Statistic (Z-test for mean):
Meaning
ᾱ₀ = hypothesized mean; = sample mean; SE = standard error
Watch out
Numerator is difference from null hypothesis μ₀, not from data mean. Use z if n ≥ 30 or σ known; use t for small samples with unknown σ.
When to use
Test if sample mean significantly different from hypothesized value; assume σ known or n large.
Test Statistic (t-test for mean):
Meaning
s = sample SD; df = n−1 degrees of freedom; use t-distribution instead of normal
Watch out
t-distribution has heavier tails than normal; critical values larger than z for same α. Always specify df when using t-tables. As n → ∞, t → z.
When to use
Small samples (n < 30), unknown population SD, approximately normal data.
P-value (Two-tailed): for normal; analogous for t
Meaning
Probability of observing test statistic ≥ |z_obs| if null hypothesis true; smaller p suggests stronger evidence against H₀
Watch out
p-value is NOT probability that H₀ is true. One-tailed test: p = P(Z > z_obs) or P(Z < z_obs); two-tailed: p = 2×P(|Z| > |z_obs|). Common error: confusing p-value magnitude with effect size.
When to use
Decide whether to reject H₀ by comparing p-value to significance level α; if p ≤ α, reject.
Confidence Interval for Mean:
Meaning
t_{α/2,df} = critical t-value; df = n−1; S = sample std dev
Watch out
Use (n−1) degrees of freedom, NOT n. For 95% CI, α=0.05 so look up t₀.₀₂₅. The interval [lower, upper] has (1−α)×100% confidence of containing true μ.
When to use
Small samples (n < 30) with unknown σ; for large samples, replace t with z.
Two-Sample Test (Independent, Unequal Variances):
Meaning
, , = mean, SD, size of sample i; Welch's t-test (no pooling)
Watch out
Degrees of freedom (Welch's approximation) more complex; see tables. Do NOT assume equal variances without Levene's test. If equal variances confirmed, use pooled t.
When to use
Compare means of two independent groups; assumes normality, not equal variances.
Paired t-test: , where
Meaning
Used when observations paired (before/after, matched pairs); n = number of pairs
Watch out
df = n−1 where n = number of pairs, NOT total observations (which is 2n). First compute differences d, then find and s_d.
When to use
Compare means when same subjects measured twice or matched pairs; eliminates variability from pairing.
Chi-Square Test:
Meaning
O = observed frequency; E = expected frequency; df = (# categories) − 1
Watch out
All expected frequencies E ≥ 5 (combine categories if needed). χ² ≥ 0 always; larger values suggest worse fit. Degrees of freedom depends on context (goodness of fit vs. contingency table).
When to use
Goodness of fit (discrete distribution) or independence (categorical variables in contingency table).
Common values
Value
α = 0.05 (5%), α = 0.01 (1%), α = 0.10 (10%)
Symbol
Significance level
Quantity
Common significance levels
Value
α=0.05: z₀.₀₂₅=±1.96; α=0.01: z₀.₀₀₅=±2.576
Symbol
z_{α/2}
Quantity
Critical values for two-tailed Z-test
Value
n=10 (df=9): t=2.262; n=20 (df=19): t=2.093; n=30 (df=29): t=2.045
Symbol
t_{0.025,df}
Quantity
t-critical for common sample sizes (α=0.05, two-tailed)
Important facts
- Increasing sample size n reduces SE and increases power; smaller p-values (stronger evidence against H₀).
- Smaller α (e.g., 0.01 vs. 0.05) reduces Type I error but increases Type II error (β); trade-off exists.
- p-value < α ⇒ reject H₀ (evidence for H₁); p-value ≥ α ⇒ fail to reject H₀ (insufficient evidence).
- Degrees of freedom (df) determine shape of t and χ² distributions; larger df → closer to normal.
Key definitions
Null Hypothesis (H₀)
Example
H₀: μ = μ₀ (mean equals hypothesized value); H₀: p = p₀ (proportion equals value).
Statement assuming no effect/difference/relationship; burden of proof on alternate hypothesis H₁.
Alternate Hypothesis (H₁)
Example
H₁: μ ≠ μ₀ (two-tailed); H₁: μ > μ₀ (right-tailed, if testing for improvement).
Statement contrary to H₀; what we try to evidence. Can be one-tailed (>, <) or two-tailed (≠).
Significance Level (α)
Example
α = 0.05 means 5% chance of false positive; 95% confidence in decision.
Maximum probability of Type I error (rejecting H₀ when true); common α = 0.05 or 0.01.
Type I Error (α)
Example
Concluding medicine works when it doesn't; concluding steel fails stress when it actually passes.
Rejecting H₀ when H₀ is true (false positive); probability α.
Type II Error (β)
Example
Concluding medicine doesn't work when it actually does.
Failing to reject H₀ when H₁ is true (false negative); probability β; power = 1−β.
Power (1−β)
Example
High power: likely to detect true effect if it exists.
Probability of correctly rejecting H₀ when H₁ true; desired power ≥ 0.80.
Diagrams to know
- Hypothesis test decision regions: acceptance region (center), rejection regions (tails defined by α, typically 2.5% each for two-tailed).
- Type I vs II error diagram: vertical line at true parameter, horizontal region of H₀ boundary; overlap shows error probabilities.
- Power curve: plots power (1−β) vs. true parameter value; higher power as effect size increases.
Reactions or equations
is standard normal (Z ~ N(0,1)) if H₀ true
Note
If computed |z| exceeds critical value z_{α/2}, reject H₀. For one-tailed, use z_α (not α/2).
Conditions
Assumes σ known and either n large or data normal
for two-tailed test
Note
Multiply by 2 for two-tailed (check both tails); single multiply for one-tailed.
Conditions
Uses standard normal table
Must remember
Population variance divides by n; sample variance divides by (n−1). This single error invalidates answers on most exams. Always check: is this the entire population or a sample?
The General Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Forgetting the subtraction term (overlap) is the #1 mistake in probability problems.
For binomial distribution, the formula is C(n,x)·p^x·(1−p)^(n−x). Missing the binomial coefficient C(n,x) is a common error; order matters for the calculation.
Standardize normal distributions: z = (x − μ)/σ. Compare to z-tables to find cumulative probabilities. Do NOT use raw x-values with standard tables.
Permutations count ordered arrangements: P(n,r) = n!/(n−r)!. Combinations count unordered selections: C(n,r) = n!/[r!(n−r)!]. Order matters? Use permutation. Order doesn't? Use combination.
68-95-99.7 Rule: In normal distribution, 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ. Memorize for quick approximations.
Confidence interval: CI = point estimate ± (critical value) × (standard error). For large samples, use z; for small samples with unknown σ, use t. Use (n−1) df for t.
p-value < α means REJECT H₀ (evidence for alternate hypothesis). p-value ≥ α means FAIL TO REJECT H₀ (insufficient evidence). The p-value is NOT the probability that H₀ is true.
Independent events: P(A ∩ B) = P(A) × P(B). Dependent events (conditional): P(A ∩ B) = P(A) × P(B|A). Not the same relationship; recognize which applies.
For Poisson distribution, mean = variance = λ. If calculated mean ≠ variance in data, distribution may not be Poisson. Use when n large, p small, np = λ (approximates binomial).
Last minute tips
Population vs. Sample Denominator — ALWAYS check whether the data represents an entire population or a sample from a larger population. Use n for population variance; use (n−1) for sample variance. This single distinction is worth 5−10% of exam points.
In probability problems, draw a Venn diagram or contingency table BEFORE writing formulas. Visual representation helps avoid the addition rule overlap mistake and clarifies whether events are independent, dependent, or mutually exclusive.
For normal distribution problems, ALWAYS standardize using z = (x − μ)/σ before touching a z-table. Write out the calculation step-by-step; skipping this step or dividing by variance instead of SD is a frequent error.
On binomial problems, ask yourself: (1) Fixed number of trials n? (2) Two outcomes (success/fail)? (3) Constant probability p? (4) Independent trials? If yes to all, use binomial. Include the coefficient C(n,x) — don't forget it.
Hypothesis testing: Write H₀ and H₁ explicitly. Identify α. Calculate the test statistic. Find the p-value or critical region. Make a decision using 'p < α → reject H₀.' Organize logically to catch algebra errors before submitting.
Comparison tables
Permutations vs. Combinations
| Feature | Permutations P(n,r) | Combinations C(n,r) |
|---|---|---|
| Definition | Ordered arrangements; sequence matters | Unordered selections; order irrelevant |
| Formula | P(n,r) = n! / (n−r)! | C(n,r) = n! / [r!(n−r)!] |
| Relationship | P(n,r) = C(n,r) × r! | C(n,r) = P(n,r) / r! |
| Example (n=5, r=2) | P(5,2) = 20 ordered pairs (AB ≠ BA) | C(5,2) = 10 unordered pairs (AB = BA) |
| Use Case | Passwords, rankings, arrangements | Committee selection, toppings, subsets |
| Count Property | P(n,r) > C(n,r) always | C(n,r) < P(n,r) always |
Population vs. Sample Statistics
| Statistic | Population | Sample |
|---|---|---|
| Mean | μ (mu) | (x-bar) |
| Variance | σ² = Σ(x−μ)²/n | s² = Σ(x−)²/(n−1) |
| Std Deviation | σ = √[Σ(x−μ)²/n] | s = √[Σ(x−)²/(n−1)] |
| Denominator | n (count of all items) | n−1 (degrees of freedom, Bessel's correction) |
| Purpose | Describes entire group | Estimates population; reduces bias |
| Bias | Biased (uses all data) | Unbiased estimator of σ² |
Discrete Probability Distributions
| Distribution | Conditions | PMF Formula | Mean μ | Variance σ² |
|---|---|---|---|---|
| Binomial | n trials, 2 outcomes, constant p | np | np(1−p) | |
| Poisson | Rare events, time/space interval | λ | λ | |
| Geometric | Trials to first success | 1/p | (1−p)/p² | |
| Uniform (discrete) | k equally likely outcomes | 1/k | (k+1)/2 | (k²−1)/12 |
Normal vs. Other Continuous Distributions
| Property | Normal | Exponential | Uniform |
|---|---|---|---|
| Support | (−∞, ∞) | [0, ∞) | [a, b] |
| Shape | Symmetric bell | Right-skewed decay | Flat rectangle |
| Mean | μ | 1/λ | (a+b)/2 |
| Variance | σ² | 1/λ² | (b−a)²/12 |
| Use Case | Measurement errors, natural variation | Waiting time, lifetime | Random in interval |
| CDF | Φ((x−μ)/σ) | 1−e^(−λx) | (x−a)/(b−a) |
Z-Test vs. t-Test
| Factor | Z-Test | t-Test |
|---|---|---|
| Sample Size | n ≥ 30 (large) or σ known | n < 30 (small) |
| Population σ | Known or assumed | Unknown; use sample s |
| Distribution | Standard normal Z ~ N(0,1) | t-distribution, df=n−1 |
| Critical Value | z_{α/2} (fixed; e.g., 1.96 for α=0.05) | t_{α/2,df} (varies by df; larger than z) |
| Test Statistic | z = (−μ₀)/(σ/√n) | t = (−μ₀)/(s/√n) |
| Interval Width | Narrower (uses σ) | Wider (accounts for uncertainty in s) |
Independent vs. Dependent Events
| Aspect | Independent Events | Dependent Events |
|---|---|---|
| Definition | P(A|B) = P(A); outcome of B irrelevant to A | P(A|B) ≠ P(A); B affects probability of A |
| Example | Roll die twice; coin flip twice (with replacement) | Draw cards without replacement; before/after |
| Multiplication Rule | P(A ∩ B) = P(A) × P(B) | P(A ∩ B) = P(A) × P(B|A) |
| Conditional | P(A|B) = P(A) | P(A|B) ≠ P(A); use conditional formula |
| Calculation | Simpler (direct multiplication) | More complex (conditions change outcome space) |
Diagrams
Engineering Data Analysis — Complete Concept Map
Mind map showing hierarchical organization of all major topics in engineering data analysis. Use to understand how concepts relate and build from descriptive statistics through inference.
Decision Tree — Which Distribution to Use?
Decision flowchart to select appropriate distribution for a given problem. Start at top with question 'Is variable discrete or continuous?' and follow branches to recommended distribution and formula. Speeds up problem identification.
Hypothesis Testing Step-by-Step Procedure
Complete step-by-step workflow for hypothesis testing. Follow in order: state hypotheses, set α, choose test (based on sample size and σ), compute statistic, find p-value, decide, and conclude. Prevents skipping critical steps.
Counting Problem Classifier — Permutation vs. Combination
Quick classifier to determine whether to use permutation or combination. Central question: Does order matter? If yes → permutation; if no → combination. Examples and guideposts for rapid identification.
Sampling Distribution Evolution (Central Limit Theorem)
Timeline showing how standard error of the sample mean decreases as sample size n increases. Demonstrates Central Limit Theorem: larger samples yield more precise estimates. SE = σ/√n is key relationship.