CELE Engineering Mathematics — Engineering Data Analysis (Probability and Statistics)Cheat Sheet

Formula

Mean: $\bar{x} = \dfrac{\sum x}{n}$

Meaning

x = data values; n = count of data points; $\bar{x}$ = 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.

Formula

Population Variance: $\sigma^2 = \dfrac{\sum(x - \bar{x})^2}{n}$

Meaning

σ² = variance; x = each value; $\bar{x}$ = 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.

Formula

Sample Variance: $s^2 = \dfrac{\sum(x - \bar{x})^2}{n-1}$

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.

Formula

Population Std Dev: $\sigma = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n}}$

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.

Formula

Sample Std Dev: $s = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n-1}}$

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.

Formula

Coefficient of Variation: $CV = \dfrac{s}{\bar{x}} \times 100\%$

Meaning

s = sample std dev; $\bar{x}$ = 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.

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

Term

Median

Example

For [2, 5, 7, 9, 15], median = 7. For [2, 5, 7, 9], median = (5+7)/2 = 6.

Definition

Middle value when data arranged in order; 50th percentile; unaffected by outliers.

Term

Mode

Example

In [1, 2, 2, 3, 3, 3, 5], mode = 3. Bimodal if two values tie.

Definition

Most frequently occurring value in a dataset.

Term

Range

Example

For [3, 7, 12, 20], range = 20 − 3 = 17.

Definition

Difference between maximum and minimum values: max − min.

Term

Quartile (Q1, Q2, Q3)

Example

Interquartile range (IQR) = Q3 − Q1; outliers flagged if > Q3 + 1.5·IQR.

Definition

Q1 = 25th percentile; Q2 = median (50th); Q3 = 75th percentile; divide data into four equal parts.

Formula

Classical Probability: $P(A) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

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).

Formula

Complement Rule: $P(A^c) = P(\text{not } A) = 1 - P(A)$

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').

Formula

Addition Rule (Mutually Exclusive): $P(A \cup B) = P(A) + P(B)$ 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).

Formula

Addition Rule (General): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

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.

Formula

Multiplication Rule (Independent Events): $P(A \cap B) = P(A) \times P(B)$

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).

Formula

Conditional Probability: $P(A|B) = \dfrac{P(A \cap B)}{P(B)}$

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.

Formula

Bayes' Theorem: $P(A|B) = \dfrac{P(B|A) \times P(A)}{P(B)}$

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.

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

Term

Independent Events

Example

Rolling a die twice: second roll unaffected by first. Sampling with replacement.

Definition

Events A and B where outcome of one does not affect the probability of the other; P(A|B) = P(A).

Term

Dependent (Conditional) Events

Example

Drawing cards without replacement: first card removes one from deck, affecting next draw.

Definition

Events where outcome of one changes the probability of the other; P(A|B) ≠ P(A).

Term

Mutually Exclusive (Disjoint) Events

Example

A coin flip: cannot be both heads and tails simultaneously.

Definition

Events that cannot both occur; P(A ∩ B) = 0.

Term

Sample Space (S)

Example

For two dice: S has 36 outcomes (1,1) through (6,6). For a coin: S = {H, T}.

Definition

Set of all possible outcomes of an experiment.

Note

Foundation of the complement rule; rearrange as P(A^c) = 1 − P(A) when complement easier to calculate.

Equation

$P(A) + P(A^c) = 1$

Conditions

Always true for any event A in any probability space

Note

Conditional form essential for dependent events (without replacement, sequential constraints).

Equation

$P(A \cap B) = P(A) \times P(B|A)$

Conditions

General form for any two events; becomes P(A) × P(B) if A, B independent

Formula

Factorial: $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$

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.

Formula

Permutations (Order Matters): $P(n,r) = \dfrac{n!}{(n-r)!}$

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.

Formula

Combinations (Order Does NOT Matter): $C(n,r) = \binom{n}{r} = \dfrac{n!}{r!(n-r)!}$

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'.

Formula

Permutations with Repetition: $P(n) = \dfrac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$

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.

Formula

Multiplication Principle: $n = n_1 \times n_2 \times \cdots \times n_k$

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).

Formula

Addition Principle: $n = n_1 + n_2 + \cdots + n_k$

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).

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)

Term

Permutation

Example

Passwords, race positions. P(5,3) = 60 ways to assign 1st, 2nd, 3rd place from 5 runners.

Definition

Ordered arrangement of r items from n distinct items; each arrangement is unique by position.

Term

Combination

Example

Committee selection, choosing toppings. C(5,3) = 10 ways to choose 3 people from 5 (no rank difference).

Definition

Unordered selection of r items from n items; order irrelevant.

Term

Identical (Indistinguishable) Objects

Example

Letters in 'BOOK': 4!/(1!×1!×2!) = 24/2 = 12 distinct arrangements (two O's identical).

Definition

Repeated elements in an arrangement reduce the count; account via division by factorial of each frequency.

Term

Mutually Exclusive Categories

Example

A student is either BSCE or BSME, not both in the same instant.

Definition

Non-overlapping groups; an item belongs to exactly one category.

Note

Rearrange: C(n,r) = P(n,r) / r!. Use to verify or derive one from the other.

Equation

$P(n,r) = C(n,r) \times r!$

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.

Equation

$C(n,r) = C(n-1,r-1) + C(n-1,r)$

Conditions

Pascal's identity; true for all valid n, r

Formula

Binomial Distribution: $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

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.

Formula

Binomial Mean: $\mu = np$

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.

Formula

Binomial Variance: $\sigma^2 = np(1-p)$

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.

Formula

Poisson Distribution: $P(X = x) = \dfrac{e^{-\lambda} \lambda^x}{x!}$

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.

Formula

Poisson Mean & Variance: $\mu = \lambda, \ \sigma^2 = \lambda$

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.

Formula

Geometric Distribution: $P(X = x) = (1-p)^{x-1} p$ (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.

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

Term

Binomial Trial

Example

One coin flip (heads = success, p = 0.5), or testing if a concrete sample fails (p = probability of failure).

Definition

Single experiment with exactly two outcomes (success/failure); probability p of success constant across trials.

Term

Random Variable (X)

Example

X = number of defective units in sample of 10; X = strength of steel rod (continuous).

Definition

Function assigning numerical value to each outcome in sample space; discrete (integer values) or continuous.

Term

Probability Mass Function (PMF)

Example

For fair die: P(X = k) = 1/6 for k = 1, 2, ..., 6; sum of all probabilities = 1.

Definition

For discrete X: function giving P(X = x) for each value x; Σ P(X = x) = 1.

Term

Expected Value (Mean) of Discrete Distribution

Example

E[fair die] = 1(1/6) + 2(1/6) + ... + 6(1/6) = 3.5.

Definition

μ = E[X] = Σ x · P(X = x); weighted average of all values.

Note

P(X < k) = P(X ≤ k−1). Use tables for binomial/Poisson to avoid repeated calculation.

Equation

$P(X \leq k) = \sum_{x=0}^{k} P(X = x)$

Conditions

Cumulative distribution function (CDF) for discrete X; sum up to and including k

Note

Binomial: with replacement (or infinite population). Hypergeometric: without replacement (finite population, significant extraction).

Equation

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$ reduces to hypergeometric if sampling without replacement

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

Formula

Normal Distribution: $f(x) = \dfrac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

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).

Formula

Standard Normal (Z-score): $z = \dfrac{x - \mu}{\sigma}$

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.

Formula

Probability from Z-table: $P(X \leq x) = P(Z \leq z) = \Phi(z)$

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.

Formula

Central Limit Theorem: $\bar{X} \approx N(\mu, \sigma^2/n)$ for large n

Meaning

Distribution of sample mean $\bar{X}$ 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.

Formula

Uniform Distribution: $f(x) = \dfrac{1}{b-a}$ for $a \leq x \leq b$

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°).

Formula

Exponential Distribution: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$

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).

Formula

Confidence Interval (Large Sample): $\bar{x} \pm z_{\alpha/2} \dfrac{s}{\sqrt{n}}$

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).

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)

Term

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).

Definition

For continuous X: f(x) ≥ 0 such that ∫_{−∞}^{∞} f(x)dx = 1; area under curve = probability.

Term

Cumulative Distribution Function (CDF)

Example

Standard normal Φ(z); F(x) ranges 0 to 1, non-decreasing.

Definition

F(x) = P(X ≤ x); for continuous X: F'(x) = f(x) (derivative of CDF is PDF).

Term

Standard Normal Distribution

Example

Z-tables give P(Z ≤ z) for standard normal; used universally for normal probability.

Definition

Normal with μ = 0 and σ = 1; denoted Z; any normal can be standardized to this.

Term

Confidence Level

Example

95% CI: z₀.₀₂₅ = 1.96 (two-tailed); 99% CI: z₀.₀₀₅ = 2.576.

Definition

Probability (1−α) that a confidence interval contains the true parameter; e.g., 95% confidence level means α = 0.05.

Note

Subtract lower z-score probability from upper to get probability in interval.

Equation

$P(a \leq X \leq b) = \Phi\left(\dfrac{b-\mu}{\sigma}\right) - \Phi\left(\dfrac{a-\mu}{\sigma}\right)$

Conditions

For normal X with mean μ, SD σ; standardize both endpoints

Note

Equivalent to P(Z > z) where z = (k−μ)/σ.

Equation

$P(X > k) = 1 - \Phi\left(\dfrac{k-\mu}{\sigma}\right)$

Conditions

Right-tail probability; uses complement rule

Formula

Standard Error of Mean: $SE = \dfrac{\sigma}{\sqrt{n}}$ or $SE = \dfrac{s}{\sqrt{n}}$

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.

Formula

Test Statistic (Z-test for mean): $z = \dfrac{\bar{x} - \mu_0}{SE} = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$

Meaning

ᾱ₀ = hypothesized mean; $\bar{x}$ = 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.

Formula

Test Statistic (t-test for mean): $t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$

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.

Formula

P-value (Two-tailed): $p = 2 \times P(|Z| > |z_{obs}|)$ 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.

Formula

Confidence Interval for Mean: $\bar{x} \pm t_{\alpha/2,n-1} \dfrac{s}{\sqrt{n}}$

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.

Formula

Two-Sample Test (Independent, Unequal Variances): $t = \dfrac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Meaning

$\bar{x}_i$, $s_i$, $n_i$ = 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.

Formula

Paired t-test: $t = \dfrac{\bar{d}}{s_d / \sqrt{n}}$, where $\bar{d} = \text{mean difference}, s_d = \text{SD of differences}$

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 $\bar{d}$ and s_d.

When To Use

Compare means when same subjects measured twice or matched pairs; eliminates variability from pairing.

Formula

Chi-Square Test: $\chi^2 = \sum \dfrac{(O - E)^2}{E}$

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).

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)

Term

Null Hypothesis (H₀)

Example

H₀: μ = μ₀ (mean equals hypothesized value); H₀: p = p₀ (proportion equals value).

Definition

Statement assuming no effect/difference/relationship; burden of proof on alternate hypothesis H₁.

Term

Alternate Hypothesis (H₁)

Example

H₁: μ ≠ μ₀ (two-tailed); H₁: μ > μ₀ (right-tailed, if testing for improvement).

Definition

Statement contrary to H₀; what we try to evidence. Can be one-tailed (>, <) or two-tailed (≠).

Term

Significance Level (α)

Example

α = 0.05 means 5% chance of false positive; 95% confidence in decision.

Definition

Maximum probability of Type I error (rejecting H₀ when true); common α = 0.05 or 0.01.

Term

Type I Error (α)

Example

Concluding medicine works when it doesn't; concluding steel fails stress when it actually passes.

Definition

Rejecting H₀ when H₀ is true (false positive); probability α.

Term

Type II Error (β)

Example

Concluding medicine doesn't work when it actually does.

Definition

Failing to reject H₀ when H₁ is true (false negative); probability β; power = 1−β.

Term

Power (1−β)

Example

High power: likely to detect true effect if it exists.

Definition

Probability of correctly rejecting H₀ when H₁ true; desired power ≥ 0.80.

Note

If computed |z| exceeds critical value z_{α/2}, reject H₀. For one-tailed, use z_α (not α/2).

Equation

$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$ is standard normal (Z ~ N(0,1)) if H₀ true

Conditions

Assumes σ known and either n large or data normal

Note

Multiply by 2 for two-tailed (check both tails); single multiply for one-tailed.

Equation

$p = P(|Z| \geq |z_{obs}|) = 2[1 - \Phi(|z_{obs}|)]$ for two-tailed test

Conditions

Uses standard normal table

Values

• Ordered arrangements; sequence matters
• Unordered selections; order irrelevant

Property

Definition

Values

• P(n,r) = n! / (n−r)!
• C(n,r) = n! / [r!(n−r)!]

Property

Formula

Values

• P(n,r) = C(n,r) × r!
• C(n,r) = P(n,r) / r!

Property

Relationship

Values

• P(5,2) = 20 ordered pairs (AB ≠ BA)
• C(5,2) = 10 unordered pairs (AB = BA)

Property

Example (n=5, r=2)

Values

• Passwords, rankings, arrangements
• Committee selection, toppings, subsets

Property

Use Case

Values

• P(n,r) > C(n,r) always
• C(n,r) < P(n,r) always

Property

Count Property

Values

• μ (mu)
• $\bar{x}$ (x-bar)

Property

Mean

Values

• σ² = Σ(x−μ)²/n
• s² = Σ(x−$\bar{x}$)²/(n−1)

Property

Variance

Values

• σ = √[Σ(x−μ)²/n]
• s = √[Σ(x−$\bar{x}$)²/(n−1)]

Property

Std Deviation

Values

• n (count of all items)
• n−1 (degrees of freedom, Bessel's correction)

Property

Denominator

Values

• Describes entire group
• Estimates population; reduces bias

Property

Purpose

Values

• Biased (uses all data)
• Unbiased estimator of σ²

Property

Bias

Values

• n trials, 2 outcomes, constant p
• $\binom{n}{x}p^x(1-p)^{n-x}$
• np
• np(1−p)

Property

Binomial

Values

• Rare events, time/space interval
• $e^{-λ}λ^x / x!$
• λ
• λ

Property

Poisson

Values

• Trials to first success
• $(1-p)^{x-1}p$
• 1/p
• (1−p)/p²

Property

Geometric

Values

• k equally likely outcomes
• 1/k
• (k+1)/2
• (k²−1)/12

Property

Uniform (discrete)

Values

• (−∞, ∞)
• [0, ∞)
• [a, b]

Property

Support

Values

• Symmetric bell
• Right-skewed decay
• Flat rectangle

Property

Shape

Values

• μ
• 1/λ
• (a+b)/2

Property

Mean

Values

• σ²
• 1/λ²
• (b−a)²/12

Property

Variance

Values

• Measurement errors, natural variation
• Waiting time, lifetime
• Random in interval

Property

Use Case

Values

• Φ((x−μ)/σ)
• 1−e^(−λx)
• (x−a)/(b−a)

Property

CDF

Values

• n ≥ 30 (large) or σ known
• n < 30 (small)

Property

Sample Size

Values

• Known or assumed
• Unknown; use sample s

Property

Population σ

Values

• Standard normal Z ~ N(0,1)
• t-distribution, df=n−1

Property

Distribution

Values

• z_{α/2} (fixed; e.g., 1.96 for α=0.05)
• t_{α/2,df} (varies by df; larger than z)

Property

Critical Value

Values

• z = ($\bar{x}$−μ₀)/(σ/√n)
• t = ($\bar{x}$−μ₀)/(s/√n)

Property

Test Statistic

Values

• Narrower (uses σ)
• Wider (accounts for uncertainty in s)

Property

Interval Width

Values

• P(A|B) = P(A); outcome of B irrelevant to A
• P(A|B) ≠ P(A); B affects probability of A

Property

Definition

Values

• Roll die twice; coin flip twice (with replacement)
• Draw cards without replacement; before/after

Property

Example

Values

• P(A ∩ B) = P(A) × P(B)
• P(A ∩ B) = P(A) × P(B|A)

Property

Multiplication Rule

Values

• P(A|B) = P(A)
• P(A|B) ≠ P(A); use conditional formula

Property

Conditional

Values

• Simpler (direct multiplication)
• More complex (conditions change outcome space)

Property

Calculation